UNIVERSIDAD DE CHILE FACULTAD DE CIENCIAS FISICAS Y MATEMATICAS DEPARTAMENTO DE GEOFISICA

NON-STEREO IMAGERY DERIVED TOPOGRAPHIC INFORMATION:

NEW REMOTE SENSING METHOD

TESIS PARA OPTAR AL GRADO DE MAGISTER EN CIENCIAS MENCION GEOFISICA

CAMILO ANDRES RADA GIACAMAN

PROFESOR GUIA: ANDRES PAVEZ ALVARADO

MIEMBROS DE LA COMISION:

RENE GAREAUD SALAZAR ANDRES RIVERA

EMILIO VERA

SANTIAGO DE CHILE ENERO 2009

RESUMEN La relación entre la elevación solar y el largo de las sombras es conocida desde tiempos remotos y usando geometría básica es posible determinar la diferencia de elevación entre una sombra y el objeto que la proyecta. De igual modo, sombras frecuentemente presentes en imágenes satelitales y fotografías aéreas contienen valiosa información acerca de la topografía del terreno observado. Muchos problemas de ciencias de la tierra, como el balance de masa glaciar, administración de recursos hídricos, estudio y monitoreo de plataformas de hielo, monitoreo de icebergs así como el estudio y monitoreo de erupciones volcánicas podrían verse beneficiados por esta información si fuera suficientemente precisa. Desafortunadamente esta información no ha sido utilizada debido a que errores de hasta algunos cientos de metros se encuentran cuando este principio geométrico básico es aplicado en su forma más simple. Esta conceptualmente simple idea es conducida aquí hasta su máximo desempeño a través de cálculos precisos de la posición solar, uso automatizado de archivos meteorológicos para resolver la refracción atmosférica, modelamiento preciso de la geometría del satélite y un detallado cálculo del perfil teórico de intensidad de las sombras, incluyendo los efectos de la forma del proyector y el obscurecimiento del limbo solar. Nosotros desarrollamos y validamos una metodología que utiliza sombras como indicadores de diferencias en el terreno, utilizando imágenes satelitales en el rango visual y sin pares estéreo, lo que sería además exportable para su aplicación en fotografías aéreas y terrestres. Usando imágenes ASTER L1A, hemos encontrado que las diferencias de altitud pueden ser calculadas con errores menores a 6 metros. Debido a la sensibilidad de los errores a la resolución espacial de las imágenes, proponemos que utilizando imágenes Landsat MSS (disponibles desde los 70’s) una se podría esperar una precisión de aproximadamente 10 metros, y menos de un metros en satélites de última generación y alta resolución. Esta metodología es insensible a los errores absolutos en la orientación del satélite y puede ser aplicada principalmente en latitudes altas o medias durante el invierno, donde sombras pueden ser fácilmente encontradas en las imágenes actualmente disponibles.

ABSTRACT Solar elevation and shadow length relationship is know from early times, and using basic geometry, the height difference between shadow and shadow’s projector can be derived. In this way, shadows often present in satellite and aerial imagery, contain valuable terrain information. Many Earth sciences problems, as glacial mass balance, water resources management, ice shelf monitoring and research, iceberg monitoring and volcanic eruptions monitoring and research would be beneficed by this information if it was accurate enough. Unfortunately this information hasn’t been used because errors up to few hundred meters are found when this basic geometric principle is applied in its simplest way. This conceptually straightforward idea is taken here to its maximum performance by very precise Sun position calculation, automated use of archived meteorological models to solve atmospheric refraction, precise modeling of satellite geometry and accurate calculation of theoretical shadow’s intensity profiles including limb darkening effect and calculated projector geometry. We develop and validate a methodology that uses shadows as terrain difference indicators, using non stereo visual satellite imagery and extensible to non stereo aerial or ground based photography. We found that using ASTER L1A imagery; height differences can be measured with accuracies better than 6 meters. Due to error sensitivity to image spatial resolution, we propose that using Landsat MSS images (available from the 70’s) an accuracy of about 10 meters can be reached, and less than one meter in state-of-art high resolution satellites. The methodology is not sensitive to absolute errors in satellite orientation and can be applied mainly over high latitude and winter mid latitude areas, where shadows can be easily found in currently available imagery.

Acknowledgments To The Omega Foundation to support this research in Antarctica and facilitate the Trimble 5700 GPS during the whole validation process and also for providing Antarctic ASTER imagery, In particular to my expedition partners Damien Gildea, Rodrigo Fica, Manuel Bugueño, Steve Chaplin, María Paz Ibarra, Jed Brown and Jarmila Tyrril. To Andrés Rivera, from CECS, Chile, who facilitate ASTER imagery and Patriot Hills GPS data in addition of very useful guidelines and advice, and for course for accepting to be the invited part of my thesis commission.To Michael Ramsey who facilitate ASTER imagery of volcanic plumes. To my thesis commission Emilio Vera, Rene Garreaud and Andrés Pavez. To my family for supporting this work in many different ways and for their advice and motivation. And finally to Natalia Martinez for her support in the central Andes validation field work and mainly for being the most wonderful source of light and energy in my life.

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INDEX

1. INTRODUCTION ..................................................................................................................................5 2. AVAILABLE AND PLANNED METHODOLOGIES TO TRACK SNOW/ICE ELEVATION CHANGES ..................................................................................................................................................5 2.1. OPTICAL SENSORS - STEREOSCOPY....................................................................................5 2.2. RADAR............................................................................................................................................6 2.2.1. REPEAT-PASS INSAR .........................................................................................................7 2.2.2. SINGLE-PASS INSAR...........................................................................................................7 2.3. LIDAR (LASER) ............................................................................................................................8 2.4. FUTURE TECHNOLOGIES ........................................................................................................9 3. SHADOWS AS TERRAIN INDICATORS..........................................................................................9 3.1. ERROR SOURCES......................................................................................................................10 3.1.1. PROJECTOR REAL POSITION .......................................................................................10 3.1.2. SATELLITE POSITION .....................................................................................................11 3.1.3. SHADOW SIZE....................................................................................................................11 3.1.4. SUN POSITION AND ATMOSPHERIC REFRACTION ...............................................11 3.1.5. SHADOW MEASUREMENT .............................................................................................12 3.2. APPLICATIONS..........................................................................................................................13 3.2.1. AREA OF APPLICABILITY...................................................................................................14 3.3. METHOD ADVANTAGES IN CURRENT DATA AVAILABILITY CONTEXT................16 4. METHODOLOGY ...............................................................................................................................17 4.1. SATELLITE POSITION .............................................................................................................17 4.2. ELLIPSOIDAL APPARENT POSITIONS................................................................................19 4.3. SHADOW AND PROJECTOR PICKING ...........................................................................19 4.3.1. PRELIMINARY PICKING.................................................................................................19 4.3.2. SHADOW CALCULATION AND PICKING ...................................................................20 4.3.3. PROJECTOR PICKING .....................................................................................................25 4.4. SHADOW PATH PREDICTION ...............................................................................................26 4.5. SUN’S APPARENT POSITION .................................................................................................26 4.5.1. SUN’S REAL POSITION ....................................................................................................26 4.5.2. ATMOSPHERIC REFRACTION ......................................................................................27 5. MULTIPLE MEASUREMENT..........................................................................................................28 6. VALIDATION ......................................................................................................................................29 6.1. SENTINEL RANGE VALIDATION DATA SET.....................................................................30 6.2. ANDES VALIDATION DATA SET...........................................................................................31 6.3. ABBOT ICE SHELF VALIDATION DATA SET ....................................................................32 6.4. RONNE-FILCHNER ICE SHELF VALIDATION ..................................................................37 6.5. SHIVELUCH VOLCANO DATA SET ......................................................................................37 6.6. VALIDATION CONCLUSION ..................................................................................................38 7. CONCLUSION.....................................................................................................................................39

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1. Introduction

Studies related to Earth dynamics often need to know present and past terrain shapes to track changes and trends. One important example of this kind of application is glacial mass balance, where changes in thickness and area are necessary in order to remotely measure net mass variations. Earth surface can be studied using DEM’s (Digital Elevation Models) generated by radar, laser, stereo imagery or direct GPS surveys within others, but often the availability of this data, its temporal and/or spatial coverage, doesn’t match researcher’s needs. That’s why a method to derive topographic information using visual bands in non-stereo imagery is so interesting, specially taking in account that this kind of imagery is the most common, the one with higher spatial resolution and more extensive temporal coverage to the past. Spaceborne imagery records extend back to the 70’s (Landsat) or even the 50’s (Corona). Possibly due to global climate variations, glacier terrain has shown to be very dynamic over the last century, with thinning rates reaching five meters a year or more. For example O’Higgins glacier in Patagonia, has been observed since 1914, and a total thinning of 220 m was estimated on 1995 (Casassa et al. 1997) at rates up to 6.7 meters a year before 1960. Even with such a big signals, there is a lack of quantitative terrain measurements, setting an additional motivation to develop methodologies to extract topographic data from early imagery. Snow and ice monitoring importance arise mainly from the fact that most of the global fresh water resources are stored as snow and ice at high altitudes and/or at high latitudes. Its dynamics determine water availability in amount (seasonal run-off) and time (non renewable water sources) for many countries, becoming a key parameter for energy management, urban planning and water resources policies. Monitoring of glaciers, on the other hand, has an important role in climate change studies, placed as priority in many scientific institutions and governments as a response to the IPCC 2007 report. The most impacting possible outcome of climate change is the sea level rise, phenomena directly linked to glacier dynamics. Previous to the extensive availability of GPS and remote sensing techniques, snow packs and glaciers were tracked by measuring changes in their front and their area. This was done using ground and air based photography or field measurements along the perimeter. Even then, glacier retreat became a well studied phenomenon. But area measurements give only a qualitative idea of changes in the key variable: glacier/snow mass balance. To assert about the changes in mass, it is necessary to perform repeated measures of glaciated area’s topography, or to monitor accumulation and ablation magnitudes over its surface. At the current point when global glacier monitoring is pursued, direct field measures become unfeasible and remote sensing techniques are needed. Glacier/snow surface topography is somehow the most challenging parameter to measure, because traditional topographic methods (aerial photogrammetry) are very expensive and often fail over featureless terrain as smooth glaciers or snowfields. Here we develop a method to take advantage of shadows often present in mid to high latitude imagery, using them as terrain change indicators, making possible to extract new useful data from the huge available set of old and new visual imagery. In several research areas where terrain time series are needed, this new information would improve series by adding new samples and extending them to the past, allowing to better constrain temporal variations or to extend studies to other areas were actual data are only available recently.

2. Available and planned methodologies to track snow/ice elevation changes

Most topographic mapping methods produce data useful for snow cover and glacier monitoring. Among these methods we can distinguish direct and remote measurements. The former is usually done trough GPS or topographic surveys directly on the field. This is accurate but very time consuming and expensive especially on remote places. Also in some cases it is dangerous or just infeasible. We will focus now on remote methodologies, using sensors over spaceborne or airborne platforms.

2.1. Optical sensors - Stereoscopy

Terrain models derived from stereo-imagery is a classical approach, topographical maps have been built this way from early in the 20th century. One of the advantages is the ability to produce multiple measurements over the whole image producing a DEM. And one important disadvantage is the necessity of a contrasted terrain, because elevation is computed correlating features in both stereo images, therefore, if there is no features in the terrain (as in smooth snow fields) no elevations can be easily computed. When features are sparse, DEM generation is difficult or impossible, but height determination of punctual features if any, can be always done if acquisition geometry is well determined. Accuracy in punctual calculations relay on orbit determination precision and ground control points (GCP).

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When aerial imagery is used, accuracies depend on scale, camera specifications, methodologies, etc. But usually range from few to few tens meters. Very high accuracies can be reached, but it become very expensive, especially over remote areas. When global, cheap and continuous monitoring is desired, satellite imagery is the best option. This can be done using multi-temporal stereo pairs (images acquired from different satellite passes, usually with time differences over 3 days) as have been widely done using SPOT-1 to 4, IRS-1C/D and Landsat imagery. Although for glaciological applications, stereoscopy performance is reduced due to changes in snow coverage, reducing the success of the image correlation process. However, DEM’s produced this way has been used for glacier monitoring (Bahuguna et al. 2004). Other satellite platforms such as ASTER, SPOT-5, Cartosat-1 and Formosat-2, offer along-track stereo channels, acquiring stereo pairs from two positions in the same pass and within less than a minute. Validations of ASTER (15 m resolution) DEM’s over non glaciated areas, have found that RMS errors can be expected to be between ±7 to 15 meters (Hirano et al. 2003), ±14 to 19 meters (Poli et al. 2004) or ±10 to 30 m (Toutin & Berthier 2007). For SPOT-5 HRS (10 m resolution) have been found accuracies of ±5.8 m at 68% confidence level (Toutin 2006), ±15 m (Bouillon et al. 2006) or ±5 to 10 m also depending on terrain roughness and processing (Kornus et al. 2005). Variability in results can be partially explained by processing differences and because errors are found to increase in rough terrain and steeper slopes at rates about 0.12 m/° in SPOT-5 and 0.50 m/° in ASTER for 68% confidence level errors (Toutin 2006, Toutin & Berthier 2007). As seen, over non glaciated areas, as a thumb rule we can say that vertical accuracies can be expected to be about the same magnitude of the image horizontal pixel size. Similar validations have been done over glaciated terrain to assert DEM applicability for glaciological purposes, finding for ASTER, RMS differences with reference heights of ±60 m (Racoviteanu et al. 2007, Kääb et al. 2002, Kääb 2002) but artifacts and punctual errors up to 500 m have also been observed. In order to assess ASTER DEM performance over glaciated and non-glaciated areas, it was compared to a reference DEM. Resulting RMS differences increased from 20 meters in non-glaciated areas to 26 meters over glaciers (Racoviteanu et al. 2007). Errors are also function of the number of ground control points used. In study of 15 meters resolution ASTER DEM’s, mean errors decrease from 32 to 16 meters increasing GCP from 30 to 65 (San et al. 2005). In a glaciological study using SPOT-5 to measure glacier elevation changes (Toutin & Berthier 2007), standard deviations around 25 meters were found compared with a reference DEM (SRTM) in ice-free areas, error analysis lead to the conclusion that only elevation change signals with amplitudes over 10 m could be detected. However glacier changes in Canada and Alaska were successfully detected. Using multiple bias corrections over SPOT-5 HRG (2.5 m resolution) multi-temporal DEM (images with one day difference), weaker signals of 3.5 m have been detected (Berthier et al. 2006). Using multiple image matching (three) and very high resolution IKONOS imagery (1 meter), vertical accuracies of 2 to 3 meters over mix terrain (including snow) have been reached, and vertical accuracies of one meter or less where found over bare ground (Zhang & Gruen 2006). It is important to note that mass balance calculations using DEM, relay on the averaging over many punctual measurements, because punctual errors are often much bigger than RMS errors. As a conclusion, using spaceborne sensors capable to acquire stereo imagery, surface change over glaciated terrain can be monitored. Sensitivity depends on GCP, Orbit determination, terrain characteristics and strongly on pixel size. For currently available data, as a thumb rule, this sensitivity seems to be enough to detect signals with amplitudes down to about twice the image pixel size.

2.2. Radar

An advantage common to all Radar systems is its all weather operation, because Radar waves can penetrate thick cloud covers. A first approach is Radar altimetry, where a downwards looking antenna sends extremely short electromagnetic pulses and distances from satellite to ground are computed as function of signal’s two way travel times. This technique is very accurate in range (up to few centimeters) but spatial resolutions range from kilometers to hundred meters, suffering huge distortions in terrains with significant and/or changing slopes. An important source of ambiguity arises from the fact that Radar waves can penetrate on dry snow or ice, potentially up to several meters (Farr et al. 2006). Therefore Radar altimetry is useful for measuring sea surface and big smooth ice sheets. Antarctic DEMs has been built from Radar altimetry using GEOSAT and ERS-1 (Herzfeld 1999, Bamber 1994) but at very low ground resolution (3 and 20 Km respectively). A second approach is given by side-looking radars or SLR. Along track resolution of a SLR depends strongly on platform height and antenna length, airborne platforms reach good along-track resolutions, but in spaceborne

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platforms it is usually unacceptably poor even for large antennas. For this reason Synthetic Aperture Radars (SAR) were developed. SAR systems enhance resolution by using a moving small antenna to simulate a big one. Side-looking radars, map the surface in a linear metric scale in the along-track direction and in a range scale across-track. Range values are a mix of target’s distance to sub sensor track and target’s height. Therefore no topography quantitative analysis can be done. To measure topography it is necessary to solve range ambiguity using two data sets covering the same area but acquired from slightly different positions and apply a technique called interferometry where phase differences are used to solve true target’s spatial position. This methodology is known as InSAR. We will discuss two InSAR systems: repeat-pass and single-pass.

2.2.1. Repeat-pass InSAR

When interferometry is applied over two SAR images acquired over the same area in different passes of the satellite, we are talking about repeat-pass InSAR. SAR data suitable for interferometry is available since late 80’s, so it constitutes a very valuable data source spanning already more than 20 years. Since the very first InSAR publications (Graham 1974, Zebker & Goldstein 1986, Goldstein et al. 1988) numerous InSAR studies have been released, at the begin orbit geometry force to deal with images acquired three or more days away, but after launch of ERS-2 in 1995 data sets acquired only one day apart became available increasing InSAR applicability and performance, making it widely applied for topographic mapping, change detection, volcanic hazards, seismic events, ground deformation and glacier flow. The main problem of repeat-pass InSAR is the lost of image coherence between both satellite passes, due to rearrangement or change of scatterers on the surface. Glaciated and mountain terrain is especially likely to coherence loss due to changes in snow coverage, density, distribution or texture due to winds, glacier movement, etc. In addition, applicability of this methodology is strongly limited by the difficulty to find a pair of acquisitions with geometry and coherence suitable for InSAR. Ground resolution of this imaging sensors limit the maximum baseline (i.e. distance between satellite positions in both acquisition), because a distance ambiguity arise if more than one phase cycle develop within one image pixel. Therefore small base lines are required. In spite of this problems, repeat-pass InSAR have been used successfully for glaciological applications including DEM generation and glacier flow velocities determination. But in some cases, strong coherence problems have been found on glaciated and/or snow covered areas even using SAR images acquired with just one day difference (Eldhuset et al. 2003). We have not found accuracy assessment regarding repeat-pass InSAR specifically over glaciated or snow covered areas. On mix terrain (glacier/snow/dry land) ERS SAR derived DEMs gave, when compared to a reference DEM, a RMS difference of about 10 to 20 meters (Eldhuset et al. 2003). Landmap is an InSAR DEM mosaic at 25 meters resolution over UK and Ireland, derived from ERS-1 and ERS-2 SAR imagery. It was evaluated to measure glacier geomorphological parameters and it was found to perform very poorly in the test area showing an unreal rough surface, probably due to low coherence in SAR source images (Smith et al. 2006). These difficulties of DEM generation using repeat-pass InSAR, and a strongly need of a global DEM motivated the development of spaceborne and airborne single-pass InSAR like the Shuttle Imaging Radar C or SIR-C (Farr et al. 2006). Some of these difficulties have been overcome by Persistent Scatterer Interferometry or PSI a relatively recent technique, a multi-image approach that identifies objects with a consistent and stable radar reflection in all the images. PSI have been successfully used mainly in urban areas (for example in Terrafirma project of the European Space Agency). Under current development, PSI will probably become in a useful glaciological tool.

2.2.2. Single-pass InSAR

The best way to solve coherence problems and geometric constrains stated in the previous section, is to acquire both SAR images simultaneously and over an accurately controlled geometry. This is what single-pass InSAR does, by using two side-looking antennas mounted over a long mast, so base line and geometry can be determined accurately. In airborne platforms, single-pass InSAR is currently emerging as the most economic technology for collecting high resolution elevation data and in spaceborne platforms, it placed a milestone in Earth sciences trough the SRTM (Shuttle Radar Topographic Mission) and the generation of a near-global coverage DEM with one arcsecond resolution. Airborne single-pass InSAR is capable to reach very high spatial resolution and vertical accuracies as function of sensor design and fly height. An example of a systematic campaign is NEXTMap Great BritainTM, at 5 m resolution and vertical accuracy between 0.5 and 1 meter. This data set has been used successfully to identifying glacial surface features (Smith et al. 2006), therefore it reaches good relative heights. Shifts in absolute heights due to radar waves penetration must be considered with caution.

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However, when change monitoring and global coverage is desired over long periods of time, airborne instruments are unfeasible, therefore we will focus on spaceborne platforms. Unfortunately, there is no single-pass InSAR sensor offering continuous observations as ERS, RADARSAT, etc. does for SAR observation. Such sensor would be probably the best option for accurate global glacier monitoring. Single-pass InSAR from space, reach typically 5 to 10 meters spatial resolution (Farr et al. 2006). The only data set available produced by spaceborne InSAR is the SRTM (Shuttle Radar Topographic Mission) that used simoultaneosly two instruments, SIR-C and SIR-X. SIR-C derived data has become in the most widely used topographic model. SIR-C mapped the globe between parallels 56° south and 60° north at a ground resolution of about 30 meters. Extensive work has been done in accuracy assessment and data validation. Shortly, vertical errors (90%) of 9 m absolute and 10 m relative have been found., maximum errors appears over steep terrain and very smooth sandy surfaces. In the SRTM spatial resolution and vertical accuracy of SIR-C DEM was sacrificed in order to achieve needed ground coverage and mission specifications, because the whole target area must be mapped in 10 days, requiring a swath width about 4 times bigger than its actual value, swath width was doubled by using two radar beams simultaneously. Then to double it again, ScanSAR technology was used, this means that the radar beam was moving over Earth surface scanning a band wider than the SIR-C footprint. ScanSAR reduces the illumination time of a target on ground producing also a reduction in resolution. X-SAR data reach a spatial resolution of 25 meters and a vertical accuracy of 16 m (90% error) absolute and 6 m (90% error) relative. Its improved quality was traded-off with coverage. Therefore X-SAR DEM coverage is only a 40% of the target area (latitudes between -56° and +60°). Extensive research has been done using STRM data, including many glaciological studies, mainly to characterize glacier geometry, but it also have been used to measure elevation changes. Evaluating SIR-C DEM at 3 arcseconds resolution over glaciated and non-glaciated areas, it was compared to a reference DEM, resulting RMS values increase from 9.5 meters in non-glaciated areas to 15.8 meters over glaciers (Racoviteanu et al. 2007). Therefore relative vertical errors show to be about 70% bigger over glaciated terrain. The same data set when evaluated to identify glacial surface structures, showed poor performance in comparison with other sources. It was attributed mainly to low spatial resolution (Smith et al. 2006).

2.3. LIDAR (Laser)

LIDAR or LiDAR is the acronym for Light Detection And Ranging. As for Radars, LIDAR sensors can work as ranging or scattering systems. Here we will review ranging application because scattering applications (or backscatter LIDAR) are intended to study properties of the medium between the sensor and earth surface. The use of LIDAR systems for Laser profiling (or Laser altimetry) is currently the most accurate remote sensing technology for topography measurements. Its it conceptually extremely straightforward: A short light pulse with a small beamwidth is emitted towards Earth surface where it’s reflected back. The returned light is detected and its two way travel time recorded. In addition with an estimated light speed, it is used to compute sensor height over ground. Then accurate GPS positions and attitudes from the air/spacecraft navigation systems are used to reduce those sensor heights over ground to surface absolute ellipsoidal elevations. Typical airborne systems, flying at an altitude of 200 m over ground might have a footprint of 0.2 m and a vertical accuracy of 0.4 m, but it can be increased by averaging over many footprints. In this way accuracy of about 0.17 m can be achieved by reducing horizontal resolution to 1 meter. Due to its accuracy, LIDAR measurements were preferred for many of the studies cited before as reference surface. An additional advantage arises from the fact that used light (usually in the visible or near-infrared range) have no penetration in snow or ice, producing ranges referred to the actual surface. LIDAR data can be used successfully for mass-balance estimation by repeating profiles on an annual or semi-annual basis, also intensity of the returned pulse can be used to find snow/ice boundary (Arnold et al. 2006). The main LIDAR’s disadvantages are: Low flight altitudes needed (usually about a quarter of typical SAR altitudes, high performance systems are capable to acquire at altitudes up to 3000 m), narrow swath (typically up to 20° off-nadir each side) and high cost, making airborne LIDAR unpractical for extensive glacier monitoring. A very interesting solution are spaceborne LIDAR sensors, as GLAS (Geoscience Laser Altimeter System) over ICESat (Ice, Cloud and land Elevation Satellite). It continuously profile ground elevations with 15 cm accuracy over snow or ice (1 to 10 meters over ground). Laser footprint is 70 meters wide, and punctual measurements are done along-track every 170 meters. Orbits can be repeated with ±35 m accuracy. ICESat is perfect for ice sheet change monitoring, by averaging measurements, signals with amplitudes as low as 1 centimeter can be detected. Sub satellite tracks are repeated every 183 days, producing track separations of 15 Km at equator, 11 Km at mid-latitudes and 2.6 Km at 80° degrees latitude (maximum reached latitude by ICESat is 86°). GLAS data is available since January 2003 and constitutes an excellent base to monitor ice sheet changes. However large ground track separation limit usability over small glaciers. Although can be mitigated by its off-nadir pointing capability.

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2.4. Future technologies

A clear indicator of the necessity of better sensors for snow/ice monitoring is the NASA ESTO (Earth Science Technology Office) Instrument Incubator Program (IIP) which since 1998 has the objective to support measurements related to climate change and topography change. But later releases of IIP (2002) start focusing more specifically to “Topography and surface change” and “Sea ice thickness and snow cover”. On 2004 “Ice topographic mapping” appears as one of the main priorities. On 2007, proposed research topics include “ice sheet height; surface deformation; snow accumulation”. Reviewing IIP awarded projects, there is a clear interest in snow/ice monitoring and proposed technologies show a strong focus on Scanning LIDAR applications for high resolution topographic mapping (Some example awarded proposals: An Electronically Steerable Flash Lidar, Efficient Swath Mapping Laser Altimetry Demonstration, Division Multi-Functional Fiber Laser Lidar (MFLL) for Ice Sheet Topographic Mapping - A Development and Demonstration Proposal, Push-Broom Laser Altimeter Demonstration for Space-Based Cryospheric Topographic and Surface Property Mapping, ) and SAR/InSAR sensors (Example awarded proposals: Glaciers and Ice Sheets Interferometric Radar, A Ka-Band Digitally Beamformed Radar Interferometer for Topographic Mapping of Glaciers and Ice Sheets, Large Aperture, Scanning, L-Band Synthetic Aperture Radar). This new technologies will offer in near future continuous high resolution observations, with accuracies of few meters horizontally and few centimeters vertical. Maybe the most promising technologies are InSAR systems that are supposed to achieve accuracies and resolutions like LIDAR sensors ones but with the Radar advantage of all weather observation capability.

3. Shadows as terrain indicators

Shadows are common in mountain terrain at high latitudes and winter mid latitudes (See section 3.2.1), where the main source of terrain change are snow accumulation or glacier thickness variations. Both factors are stronger at the bottom of valleys and basins surrounded by mountains. And evolve in time scales much shorter than mountains changes in shape and altitude. If we assume that the projectors of shadows (mountains or just projectors from now on) have a fix altitude, then knowing the Sun position at the moment the image was acquired, we can use the length of those shadows to measure height differences between the projector and the surface where the shadows are projected, then any changes in that difference can be consider as a surface absolute altitude change, due to glacier thickness changes, snow accumulation, etc.

Figure 1: Schematic geometry of the height difference calculation using shadows.

Shadow S, is know to be in the line Sun-Projector, and its 3D position (S) is calculated by using the angular size

of the shadow from the satellite φφφφ.

With the help of Figure 1 we can explain better the way this method works.

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Let’s say that both, Sun () and satellite (O) positions can be known accurately and that the angular size φ of the shadow (from the satellite) can be measured in the image. Then if we know projector position (P), shadow’s position (S) can be easily calculated as the point in the line -P at an angular distance φ from P seen from satellite position. Then, knowing the exact three dimensional position of S relative to P, the height difference ∆h can be easily calculated. This is an extremely simple idea, but there are many technical obstacles to its application. Here we develop a methodology to perform this calculation as accurate as possible, making possible to extract new useful data from the available imagery.

3.1. Error sources

The accuracy of the measurement of the height difference, relay on the precise determination of the Sun, Satellite and projector positions, atmospheric refraction, pointing stability and shadow measurement. The following section, explain and analyzes these error sources in order to evaluate their magnitude and assert a total theoretical error for different scenarios and satellite platforms. From now on, all referred terrain elevations, altitudes or heights, are ellipsoidal. Errors regarding projector, satellite or Sun positions were calculated by introducing an error on input values and then evaluating differences in multiple output values obtained iterating over all possible geometric configurations.

Figure 2: Relative height difference error per kilometer in projector’s height error as function of the solar elevation. The curve is valid for a satellite in an orbit 725 Km high in the worst solar-satellite geometry.

Note that percentage errors are very small even with kilometer-size errors in estimated projector position.

3.1.1. Projector real position

If the projector position to be used in the calculation of the height difference is accurate the computed shadow absolute altitude and position will be accurate too. But if only relative height differences are need (Ice sheet/Iceberg thickness, volcanic ash plume height, etc.), is possible to use an approximate position ansdaltitude for the projector, this can be taken from the internal georeference of the image and a DEM (Digital Elevation Model), or stereo heights if stereo images are available, from a Map or from other images, etc. We will see that error in relative height difference due to errors in projector position are very small, then, accurate projector positions will be necessary only if absolute position of the shadows is needed. The sensibility of the relative height difference to errors in the projector position depends on the Sun elevation, being smaller for small elevations. This method is useful with Sun elevations below 45 degrees, because mountain slopes are rarely steeper than that, and no shadows will be projected with the Sun elevations higher than the slope inclination. Also the relative height difference error will be proportional to its absolute value (i.e. proportional to the shadow length), that’s why we express it as a relative percent error. Figure 2 shows the error in the height difference per kilometer of error in projector’s altitude and. This relationship change slightly for different shadow lengths, that’s why Figure 2 show a range instead of a single line. The uppermost boundary of the range corresponds to a shadow 10 meters long, and the lower to one 10.000 meters long. We can see that at Sun elevations of 45 degrees the error due to one meter of projector’s altitude error is always lower than 0.00015% of the absolute height difference (0.15% if the error in projector position is 1000 m as shown in Figure 2, a value extremely high). This means that if we have a shadow one kilometer long, and we misplace the projector by 100 meters in altitude, the error in the computed relative height difference will be smaller than 15 centimeters.

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This small variation means that when heights differences are needed, is not necessary to know the precise projector position. An example of this can be seen in Figure 3, where the height difference was calculated from an ASTER image over Antarctica, with Sun elevation of 24°. Using the 3.2 kilometers long shadow of Mount Ryan (3.808 m), and setting projector altitude ranging from 0 to 9000 meters, the calculated relative height differences were between 1478 and 1496 varying at -0.002 meters per meter in projector altitude error. This mean that a 9000 meters error lead to a 18 meters error in the height difference, and a 100 meters error lead to a 20 centimeters error in the height difference. In the Figure 2 and Figure 3, only errors in altitude were taken into account. If we calculate error sensitivity of the results for projector misplacing in any direction, we found that the worst geometry is when Sun and Satellite have the same azimuth and same elevation. This situation will never happens because satellite off-nadir angles range usually up to 25° (the case of the backward ASTER band 3B), and this method is useful with imagery acquired when Sun elevation is under 45°.

Figure 3: Example of how shadow’s calculated height difference varies when projector absolute altitude changes.

If we consider the extreme case with satellite off-nadir of 25°, Sun elevation of 45° and exactly the same azimuth, the error sensitivity can climb up to 0.0004% per meter of projector positioning error. In this case, for an error of 150 m in the projector 3D position, we can wait for a maximum error in the height difference of 0.06 %. In more favorable geometries, this error can be considerably smaller. All the numeric values calculated before are under the supposition of a satellite orbit height of 725 Km like ASTER or LANDSAT sensors.

3.1.2. Satellite position

Every satellite platform is tracked by a Ground Control Segment, in order to keep a precise knowledge of the position, velocity and orbital parameters of the satellite at any time. The precision at witch those values are know varies on each platform. For instance, ASTER (over Terra Satellite) is supposed to have an error (3σ) in the satellite positioning of 150 m, but this value drop down to 10 meters or less in state-of-art high resolution satellites (<7.6 meters in WorldView-1). By modeling problem’s geometry, we found that in the worst case, (most error sensitive geometry) the errors in the height difference determination can climb up to 0.00015% per meter of error in the satellite position. For ASTER this means that satellite position indetermination can lead to errors in measured height differences up to 0.06%. Or 0.6 meters for a 1000 meters measured height difference.

3.1.3. Shadow size

Absolute variation of satellite’s LOS (Line Of Sight), have no influences in the height difference calculation, because relative LOS from one pixel to other are use in the calculation. Relative LOS errors can appear as result of jitter instability and telescope or satellite short term orientation instability. The combined effect of this factors is carefully controlled, because have big importance in the BBR process (band to band registration) done in much of the satellite platforms in order to have good matching between bands. In ASTER this value is keep as to ensure a pixel to pixel matching best than ± 0.2 pixels in any 8 minutes interval. In this way, the error in the relative positioning in the image of the pixels at begin and end of the shadow would be about ± 0.14 pixels equivalent to 2 meters in the apparent length of the shadow. The influence of this error over the measured height difference depends on the geometry, but in the worst case (high solar elevation) it is up to 1.7 meters. But with a solar elevation of 20° (common at latitudes about ± 75°) would be of only 70 centimeters.

3.1.4. Sun position and atmospheric refraction

Modern celestial mechanics is capable to solve celestial bodies’ positions with accuracies up to hundredths of arcseconds (Meeus 1998), leading to errors in height differences smaller than few centimeters in a few kilometers long shadow. For our proposes the whole problem is to find the value of atmospheric refraction,

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which is about one arcminute at an elevation of 45°, which would produce an error of few meters in the measured height difference. Atmospheric refraction can reach up to 35 arcminutes at the horizon. Refraction effect is to increase apparent elevations of the stars and the Sun, leading to a systematic underestimation of height differences if not corrected. In observational astronomy, many formulas have been developed to calculate this effect, some of them considered accurate up to a tenth or even a hundredth of arcminute. A simple and accurate one is Sæmundsson formula (Meeus 1998), a good option in order to use one single equation to accurately compute refractions in the range 0° to 60°. Here we present a version corrected by temperature and pressure at the observation point (Meeus 1998). According to it, the refraction R [arcminutes] as function of the real elevation h [degrees] is:

+

++

=T

P

hhTan

R273

2831010

11.53.10

02.1

Eq. 1 Where R (arcminutes) is expressed as function of true elevation h (degrees), pressure P (millibars) and temperature T (Celsius degrees). This formulation can be considered accurate within 0.1 arcminute. Refraction effect increase with pressure or low temperature, for us will be common to observe thin atmospheres (lower pressures) at low temperatures. Figure 4, shows atmospheric refraction as function of the atmospheric pressure for different solar elevations (10°, 20°, 30°, 40° y 60°). In each case a colored band instead a line is plotted. This band is formed by many lines expressing this relationship for temperatures ranging from -50 °C (upper blue border) and +30 °C (lower red border). We can see that refraction can reach up to 7 arcminutes as maximum in our range of interest. Refraction is in fact an integrated effect along the whole atmosphere, so temperature and pressure profiles along the atmosphere as well as the relative humidity are needed to know it accurately. Accuracy of Sæmundsson formula can be badly affected by anomalous atmospheric conditions, errors produced are very small but systematic, in order to minimize this kind of errors we implement a more complex refraction model, taking into account relative humidity and temperature lapse rate (average temperature change with altitude), which is explained in detail in section 4.5.2. Given that we don’ t know exact pressures and temperatures in the places and moments the shadows are observed, no model will give us an exact value and refraction indetermination will be bigger than the accuracy of the model. Let’ s say that our errors in the knowledge of pressure and temperature are of 50 mb and 20 °C respectively (which are overestimated values).

Figure 4: Atmospheric refraction as function of pressure for different solar elevations (10, 20, 30, 40 and 60 degrees). Each colored band show the range for

temperatures between -50°C (blue upper border) and +30°C (red lower border).

In this case with a shadow at an approximate altitude of 2000 meters, where refraction varies linearly at a rate of 0.02 arcminutes/°C (1.4 arcminutes from -50°C to +30°C), we will have a error in the refraction determination of about 0.7 arcminutes, leading in a bad geometry to a error in the height deference of 0.12%.

3.1.5. Shadow measurement

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One of the main challenges is to be able to precisely point shadow’ s begin and end. This can be made fitting modeled intensity profiles along the shadow with the actual intensity profiles measured in the image. To pick the begin of it, which is the projector, one should know its shape along the shadow direction, but we will always approximate it with an ideal projector with triangular section, so in it’ s intensity profile we should see illuminated white pixels in one side, dark black pixels in the other side and a gray pixel in between. This gray pixel is the position of the projector, and its intensity difference with the illuminated and dark pixels, can be used to assert the sub pixel position of the projector. This process can give us a precision of half pixel or less. Picking the end of the shadow is more complex but it can be done more precise than with the projector, because it’ s diffuse shape produce a multiple pixel profile which can be matched with a computed theoretical profile. This matching gives us confidence about the correct picking of the shadow end within 0.2 pixels or less. So the shadow length can be know with a precision of about 0.7 pixels, error that will conduce to height difference errors smaller than 5.7 meters for ASTER or Landsat 7, but as small as 40 centimeters in high resolution state-of-the art commercial satellites. Also this should be a random error, thus multiple measurements in a single shadow can considerably reduce it.

Table 1: Errors by source in two different scenarios. Percent errors are relative to the absolute value of the height difference, producing very small errors in short height differences (i.e. Ice shelves thickness) and larger errors in bigger height differences (i.e. big mountains, volcanic plumes).

Error source ASTER platform Error Ultra-high resolution

Satellites + Field GPS measurements

Error

Projector position With a 200 m error 0.08% Differential GPS

(0.1 m accuracy) 0.000%

Satellite position 150 m error (ASTER) 0.06% 10 m error 0.004% Shadow’ s angular size 0.2 pixel (ASTER) 1.7 m 0.2 pixel, pixel size 1 m 0.12 m Shadow length 0.7 pixels, 15 m/pixel 5.7 m 0.7 pixel, pixel size 1 m 0.38 m Atmospheric refraction Standard P & T

(0.7’ error) 0.12% Archived P & T model (0.4’ error)

0.07%

TOTAL (quadratic sum) Less than 5.9 m + 0.16%

Less than 0.5 m + 0.09%

NOTE: The combined total error shown here is an overestimation, because the worst geometries for one case is good for others.

3.2. Applications

Is possible to find many applications for this methodology some of them are:

1. Terrain level changes due to glacier thickness changes or snow accumulation. This can be done in different ways. Complete remote: Use a set of images to track relative surface changes. One limitation is that measured relative altitudes probably will belong to different points in surface, although close each other. A reference DEM would solve comparison problems in different times and shadow positions. Uses: - Track changes in extremely remote glaciers. - Measure approximate precipitation in remote cold areas (this is an important input for meteorological models). Ultra high resolution imagery would be needed. Tracking current situation: Use recently acquired imagery to track a known area. This can be useful to know current snow charge (or amount of fallen precipitation) in high mountain ranges with difficult winter access. However usability might be strongly limited because to reach needed accuracy, ultra high resolution imagery is needed, consequently very low temporal resolution will be available. Uses: - Measure the water resources available in a valley. Useful in water use management and electric production planning. - Measure winter precipitation in snowy areas (this is an important input for meteorological models).

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Field control of past terrain levels: Use a set of images to calculate past absolute heights (precise projector position must be measured) on different points over a glacial body. Then on field work, measure current heights on those points, then surface level trend can be estimated. Uses: - Extend current glacial mass balance studies to the past, before begin of field monitoring.

2. Height measurement of big changing features. Uses: - Volcanic plume height monitoring in current or past eruptions. - Calibration of other plume height estimation techniques. - Clouds height measurements (especially useful in non-stereo platforms with no thermal bands, as all the high resolution modern commercial satellites.

3. Sea ice freeboard height. Iceberg, ice shelf or any floating object height can be measured accurately in high latitudes and then, assuming hydrostatic equilibrium, total mass can be calculated. Uses:

- Antarctic ice shelf monitoring. - Pats Antarctic ice shelf status characterization. - Calving glaciers wastage measurements. - Icebergs coverage prediction for navigation planning and forecast.

3.2.1. Area of applicability

Most of the useful remote sensors for this method are onboard of spacecraft placed in sun-synchronous orbits, visiting each latitude at the same local time on every pass.

Figure 5: Observation local time by latitude for Sun-synchronous orbits. Blue thick line represent all the

relationship values for orbital geometries in the range defined by seven selected satellite platforms (Terra,

SPOT, Landsat, Ikonos, QuickBird, IRS and WorldView), with orbit inclination ranging from 97.2 to 98.7°, orbital period between 93.5 and 101.4 minutes and equatorial crossing time between 10:15 and 10:30 A.M.

Is interesting to note that in local time, imagery acquisition is done backwards in time.

Figure 5 shows the local observation time at each latitude for a set of available satellite platforms. Times are computed for at-nadir observations. Equatorial crossing time is usually at 10:30 A.M. and the highlighted area show how most latitudes (between -75° and +75° approximately) are observed in a range of thee hours around this value. During a daytime acquisition the satellite fly from north to south, therefore northern latitudes are observed before than southern ones. But the shift in longitude produce that in local time, the satellite is going backwards in time. This means that southern hemisphere latitudes are observed earlier in the morning and northern latitude later, closer to local midday, when shadows are shorter.

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Figure 6: Sun elevation at observation time for Sun-Synchronous orbits by latitude. Curves for Northern summer solstice (blue), Northern winter solstice (red) and equinoxes (green). Greenish latitude ranges show

the applicability area of this method. Curves appears as thick lines because they represent all the values for

orbital geometries in the range defined by seven selected satellite platforms (Terra, SPOT, Landsat, Ikonos,

QuickBird, IRS and WorldView), with orbit inclination ranging from 97.2 to 98.7°, orbital period between 93.5

and 101.4 minutes and equatorial crossing time between 10:15 and 10:30 A.M.

Figure 6 show solar elevation for each value in Figure 5 for three dates: Northern solstice (blue), Southern solstice (red), and equinoxes (green). The solar elevation upper limit of practical applicability of the method is about 45°, and the lower would be 0° but most satellite platforms do not collect daytime data when Sun elevation is under a certain value (5° for Landsat 7). In the figure, three colored ranges are defined. A reddish one covering the area where this method is useless using sun-synchronous satellites (-13° to 17° in latitude). A yellow range, where usable data would be sparse (-82° to -77°, -33° to -13° and 17° to 42°). And finally, a green area, covering the ranges of maximum applicability, because images acquired under usable solar conditions will be abundant. This range is, in the southern hemisphere from 77° to 33° south (44° wide) and, in the northern hemisphere from 42° to 82° north (40° wide).

Figure 7: Solar elevation for sun-synchronous

acquisitions trough one full year. Red curves show values for northern latitudes and Blue curves for

southern latitudes. Note that at latitudes inside polar circles, positive solar elevation are found in ascending

(nighttime) orbit sections.

Solar elevation in sun-synchronous acquisitions by date and latitude can be seen in Figure 7. We can see there that latitudes south of -60° and north of 70° are observed the whole year under good solar conditions for shadow methodology application. The availability of usable data at each latitude can be seen easier in Figure 8 where the number of days in a year (year 2008 was used for calculations) at which sun-synchronous satellites observe the ground when solar elevation is under an usable threshold (45° to 10° thresholds in steps of 5° are shown).

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Figure 8: Number of days at which sun-synchronous

satellites observe each latitude with solar elevation under a elevation of 45° (upper line), 40° (second upper) and so on down to 10° (lower line). A lower elevation threshold of 5° was set. When a given latitude where observed in

the target elevation range in the ascending and descending passes, it was count as two days.

Main features of the curves are:

- Increasing usable numbers of days with increasing latitudes - Acquisitions with solar elevations under 45° start at 14° S and 17° N in southern and northern hemispheres

respectively. - A local maximum is reached at 56° S and 61° N. After it number of days decrease because some

acquisitions are produced at solar elevations under 5°. - Numbers of days start increasing again at 71° S and 66° N because usable acquisitions happens during

ascending (nighttime) passes. We can see that observations during more than half year are done under usable solar elevation at latitudes higher than 36° in the southern hemisphere and 44° in the northern one. Also, more than 220 days of usable solar conditions are found at all latitudes south of 43° S and north of °52 N.

3.3. Method advantages in current data availability context

As we saw in section 2, many terrain elevation sources are currently available. Looking to the past, airborne stereophotogrammetry begins in the early 20th century, even before airplanes were developed. In the 50’ s military satellite image acquisition start with Corona missions and in the 70’ s Landsat family satellites start imaging Earth surface continuously and systematically. In the 80’ s spaceborne SAR and Radar altimetry acquisition began, and in the 21th century a new era of accurate extensive measurements began with the use of spaceborne single-pass InSAR and LIDAR technologies. We can see that as we go farther to the past, elevation-rich data became sparser and even today it doesn’ t match current research needs. Shadow derived height differences can for instance contribute to make denser spatially and temporally the net of elevation control points currently used for glacier mass balance tracking and climate change modeling. Additional control points derived from this methodology became especially valuable in the past, before extensive acquisition of spaceborne stereo imagery starts. Ice sheets thickness could be estimated from old or new imagery, and valuable accurate volcanic plume heights could be derived, to be used as main data source or even for calibration of other methodologies. We will demonstrate that even nowadays spaceborne shadow derived height differences are very likely able to produce elevation control points with a vertical accuracy better than spaceborne stereoscopy and horizontal resolution better than spaceborne LIDAR. Is important to highlight that applicability problems (solar geometry, cloud cover and terrain requirements) make this methodology suitable mainly to make denser a given set of elevation control points, but in most applications it doesn’ t constitute a practical change tacking data source by itself due to the scarcity of suitable images and the fact that we can’ t freely choose shadows positions.

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Figure 9: Schema of the shadow position calculation process.

4. Methodology

The following methodology was designed to be as accurate as possible without reducing its field of action by requiring information and/or accuracies difficult to obtain. The simple and obvious calculation shown in Figure 1 is not useful for satellite imagery because it suppose that projector and shadow were observed from the same point, this is true for aerial photography but in satellite imagery, the satellite moves during the acquisition, and both observing points will be away each other a distance bigger (about 10% for ASTER or Landsat) than the along track length of the shadow. So if we have a shadow as the one used in Figure 3 (3.2 Km long) in an angle of 45°, the along-track length is 2.2 Km, and the satellite displacement during the acquisition of the full shadow would be about 2.5 Km. In order to do accurate calculations taking in account this movement, we develop a method consisting in the following steps.

1. Find satellite position at projector observation time (Sp). 2. Find projector apparent position over the ellipsoid using internal satellite orientation parameters (Pe). 3. Define the line Sp-Pe, which is the visual from the satellite to the projector. 4. Find satellite position at shadow observation time (Ss). 5. Find shadow apparent position over the ellipsoid using internal satellite orientation parameters (Se). 6. Define the line Ss-Se, which is the visual from the satellite to the shadow. 7. Find Sun apparent position taking into account atmospheric refraction (). 8. Set the Projector position to be used in the calculation (P). 9. Define the line -P, which is the line defined by the light ray coming from the center of the Sun that

pass exactly over the projector toward the shadow. 10. Find the point of Sp-Pe closest to -P (Pe’ ). 11. Find the point of Ss-Se closest to -P (Se’ ). 12. Find the distance D between Pe’ and Se’ . 13. Find the shadow position S, as the point in -P at a distance D from P in the Sun’ s opposite direction.

The following sections, describe how to find the different position mentioned in the points 1 to 13.

4.1. Satellite position

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Although satellite position can be calculated at any time from orbital parameters, many corrections must be done, also orbital parameters change with time due to external perturbations. Low level satellite data usually include tables with the exact acquisition time and satellite position for selected pixels. The calculation process might be different for different observation platforms. Here we show this process for the instrument ASTER, on the Terra Satellite platform. Information shown was taken from a ASTER L1A product.

Figure 10: Lattice point (red crosses) distribution in an ASTER VNIR normal band. The black arrow shows the

satellite flying direction.

ASTER metadata include satellite position and acquisition time for selected image points called lattice points. Figure 10 shows its distribution along the image. Values at any pixel can be computed interpolating the values given for lattice points. ASTER sensor consists in a linear array that acquires each line of data simultaneously, then the image is constructed by a sequence of lines acquired as the platform moves in its orbit. That’ s why all the pixels in each line of the image are acquired simultaneously, so satellite position and acquisition time is the same along each line. Time is given with one millisecond resolution (the interval between lines is about 2.2 milliseconds). Figure 11 show the acquisition time of each lattice points in seconds referred to the time for the first line.

Figure 11: Acquisition time by line. Dots are times

given for lattice points.

Satellite velocity is about 30,000 Km/h and positions are given at one meter resolution every about 0.88 seconds. Figure 12 shows satellite positions during one acquisition. Satellite position for projector and shadow’ s observations are computed by interpolation at the corresponding line numbers the positions given at lattice points. Different interpolation methods can give position differences up to 0.8 meters.

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Figure 12: Black dots point satellite positions relative to

the first lattice point. The line show the best order 2 polynomial fit.

4.2. Ellipsoidal apparent positions

The ellipsoidal apparent position of a point in the image is the intersection between the instantaneous visual from the satellite to that point, and earth’ s ellipsoid. The difference with the real position will be proportional to the ellipsoidal height of the point and the off-nadir observation angle. Errors in this position are related to errors in satellite orientation parameters. Ellipsoidal apparent positions must be calculated using satellite position and orientation data and following each platform specifications. The ASTER L1A product, include ellipsoidal apparent positions at lattice points. With them, ellipsoidal apparent positions at any point can be calculated by interpolation. Different interpolation methods can give differences up to 5 meters against linear interpolation in some cases; bilinear interpolation is used to improve accuracy.

4.3. Shadow and projector picking

The most important step and likely to errors, is the projector and shadow picking in the image. In order to do it accurately, theoretical illumination profiles must be fitted to the image data. Shadows and projectors illumination profiles are complete different. The following two sections describe each case.

4.3.1. Preliminary picking

Preliminary picking of the projector and shadow tip positions in the image is done by just pointing their positions in the image. In some cases this is very simple, but in others, the geometry of the Sun, projector and surface lead to big distortions in the projected shapes of the mountains, making easy a mismatching of the projector with its corresponding shadow. In order to help the matching the graphic interface performs and display the following calculations:

1. Once the projector is set, the trace of the shadow is computed and displayed. 2. Once the shadow tip is set, the trace of the reverse shadow is computed and displayed.

ASTER sensor acquire a second image of the target area from a backwards looking perspective for stereoscopy proposes. We don’ t use stereo capability for altitude calculations, but we do use this stereo pair as an independent data set. Therefore, just in order to facilitate the task of picking the same projector and shadow in normal and backward bands, the following two steps were added. They are not mandatory and would become senseless in single perspective sensors as Landsat.

3. Once a point (projector or shadow) is set in a Normal band, its visual is calculated and displayed in the backwards bands.

4. Once a point (projector or shadow) is set in a backwards band, its visual is calculated and displayed in the Normal bands. To compute shadows (and reverse shadows) traces. Is necessary to:

1. Find the apparent 3D position of the projector or shadow for reverse traces (from now on called “ target” ). This is done by intersecting the line between the Satellite and target’ s ellipsoidal apparent position with an earth ellipsoid at the approximate projector height (taken from stereo imagery, DEMs, maps, etc.).

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2. Compute the line between the Sun and target’ s apparent 3D position using the apparent (refracted) position of the Sun.

3. Find a set of points in the space given by the intersection of the former line with the earth ellipsoid at different heights (ranging from -1000 to 9000 m for example).

4. Find the ellipsoidal position of each point seen from the satellite. This is done by intersecting the line Satellite-Point with earth’ s ellipsoid.

5. Use satellite orientation data to find the position in the image of each calculated ellipsoidal position. This is done by reversing the process described in section 4.2.

6. Plotting the curve defined by the points in the image. To compute visual traces in reverse or normal bands seen from opposite’ s perspective. Is necessary to:

1. Find target’ s apparent 3D position in the origin band. 2. Compute the line between the Satellite and target’ s apparent 3D position. 3. Find a set of points in the space given by the intersection of the former line with the earth ellipsoid at

different heights around target approximate height. 4. Find the ellipsoidal position of each point seen from the satellite in the final band. This is done by

intersecting the line Satellite-Point with earth’ s ellipsoid. 5. Use satellite orientation data for the final band to find the position in the image of each calculated

ellipsoidal position. This is done by reversing the process described in section 4.2. 6. Plotting the curve defined by the calculated points in the image.

4.3.2. Shadow calculation and picking

The first step in the shadow calculation is to know the intensity distribution into the solar disk. As long as we need only relative intensities, the only important effect to take into account is limb darkening (differential atmospheric extinction is neglected, because it is too small within angular differences of the order of Sun’ s angular diameter ~ 0.5°). Limb darkening is an effect product of solar atmosphere radiance distribution that produces a darkening of the solar disk towards the borders. Figure 13 b. shows a photographic image of the sun where the limb darkening effect is evident. Limb darkening was modeled using the formulation presented by Cox 2000 where the intensity of a point is given by

=

ΩΩ−=

n

k

k

k SinCosCos

aII0

22

0

θ

Eq. 2 Where Ω is the solar angular radius (semidiameter) and θ is the angular distance from the center of the Sun to the point. I0 is the central intensity and ak are scalar coefficients. For the Sun at 550 nm, limb darkening is well expressed with n = 2 and: a0 = 0.30 a1 = 0.93 a2 = -0.23

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Figure 13: Upper figure (a.): Calculated solar disk’s

intensity distribution. Lower figure (b.) Real image of the solar disk showing limb darkening effect (adapted gray scale version from www.budgetastronomer.ca).

Using equation Eq. 2, I0 = 1 and these coefficients, a theoretical Sun disk is calculated and normalized, so the sum of intensity values over the whole disk is equal to one. Figure 13a show a calculated solar disk with 100 pixels radius. Then the shape of the projector must be estimated in order to calculate the relative intensity received at a point in the surface. This quantity is function of the Sun elevation over or under the projector, the shape of the projector and the inclination of the surface in the shadow direction. The shape of the projector can be estimated by measuring the angle at the tip of the shadow (see Figure 14e) and by considering the projector as a triangle and its shadow as an elongated triangle with the same base length. In this case, the real shape angle ψ of the projector (see Figure 14f) is given by:

( )

= −

εψψ

tan2'tan

tan2 1

Eq. 3 Where ψ’ is the apparent angle measured at the tip of the shadow (see Figure 14e) and ε is Sun’ s elevation. Figure 14 shows an example of this process performed over a pyramidal shaped peak in Antarctica. Using the values for ψ’ given in panel e. the corresponding Sun elevation of 12.395° and equation Eq. 3, the resulting value for the real angle ψ is 123°, a good approximation to the 128° measured in Figure 14f for our needs. Then, moving this simplified triangular projector over the calculated solar disk, and integrating pixel intensities outside the projector, we can obtain the relative intensity received by any point in the surface under the shadow relative to the fully illuminated area. The amount of illumination received by a point in the surface will be function of the elevation difference between the projector and the Sun seen from that point. This process is illustrated in Figure 15.

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Before the fitting process, the theoretical shadow illumination profile is calculated for a flat horizontal surface at an approximated altitude, using altitude values calculated from stereo pairs (possible in ASTER and other stereo sensors), DEMs or from shadows iterating this method. Is necessary to keep tracking of the position of the point corresponding to a elevation difference equal to zero in the shadow illumination profile. Figure 16 shows three theoretical shadow illumination profiles using different projector geometries and considering and neglecting limb darkening phenomena. We can see that profile shape is sensitive to projector shape, and more sensitive to limb darkening for sharp projectors.

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Figure 14: a. Shaltz Ridge Peak in the Sentinel Range,

Antarctica (Latitude 78.5° South). This Peak was used as one of the control points for this method.

b. Shaltz Ridge Peak at the moment of shadow measurement from shadow’s end.

c. Close-up to the projector summit with a solar filter from shadow’s end. See that limb darkening is

noticeable.

d. Shaltz Ridge in a ASTER scene taken on December 23rd 2004, with a Sun elevation of 12.395°.

e. Close-up to d. with the value of the angle ψψψψ’.

f. Same as c. with the value for the angle ψψψψ.

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Figure 15: Schematic process of the theoretical shadow

profile construction. The intensity at any point is computed using the elevation difference to place the

simplified triangular projector over de calculated solar disk and then integrating the intensity in the solar disk

visible from the point.

Figure 16: Theoretical shadow illumination profiles for different projector geometries: A 45° pyramidal Peak

(red), a flat ridge (blue) and a 45° triangular notch (green). Black lines correspond to the same geometries

but neglecting limb darkening.

Fitting process can be done automatically, but human supervision gives much better results. But completely automated methods are necessary to compute multiple height differences along a shadow. The fitting process consist in superpose the intensity profile measured in the image with the theoretical one, and find the parameters that reach the best match. This fitting process goes along the following steps:

1. Find the position of the profile that goes over the real tip of the shadow by analyzing different image profiles taken perpendicularly to the apparent shadow trace in the image (see section 4.3.1 for shadow traces calculation). If the projector is a summit or a notch, this can be done by looking for the profile with less (or more for a notch) integrated intensity.

2. Scale and add a base intensity to the theoretical shadow profile to make its base and roof match the intensities in the darker area of the shadow and the completely illuminated area respectively.

3. Move the theoretical profile along the measured one in order to find the displacement with best match (match quality can be measured by sum of square differences).

4. Change surface inclination in order match the length of the shadows in measured and theoretical profiles.

5. Repeat steps 3 and 4 until the best match is found. When the fitting process is done, the position of the point in the image corresponding to the “ center” of the theoretical profile is set as the accurate position of the shadow in the image. We understand as “ center” of the theoretical profile, the point that corresponds to an elevation difference between Sun and projector of zero degrees. Appendix A shows the graphic interface designed to find the precise position of the shadow.

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Shadows detection and fitting problems derived from shadows projection area properties have been detected: Irregular or very low reflectivity. The former is common in areas with irregular snow coverage; reflectivity irregularities produce distortions on the shadows intensity profile, making matching process more difficult and less accurate. The second is common when shadow is projected over water, as often happen in Ice shelf or iceberg freeboard height measurements. This is due to the low reflectivity of water in visual and infrared bands. Maximum reflectivity is found in the blue band. Usually when no blue band is available (as in ASTER), is impossible to detect shadows over water.

Figure 17: Iceberg shadow seen in Landsat 7 ETM+

bands 1 or blue (left), 2 or green (center) and 3 or red (right). Contrast enhancement has been applied to each image, and final color was set in band 1 and 3 to match

band 2. Is clear how shadows (in the bottom right corner of icebergs) are well contrasted in the blue band, but gets

fainter when observed wavelength is displaced to the red.

Figure 17 shows icebergs shadows in thee Landsat ETM+ bands (blue, green and red). There we can see how shadows appear well contrasted in the blue band, fainter in the green one and almost imperceptible in the red one.

4.3.3. Projector picking

Using the approximation of a sharp projector, the illumination profile along the projector is much simpler, because we will have a illuminated side, a dark side an a transition gray pixel in between.

Figure 18: Left: Simple image model with a projector along the center. Five different profiles are displayed.

The intensity in the transition pixel depends on the sampling shift. Right: Intensity profiles for each case, we can see how the intensity of the transition pixel is related

to the sub pixel position of the projector (pointed b y black triangles).

If there is no transition pixel, the projector position is well know and is exactly in between illuminated and dark pixels. But this case is very unlikely. Most times there will be a gray pixel in between, and its intensity relative to bright and dark sides can be used to find the sub pixel position of the projector. Figure 18 shows how the intensity of the gray transition pixel can be used to point the sub pixel position of the projector, so the accuracy of half pixel mentioned in section 3.1.5 is feasible and even likely to be surpassed. One of the extra complications is the fact that the profile trace in the image is almost always diagonal in relation with lines and columns, so the intensity steps in the profile have different lengths and more than one transition pixel can be found. So the precise projector positions have to be pointed by inspection of the profiles and the image.

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4.4. Shadow path prediction

In most cases, we will be interested in nor prominent neither sharp projectors, making confuse which shadow belong to which projector. To overpass this ambiguity we developed an accurate shadow path prediction algorithm, because by only using Sun’ s azimuth rotated to correct image rotation, we found very big errors up to hundreds of meters per kilometer of shadow length. To correct it, it is necessary to take into account acquisition geometry, satellite displacement during acquisition and light rays curvature when penetrating the atmosphere. The algorithm works as follow:

1. Given a x,y pixel position in the image, and a estimated pixel height h, the 3D position of the point is computed (from now on the Shadow Tie Point or STP), by intersecting the ellipsoid of height h with the visual from the satellite to the pixel xy (satellite position must be calculated for y line observation time).

2. Using archived meteorological model (specifically altitudinal pressure and temperature profiles) refraction and solar apparent position is calculated for STP (details in section 4.5).

3. To find the 3D position of the next point of shadow’ s path, the line Sun-STP is intersected with a ellipsoid h+∆h (∆h can be positive or negative if we are extending the shadow downward or backwards).

4. Solar apparent position is recalculated for this point (repeating step 2) and the third point of shadow’ s path is found repeating step 3 using previous point instead of STP. In this way, discrete 3D positions of the whole shadow’ s path can be calculated.

5. Each 3D position is transformed back to x,y pixel positions. The transformation in step 5 must be done by iteration, because to perform the transformation is necessary to know satellite position when the given 3D point was observed. To start the iteration, we do the transformation with an approximate satellite position and a x,y position is found, then we recalculate satellite position at the observation time of the computed pixel line y. This satellite position is then used to compute new and more accurate x,y values. With this new y value, a even more accurate satellite position is computed and using it, new x,y values and so on. Iteration converges quickly, and after three or four steps, corrections made by following cycles are negligible (about 10-7 pixels). Testing predicted shadow path in ASTER images with sharp and prominent projectors (i.e. projector-shadow matching can be done easily by visual examination) we found errors always lower than 2 meters per kilometer of shadow length. This means that errors of one pixel can be found only in extremely long shadows (seven kilometers or more). Computed shadow path is almost not sensitive to the approximate initial height h (in step 1), height errors up to few hundred meters can be done without varying shadow path in a perceptible amount. Found differences are probably due to rotation errors in satellite attitude, inaccuracies in meteorological models or refraction modeling errors. Meteorological and refraction errors should be negligible on observations made with small off-nadir angles.

4.5. Sun’s apparent position

To compute Sun’ s apparent position is necessary to know: Sun’ s real position and atmospheric refraction at observer’ s position, date and time. With this data, sun’ s apparent position is computed as the point at the same distance and azimuth than real Sun but with a elevation equal to the real elevation of the Sun plus refraction angle. Following sections explain how Sun’ real position and atmospheric refraction can be accurately calculated.

4.5.1. Sun’s real position

The Sun position calculation used is a simplified version of the VSOP87 theory, accurate up to 0.01 arcseconds using 2425 periodic terms. We use the simplified version given by Meeus 1998, accurate up to one arcsecond (0.0003 degrees) between the years -2000 and +6000. This algorithm was adapted to provide positions in geocentric rectangular coordinates in meters. To compute accurate positions is necessary to know the Terrestrial Time (or terrestrial dynamic time or ephemeris time) at the observation moment. The difference ∆t with the Universal Time (UT) is an empirically determined quantity that varies from about 43 seconds in 1973 to 66 seconds in 2008. To compute its value we interpolated the monthly spaced data from February 1973 to July 2008 provided by the U.S. Naval Observatory at http://maia.usno.navy.mil. Shortly, the calculation process goes along the following steps:

1. Compute Julian Day and Ephemeris Time for the observation moment.

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2. Compute Earth heliocentric longitude, latitude and radio vector. And then transform them to geocentric coordinates of the Sun.

3. Compute nutation in longitude and obliquity 4. Compute the true obliquity of the ecliptic. 5. Compute apparent geocentric longitude of the Sun, correcting by aberration and nutation. 6. Compute apparent sidereal time at Greenwich and then the rotation angle of the Greenwich meridian

respect to the instantaneous true equinox. 7. Transform and rotate spherical coordinates to rectangular geocentric coordinates in meters relative

Earth rotation axis, equator and Greenwich meridian.

4.5.2. Atmospheric refraction

As we see in section 3.1.4, atmospheric refraction can be calculated using equation Eq. 1, solar real elevation, temperature and atmospheric pressure are needed. We use Eq. 1 for some calculations where many refractions values are needed but not extremely accurate (i.e. shadow path prediction), we will describe here a precise method where in addition to solar real elevation, temperature and atmospheric pressure, also relative humidity and temperature lapse rate are needed. Sun’ s true elevation can be easily calculated at any position using Sun’ s real position, pressure can be approximately calculated from terrain altitude and temperature can be estimated too or measured using thermal bands if available. All this approximations induce errors in the refraction estimation. In order to reduce errors we used archived global atmospheric models at NOAA to find pressure, relative humidity and temperature atmospheric profiles. We create a custom computer interface with NOAA Ready website, to perform request of meteorological data for the date and time of observation, and then interpolate them for the exact observation time. Meteorological data is available at intervals of 6 hours since January 1997 using FLN database, and at intervals of 3 hours since December 2004 using GDAS1 database. Data is provided as altitude and temperatures at specific pressures, so it is inverted to obtain temperature and pressure as function of altitude. Figure 19 show an example of atmospheric profiles extracted to calculate refraction in an ASTER scene over Antarctica. Using this atmospheric profiles, temperature relative humidity and pressure are computed at the approximate altitude of the shadow, and temperature lapse rate is computed over the observation site. With this data atmospheric refraction is calculated and stored to be used in Sun’ s apparent position calculation. Refraction calculation is done using the precise algorithm proposed by Kenneth 1992 in the “ Explanatory supplement to the Astronomical Almanac” . It is a numerical algorithm using quadrature and based in a physical model. In this model, atmosphere is assumed to be spherically symmetric and in hydrostatic equilibrium, and to obey the perfect gas law for the combined mixture of dry air and water vapor, and also for the dry air and water vapor separately. Is consider two atmosphere layers: The troposphere from the Earth surface to the tropopause assumed to be at a fixed altitude of 11 Km. In this region temperature is assumed to decrease at a constant rate α, Kenneth 1992 use a fixed value for α (in a standard atmosphere α = 0.0065 K/m) but in order to keep accuracy performance in non standard atmospheres, we omit this assumption, thus considering α as a calculation input. At the troposphere, relative humidity is assumed to be constant and equal to its value at the observer. Over the tropopause, the second layer is the stratosphere, where temperature is assumed to remain constant and equal to its value at the tropopause and no pressure due to water vapor is considered. Stratospheric refraction is neglected over an altitude of 80 Km. The total bending of a ray is given by:

+−=

0

0

z

dzdrdnrn

drdnrε

Eq. 4 Where z is the zenithal angle, n the refraction index and r the distance to Earth’ s center. This integral is a transformation of the usual refraction integral more suitable for numerical quadrature, because is a more slowly varying function over the whole range of z and removes the problem at z = 90°. Due to the discontinuity at the tropopause, integration is made separately in each atmospheric layer.

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Figure 19: Pressure (upper panel) and temperature (lower panel) profiles from 0 to 8000 meters, extracted from archived meteorological models at NOAA, for an ASTER scene centered at 78° 23’ South and 86° 14’ West acquired on January 9th 2003 at 12:44 h UT.

5. Multiple measurement

The described methodology can be applied multiple times (maybe automated) along one shadow in order to obtain hundreds of measurements over the same zone. Now if we assume that all the area were the shadow is projected is subject to the same terrain change (plausible for snow accumulation and for some cases in glacial surfaces), we can average all the measurements in order to minimize random errors.

Table 2: Errors in height difference calculation by source and type (Syst. = Systematic, Rnd. = Random). An absolute height difference of 1000 m was assumed for this example.

Error source

Original magnitude

Error [m]

Type

Projector pos. 100 m 0.8 Syst. Satellite pos. 150 m 1.2 Syst. Shadow’ s size 0.2 pix 1.7 Rnd. Shadow length 0.7 pix 5.7 Rnd. Refraction 0.4’ 0.7 Syst. TOTAL Rnd. 6.0 TOTAL Syst. < 1.6 TOTAL < 6.2 Table 2 show a modified version of Table 1 with errors for calculations over ASTER images, with big errors in projectors position and refraction computed using archived meteorological data. Also errors have been separated between random and systematic. Then if we want to calculate the average height difference between a projector (a ridge for example) and the whole area were it’ s shadow is projected, the error in this average would be:

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nRandom

Systematic

δδδ +=

Eq. 5

Where δ is the total error, δSystematic is the systematic error, δRandom is the random error and n is the numbers of measurements done, and that can be as big as the width of the shadow in pixels or even bigger. Now we can see that if we have one hundred measurements, using the values in Table 2 and the equation Eq. 5 the final error drop to less than 2.2 meters (due considerations in the note on Table 1) and less than 1.7 meters with one thousand samples. Finally snow accumulation/ablation, or average surface changes can be estimated as the difference between two measurements of average height difference in two different images. Taking in account the error in both measurements, the change will be know with an error expected to be 3.1 meters and 2.4 meters for 100 and 1000 samples respectively (1.41 = 2 times bigger). In this way using average height differences in ASTER imagery, errors can be reduced to one third; this is less than 2 meters. Using high resolution imagery, errors in average height differences would drop to about ten centimeters. Is important to say that in that this procedure would be directly applicable only in flat horizontal surfaces or if we have two images with exactly the same solar elevation and altitude (i.e. almost never) because the shadows in each case will be projected in different areas. A practical way of application would need a good knowledge of the reference surface trough a DEM and a reference image (or projector geometry knowledge). The main application scenarios would be:

1. Snow cover monitoring: A DEM of the dry terrain and a reference image when dry would be needed. 2. Glacier monitoring: A reference DEM of the glacier would be needed in addition to a reference image

contemporaneous to the DEM or a GPS survey of the projector.

6. Validation

Validation was done in five areas:

- Sentinel Range, Antarctica: At latitude around 75° 30’ south and a base height about 2100 m. 9 shadows belonging to 5 mountain peaks were measured. Field measurements were done.

- Aconcagua Valley, Andes Cordillera: At latitude around 32° 50’ south and a base height of about 2500 m. 9 shadows belonging to 3 mountain peaks were measured. Field measurements were done.

- Abbot ice shelf, Antarctica: At latitude around 72° 12’ south and a base height of 0 meters. Multiple shadows were measured. No field measurements were done, ICESat data is used for comparison.

- Ronne-Filchner ice shelf, Antarctica: At a latitude about 78° south and a base height of 0 meters. Multiple shadows were measured. No field measurements were done, ICESat data is used for comparison.

- Volcanic plume of Shiveluch Volcano, Kamchatka, Russia: At latitude about 56° 59’ north and a base height of about 300 m. 5 shadows were measured. No field measurements were done, ASTER stereo heights are used for comparison. Were field measurements are available, the validation process was done by measuring accurate projector positions, then calculating shadows absolute positions (altitudes and horizontal coordinates) using the methodology over ASTER imagery. Calculated position where reached using a handled GPS and then an accurate measurement of the altitude at that point was done. After postprocessing GPS data, horizontal differences between calculated and measured coordinates were used back in the field, to estimate altitude difference between calculated shadow position and measured reference point. Accurate position were measured with a Trimble 5700 double frequency GPS receiver and then postprocessed by two or more long baselines services: AUSPOS, SCOUT, Auto GIPSY and OPUS if available. Output positions were compared to check solution stability, final positions used, are considered accurate within 0.4 meters in the worst cases. Were field measurements were not possible, independent remote sensed data were used: ICESat over ice shelves and stereo derived heights over volcanic plumes. Validation areas are placed in the full southern hemisphere range of applicability shown in Figure 6.

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6.1. Sentinel Range validation data set

Antarctic validation points were measured as part of The Omega Foundation mapping expedition sixth season in the Sentinel Range. From November 2007 to January 2008. Data acquisition in each point was done during at least one hour at 15 second interval (Only Peak Ryan SW was measured only 35 minutes), but many of them have more than 10 hours of data. The measured peaks used were:

- Mount Ryan: Latitude -78.369987° Longitude -86.024942° Altitude 3808.06 m

- Mount Ryan Southwest: Latitude -78.374103° Longitude -86.032424° Altitude 3769.77 m

- Shaltz Ridge Peak: Latitude -78.481736° Longitude -86.031342° Altitude 2633.93 m

- Knutsen West ridge peak: Latitude -78.499830° Longitude -86.044716° Altitude 2488.70 m

- Gardner West ridge peak: Latitude -78.402533° Longitude -86.190158° Altitude 2540.63 m Absolute height differences range from 292 to 1495 meters. Four ASTER L1A (v.3) scenes were used:

- SC:AST_L1A.003:2010581107 Date: 09 January 2003 12:44 h Sun elevation: 24.5°

- SC:AST_L1A.003:2027142602 Date: 23 December 2004 06:54 h Sun elevation: 12.4°

- SC:AST_L1A.003:2027171920 Date: 25 December 2004 13:08 h Sun elevation: 27.4°

- SC:AST_L1A.003:2032500227 Date: 04 January 2006 13:16 h Sun elevation: 26.5° Shadow measurements were done before the final state of this methodology, so the measured altitudes are placed in horizontal positions different than the calculated ones by the last version of the methodology. So the reference true altitudes of the shadows were estimated using interpolation, distances and slopes, so small errors in these references are likely to increase the measured error of the method showed in this section. Other error sources for the reference altitudes are terrain changes between measurement and image acquisition (images used range between 2003 and 2006). Shadows were projected over snow fields, but fortunately for the validation process, snow accumulation and ablation is very small at this latitude and altitude. To estimate the magnitude of this terrain change, a point close to Mount Ryan base was measured and compared with the measurement done in the same position two years before, showing a terrain change of 63 centimeters upwards. So probably the errors we estimate for the measurements done with this method, are slightly overestimated.

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Figure 20: Errors (differences) of the absolute altitudes measured with shadows and GPS for the 9 validation points in Antarctica. Crosses are punctual differences

from the measurement done over normal bands (Blue), reverse band (black) and the average (green).

Continuous lines show averages over the nine samples (same color codes). Also maximum average errors are

shown (doted green line), and global standard deviation (dotted red line).

Figure 20 shows the errors (differences) between absolute altitudes measured by shadows and GPS. We can see that errors are always under 6 meters, and standard deviation less than 4 meters. Is important to say that the three points in the upper error range, belongs to the three measurement of Mount Ryan’ s shadow, therefore maybe an extra difference is due to an error in the GPS measurement of the summit.

6.2. Andes validation data set

Central Chilean-Argentinean Andes validation points were measured during September and October 2008. Data acquisition in each point was done during at least two hours at 2 second interval. Four ASTER L1A (v.3) scenes were used:

- SC:AST_L1A.003:2006547875 Date: 18 May2000 15:02 h Sun elevation: 33.3°

- SC:AST_L1A.003:2012745748 Date: 09 April 2003 14:48 h Sun elevation: 42.2°

- SC:AST_L1A.003:2016330216 Date: 15 August 2003 14:48 h Sun elevation: 36.1°

- SC:AST_L1A.003:2022604196 Date: 11 April 2004 14:48 h Sun elevation: 41.3° 9 shadows belonging to 3 peaks were measured, absolute height differences from 308 to 656 meters. The peaks used as projectors were:

- West sub peak of Cerro Mario Ardito Latitude: -32.8416325° Longitude: -69.8130563° Altitude: 3199.43 m

- Southwest sub peak of Cerro Tolosa Latitude: -32.8019409° Longitude: -70.0564407° Altitude: 3863.28 m

- Southeast sub peak of Cerro Agua Salada Latitude: -32.8157170° Longitude: -69.9513586° Altitude: 3296.58 m

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Images with and without snow was used, for comparison purposes snow depth was estimated at acquisition dates, based on snow coverage. Errors in those estimations are believed to be less than one meter, based on knowledge of the area and typical snow cover.

Figure 21: Errors (differences) of the absolute altitudes measured with shadows and GPS for the 9 validation

points in Andes. Crosses are punctual differences from the measurement done over normal bands (Blue),

reverse band (black) and the average (green). Continuous lines show averages over the nine samples (same color codes). Also maximum average errors are

shown (doted green line), and global standard deviation (dotted red line).

Figure 21 shows the errors (differences) between absolute altitudes measured by shadows and GPS. We can see that, as in Antarctica validation, errors are always under 6 meters, and standard deviation less than 4 meters (3.2 meters).

6.3. Abbot Ice shelf validation data set

Validation over Abbot ice shelf was done using two ASTER L1A (v.3) scenes. This satellite platform was chosen to make this validation comparable to the ones in Antarctica, Andes and Kamchatka. Due to the lack of blue band in this platform and the reflectivity problem detailed at the en of section 20, scenes with shadows projected over frozen sea were selected. Scenes used:

- SC:AST_L1A.003:2005797088 Date: 14 January 2002 Sun elevation: 3.7°

- SC:AST_L1A.003:2027212156 Date: 04 January 2005 Sun elevation: 5.1° ICESat LIDAR profiles (described in section 2.3) were used as comparison elevation data. GLAS12 v.28 product was used in the processing, 8 profiles over two different ground tracks were available. 7 profiles were used for analysis, because one of them has almost no data.

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Figure 22: One of the scenes (January 4th 2005) used for Abbot ice shelf validation, ICESat ground tracks with data are shown in

red together with available profiles dates. Validation measurements were done on both ground tracks. Validation Zone 1 and Zone 2 (blue boxes) were used.

Figure 22 show the ice shelf, ICESat ground tracks with data and validation zones. Reference freeboard heights (thickness of emerged ice) were calculated from ICESat elevation profiles, by averaging several values over the sea to set base height and the first three values over the ice sheet to set ice shelf surface height. Then freeboard height was determined by subtracting base height to surface height. Note that this freeboard height is not sensitive to tidal variations. Profiles over zone 2 are shown in Figure 23, variation on base and surface heights are due to snow accumulation/ablation and tides. Wider variation in base height is probably due to small icebergs and sea ice presence on the sea, close to the ice shelf boundary. Note that sub zero elevations are due to geoid – ellipsoid differences in the area. Calculated freeboard heights range from 20.46 m to 21.33 m (0.87 m difference), with a standard deviation of 0.30 m and a mean value of 21.01 meters. No temporal or spatial trends were found. For comparison with shadows derived measurements we will assume a fix freeboard height of 21 meters for Zone 2, this assumption probably will lead to a small overestimation of errors (i.e. method accuracy would be slightly better than reported). For Zone 1, only one elevation profile is available, same procedure gives a freeboard height of 21.69 m.

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Figure 23: Available ICESat elevation profiles over zone 2 (seven profiles). Horizontal distance is referred to the first point over the ice shelf in each profile. Calculated

surface and base heights for each profile are shown (Black horizontal lines).

For Zone 1, Figure 24 show differences between shadows derived freeboard heights and the assumed reference height of 21.69 m. Ten measurements were done over 370 m of the ice shelf boundary around ICESat profile ground track. The mean value was -1.46 meters, with a standard deviation of 0.34 m in normal bands, 0.28 m in reverse bands and 0.26 m in the average of all bands. Small standard deviations and a big difference between mean and reference values are probably produced by real freeboard height changes between ICESat profile on October 2003 and image acquisition on January 2005. Given the dates, variation due to seasonal snow cover variability could be expected. Assuming no variation and constant freeboard height along the margin, the error in the shadows derived measure was 1.46 meters, which due to previous assumptions can be confidently considered as an upper limit for real error.

Figure 24: Errors (differences) of the absolute altitudes measured with shadows and LIDAR estimates for 10

measurements around ICESat profile ground track for Abbot ice shelf Zone 1 over January 2005 ASTER scene. Upper panel is keep with the same error scale than used in Andes data set for direct comparison. Lower panel

show the same data with Error axis adjusted for Abbot Zone 1 data. Color codes and symbols are the same than previous figures.

Unfortunately, shadows derived measurements on Abbot Zone 1 profile were not possible on the second ASTER scene (January 14th 2002), because sea surface was unfrozen and contrast too low for shadow detection. For Zone 2, seven ICESat profiles were found, they cover a cross-track band 140 meters wide, and an average freeboard height of 21.01 meters was found following the procedure already explained (see Figure 23).

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Figure 25: Errors (differences) of the absolute altitudes measured with shadows and LIDAR estimates for 12

measurements around ICESat profiles ground tracks for Abbot ice shelf Zone 2 over January 2005 ASTER scene. Color codes, symbols and axes scales are the same than

previous figure.

Figure 25 show error results over Zone 2, on January 2005 ASTER scene. Mean error (difference) is 0.71 m and standard deviations of 0.27 m for normal bands, 0.30 m for backwards band and 0.21 m for average elevations. We can see that the mean value (21.72 m) is almost 0.4 meters higher than the highest value found on LIDAR record. As before, small standard deviations suggest that freeboard height during image acquisition was actually higher than reference height used. If, not an error of 0.7 meters was found. On January 2002 scene, Zone 2 was only visible on normal bands, because the coverage shift in the backward band left it out of the image. Figure 26 show measured differences, the average value was -0.48 m with a standard deviation of 0.72 m. Average measured freeboard height is 21.49 m, slightly over the biggest LIDAR derived value (21.33 m). If possible spatial and temporal freeboard heights variations are neglected, and error of 0.48 meters with respect to reference height is found for this dataset.

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Figure 26: Errors (differences) of the absolute altitudes measured with shadows and LIDAR estimates for 11

measurements around ICESat profiles ground tracks for Abbot ice shelf Zone 2 over January 2002 ASTER scene.

Color codes and symbols are the same than previous figures, but there is only measurements over normal

bands. Note that lower panel is plot in a different scale than in previous figures.

Bigger standard deviation of measured values, is probably due to shadow matching problems derived from irregular surface reflectivity in January 2002 ASTER image. This can be clearly seen on Figure 27, where sea surface in the upper panel looks smoother than in the lower one, producing better shadow matching and therefore, smaller standard deviation of measured points.

Figure 27: Zone 2 detail. Upper panel sows ASTER scene from January 2005, and lower panel from January 2002. Also shown are: LIDAR elevation measurements

(red dots), projectors chosen for shadows derived measurements (brown triangles) and its respective

shadows centers (black dots).

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6.4. Ronne-Filchner ice shelf validation

Two scenes studied, results in process

6.5. Shiveluch volcano data set

We found many ASTER images showing volcanic eruptions, and much more would be available if we look into lower resolution data sources as MODIS sensors, that with 250 m resolution would be enough to compute volcanic plumes heights with 100 meters accuracy. Figure 28 shows Shiveluch Volcano and its plume. From this image we calculate height differences between five plume features and its shadows, and a DEM (SRTM) was used to compute plume absolute elevations. As reference values, stereo elevations using ASTER bands 3N and 3B could be used. But, stereo altitudes were found very difficult to obtain, because plume features seem to have moved about 400 meters during the 55.4 seconds elapsed between normal and backwards acquisitions (i.e. velocity ~ 26 Km/h). In this dataset, validation was not possible due the lack of reference heights, but if we assume that errors would be similar than in previous validation areas, the only fact that measurements are possible would be an indicator that shadow derived elevation are feasible over volcanic ash plumes, giving plume heights with an extraordinary accuracy. It is interesting to note that along-tack stereo pairs could be useful to estimate also vertical plume velocities if an specific feature is identifiable in both, images.

Figure 28: A portion of an ASTER scene acquired on February 17th 2008 showing Shiveluch Volcano with an

ash plume and its shadow.

The availability of identifiable plume features obviously depends on plume characteristics. Very diffuse plumes will not be suitable for shadow derived heights estimations; however compact plumes will provide many identifiable and even traceable features. Identification of feature’ s shadow by visual inspection is desirable to get maximum accuracy, but predicted shadow path can be used to pick the correct shadow of a specific feature, or to find the plume point corresponding to a specific shadow. By using predicted shadow paths (see section 4.4) for projector/shadow picking, we might be incurring in a small uncertainty of few pixels in a direction perpendicular to shadow’ s path, but the error in the shadow length (which is the important measurement) will be much smaller probably only a fraction of pixel. Figure 29 show close-up views of Shiveluch plume and its shadow. In this case is clear that feature-shadow matching can be done by visual inspection in some areas.

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Figure 29: Comparison between plume’s and shadow’s

shapes.

Shadow derived altitudes were computed for five features in normal and backwards bands. Projectors and shadows picking could be done with no major problems in the way described before (section 4.3), difficulties arise from reflectance irregularities in the shadow projection areas. Also feature identification in backward band was complicated by perspective and real transformation of the plume.

Figure 30: Plume’s measured features in normal bands

(Left) and backwards looking band (right).

Figure 30 shows measured features. Once projectors and shadows were accurately positioned, shadow’ s height was extracted from a Digital Elevation Model (SRTM) and projector real position was computed from it. From computed absolute plume positions, wind direction, horizontal and vertical velocities can be calculated, typical time delay between normal and backwards acquisition is slightly under one minute.

Table 3: Plume features heights and vertical velocities.

Feature number

Height 1 (N. bands)

Height 2 (B. band)

Vertical velocity

#1 5.822 m 5.740 m -1.48 m/s #2 6.214 m 6.162 m -0.96 m/s #3 6.046 m 6.161 m +2.07 m/s #4 6.690 m 6.613 m -1.39 m/s #5 6.584 m 6.465 m -2.14 m/s Table 3 shows computed plume’ s heights and vertical velocities using a mean time delay of 55.4 seconds.

6.6. Validation conclusion

As we see in Table 1, errors are expected to be proportional to the absolute height difference being measured plus a base error. In that table, values were estimated assuming the worst Sun-satellite geometry. Real typical errors should be smaller than those in Table 1.

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Figure 31: Absolute errors measured in all the validation points as function of the absolute real height difference.

Figure 31 show absolute errors of all the validation points in Antarctic, Andes and Ice shelves datasets as function of the real height difference being measured. Solid green line shows the mean trend of measured errors. Using this approach typical base error would be 1.1 meters and it would increase 0.0024 m per meter of height difference, which means a 0.24% of the measured height difference. Errors behave as expected in Table 1, and from the a total of 108 independent punctual measurements we can conclude that in the 95% of the cases errors are smaller than

( ) [ ] hm ∆+= 0024.05.3%95ε

Eq. 6 Where ∆h is the magnitude of the measured height difference. Large variability of errors is partially due to different geometries leading to different sensitivities to errors in satellite position, refraction, etc. As shown, in the current state of development, this shadow based methodology applied over ASTER L1A images, is capable to give a precision of 6 meters in punctual measurements over large shadows (mountains, ~1000 m height difference), consistent to the theoretic errors calculated in Table 1 and Table 2, this error is of the order of one third of the image pixel size. Over short shadows and especially simple projectors as in ice shelves, accuracies of about one meter could be expected. Is important to remark that all this errors are very likely an over estimation of real errors, because uncertainties in snow accumulation on image acquisition date, GPS positioning errors and LIDAR errors will lead in average to an increase in the difference between computed and measured values.

7. Conclusion

Height differences can be computed using shadows that are present in satellite imagery. Calculated heights accuracies depend on multiple parameters including geometry, satellite orbit determination and atmospheric refraction. In the proposed methodology we used a geometric approach that takes in account satellite platforms movements together with an accurate shadow and projector modeling, and precise refraction calculation based on archived meteorological data. Using this methodology we demonstrated that height differences within an image can be successfully computed with sub pixel vertical accuracies over a wide range of conditions, latitudes and geometries. We performed several validation surveys with a total of 108 independent measurements. Computed values from ASTER imagery (15 meters ground resolution) were compared against field DGPS measurements or Satellite LIDAR data. Errors found are very likely an over estimation of actual errors. Found relative errors are in 95% of the cases less than 0.24% plus a base error of 3.5 meters. Higher spatial resolution satellite imagery will probably produce much smaller errors as two thirds of computed errors are due to sources that are sensitive to pixel size. Furthermore, sensor stability and orbit determination are better in such platforms. We evidenced in this study that calculated heights accuracies are good enough to produce useful data for several earth science problems, as long term glacier thickness variations; volcanic plumes heights, volcano structure and dynamics; and ice shelf/icebergs freeboard height. We propose that this methodology could be extended to platforms such as Landsat 1-7, Corona, Aerial imagery, etc. This would contribute to make use of these data sets to build a valuable source of time-series data sets to

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better constrain past as well as present elevation data. This would then certainly improve our understanding of long-term dynamic changes in such earth science problems.

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Appendix A

Screenshot of the graphic interface designed to find the precise position of the shadow. Upper panels show close-up images of the shadow in the ASTER bands 1, 2, 3N and 3B. The upper left panel, show the band currently being fitted. The red line is the profile trace in the image, green cross is the current shadow position and blue crosses point profile’ s begin and end. This profile is shown in the main panel. Dotted blue lines show the apparent shape angle. The main panel show the image measured intensity profile (blue), the theoretical shadow profile (red), and its center (green). Horizontal axis shows the position in pixels and the vertical axis show normalized intensities. Lower fields show fitting parameters.