successive-correction methods applied to mesoscale meteorological analysis

17
IL NUOVO CIMENTO VOL. 17 C, N. 6 Novembre-Dicembre 1994 Successive-Correction Methods Applied to Mesoscale Meteorological Analysis. F. UBOLDI (1) and A. BuzzI (e) (1) c/o ENEL-DSR-CRAM - Milano, Italia (2) FISBAT-CNR - Bologna, Italia (ricevuto il 2 Maggio 1994; approvato il 26 Maggio 1994) Summary. -- Successive-correction methods applied to meteorological analysis have received renewed attention. We consider here different algorithms recently proposed in the literature. We compare them in a consistent notation and discuss their properties by analysing the different formulations and by examining results obtained in a specific application to meteorological data. PACS 92.60 - Meteorology. 1. - Introduction. The problem of objective analysis of meteorological data is the problem of defining the ,,best, values of a certain meteorological field on a suitable grid, starting from observations taken at ,,stations-. These observations are usually scattered (in space and time) and affected by both measurement errors and representativity errors. The latter are due to the fact that meteorological fields contain all scales of motion, including small-scale variations that have to be filtered out because either they cannot be properly represented (due to errors introduced by aliasing) or they do not pertain to the range of scales the analysis is intended to describe. Meteorological analysis, that is routinely carried out to prepare initial conditions for numerical models of the atmosphere, normally relies upon other information in addition to station observations. For example, statistical information, like error covariances, climatology etc. is widely used to assure minimization of analysis errors in a statistical sense (,,optimum interpolation--OI, or ~statistical interpolation,). Moreover, a ,,first guess- (or ,,background field,), typically provided by model forecast or climatology, is normally used because it has a large impact on analysis quality, especially where data are sparse. Some analysis methods strictly require the existence of a first-guess field. Multivariate analysis can be performed in meteorology by taking into account statistical and/or dynamical relations between different variables. 7A.5

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Page 1: Successive-correction methods applied to mesoscale meteorological analysis

IL NUOVO CIMENTO VOL. 17 C, N. 6 Novembre-Dicembre 1994

Successive-Correction Methods Applied to Mesoscale Meteorological Analysis.

F. UBOLDI (1) and A. BuzzI (e) (1) c/o E N E L - D S R - C R A M - Milano, I tal ia (2) F I S B A T - C N R - Bologna, I tal ia

(ricevuto il 2 Maggio 1994; approvato il 26 Maggio 1994)

Summary. - - Successive-correction methods applied to meteorological analysis have received renewed attention. We consider here different algorithms recently proposed in the literature. We compare them in a consistent notation and discuss their properties by analysing the different formulations and by examining results obtained in a specific application to meteorological data.

PACS 92.60 - Meteorology.

1 . - I n t r o d u c t i o n .

The problem of objective analysis of meteorological data is the problem of defining the ,,best, values of a certain meteorological field on a suitable grid, starting from observations taken at ,,stations-. These observations are usually scattered (in space and time) and affected by both measurement errors and representativity errors. The latter are due to the fact that meteorological fields contain all scales of motion, including small-scale variations that have to be filtered out because either they cannot be properly represented (due to errors introduced by aliasing) or they do not pertain to the range of scales the analysis is intended to describe.

Meteorological analysis, that is routinely carried out to prepare initial conditions for numerical models of the atmosphere, normally relies upon other information in addition to station observations. For example, statistical information, like error covariances, climatology etc. is widely used to assure minimization of analysis errors in a statistical sense (,,optimum interpolation--OI, or ~statistical interpolation,). Moreover, a ,,first guess- (or ,,background field,), typically provided by model forecast or climatology, is normally used because it has a large impact on analysis quality, especially where data are sparse. Some analysis methods strictly require the existence of a first-guess field. Multivariate analysis can be performed in meteorology by taking into account statistical and/or dynamical relations between different variables.

7A.5

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746 F. UBOLDI and A. BUZZ/

In this paper we examine and compare different analysis methods that are based on the successive-correction (SC) algorithm. This algorithm has recently received renewed attention because of its flexibility and easy-to-use characteristics, that make it suitable for application to modern analysis methods. For simplicity, we focus our discussion to the basic univariate case, though an extension to the multivariate case is presented.

The SC method can be formulated in different ways to provide solutions to a number of analysis problems. This method was initially introduced as a single-step correction of a field estimate based on empirically determined weighting functions [1-3]. Barnes method has been widely used and received much attention even in recent years (see, for example, [4-6]). However, Barnes and similar methods, in which isotropic distance-weighting coefficients are applied for a very small number of iterations, may introduce serious analysis errors where observations are irregularly spaced, as pointed out, among others, by Daley [7] and Buzzi et al. [8]. The latter authors have proposed a method to reduce such errors in the Barnes scheme.

Other authors, in the meantime, have developed different SC algorithms and have explored their properties for medium or large number of iteration steps [7,9-12]. These authors have shown that the above-mentioned weakness of Barnes (or similar) method can be avoided by applying multiple iterations.

Here we discuss the most recent formulations of the SC method by presenting the different algorithms in a homogeneous notation context. The results obtained by applying the algorithms to the same meteorological data set are then critically examined.

In sect. 2 of this paper we review and compare the general properties of the most recently proposed versions of the SC schemes. In sect. 3 we show examples of the application of multiple-iteration SC applied to a particular meteorological field. In sect. 4 we present an extension to the multivariate case. Conclusions are derived in sect. 5.

2. - S u c c e s s i v e - c o r r e c t i o n a l g o r i t h m s .

In the formulas below, o is the vector in which the values observed at the locations of M stations are stored; in s and g the estimates at stations and at L grid point locations are kept (hereafter the word ,,station, indicates an observation point):

0 = (O(1) . . . . O(M)) T , 8 ---- (8(1) , ...8(M)) T , g = (g(1), . . .g(L)) T .

2"1. B a r n e s method. - The original work by Barnes [2] proposed a fn'st step of estimate based on a weighted mean of the observed values (indices indicate step):

gl = Q G - I " W G . o, at grid points ;

sl = Q -1. W. o at stations ;

where W G is a (L • M) matrix and W is a symmetric (M • M) matrix, both storing the weights. The weights are calculated as a Gaussian exponential of the distance d(r~, rk) between the position of the estimate, ri, and the position of each observation,

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SUCCESSIVE-CORRECTION METHODS APPLIED TO ETC. 747

rk:

[WG]i,k = e x p [ - ( 1 / 2 ) ( d ( r g i , r k ) /R )2] , [W]j,k = e x p [ - ( 1 / 2 ) ( d ( r j , rk)/R)2].

rgi is the position of the i-th grid point; rk is the position of the k-th station; R is a scale parameter, here called ~radius of influence-; QG and Q are diagonal matrices (L • L and M • M, respectively) storing

normalization factors, each calculated as the sum over k of the unnormalized weights

[QG]~ = • [WG]~, k, [Q]j = E [W]j, k . k k

In this way the sum of the elements in each row of both matrices QG 1. W G and Q-1 . W is equal to 1.

After the first step, just one correction step is performed in Barnes scheme, with a reduced R suggested in order to achieve, at observation points, an estimate closer to the observed value, in spite of the truncation of the iteration at the second step:

g2 = gl + QG -1. W G " (o - sl ) , s2 = sl + Q -1" W . (o - Sl ) .

This scheme represents theoretically a smoothing filter, the small scales being suppressed below a cut-off wavelength which depends on the radius of influence. The filtering properties of the numerical scheme are valid in the idealized case of continuous distribution of ,,observations-; they are also valid in regions in which the station distribution is approximately homogeneous (see, for example [6-8]).

2"2. E x t e n s i o n s o f B a r n e s me thod to an N-s t ep i teration. - Instead of just one correction step, several (N) steps are performed:

g n + I = g ~ + Q G - I " W G ' ( o - s n ) , S n + l = S n + Q - 1 . W . ( o - S n ) ,

for n = 1, . . .N. The last recursive formulas are equivalent to

g ~ + I = Q G - I ' W G �9 ~, ( I - Q - 1 . W ) k "0, S n + I = Q - 1 . W �9 ~ ( I - Q - I " W ) k "o, (k = O . . . n ) (k = 0 . . . n)

where I is the identity (M • M) matrix and (I - Q -1. W)k indicates the k-th power of the matrix ( I - Q-1.W). The sum at each n can be calculated (geometric series):

g~ = Q G - i . W G . ( Q - 1 . W ) - i . ( I - ( I - Q - I . w ) n ) .o , S n = ( I -- ( I - Q - 1 . w ) n ) . o .

The effect of carrying on the iteration is to achieve an estimate, at stations, closer and closer to the observed values. This means that the cut-off wavelength is gradually shifted toward smaller values. Moreover the shape of the response function becomes steeper and steplike, while the iteration goes on, resulting in an increased sharpness of the filter cut-off. This means that the choice of the number Nof iteration steps influences the filter properties.

It can be shown that the succession converges if, for each eigenvalue 2 of W

I1- ;~ l < 1 .

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748 F. UBOLDI and A. BUZZI

The limit for n going to infinity is easily calculated:

g~ = Q G - 1 . W G . ( Q - 1 . W ) 1 . o , s~ = o .

The filtering properties of the algorithm change with the value of N, so that no filter is achieved in the limit. The analysis obtained in the limit is an exact interpolation: the analysed value at station points is equal to the observed value. In order to obtain a smooth analysis field, a finite N has to be chosen.

In practice, for a fLxed value of N, the choice of the cut-off wavelength is performed by choosing the radius of influence.

2"3. B a c k g r o u n d f i e l d i n B a r n e s m e t h o d . - In the common practice of objective analysis a reasonable first guess, or background field (BF), can be used, usually being a forecast achieved through a meteorological model. We assume that such a background field can effectively provide valuable information at least in regions where the station density is low.

The iteration previously shown can be started from the difference between the observed values and the BF values at station positions; in the same way, g and s have now to be interpreted as the differences between the analysis estimate (ag and a) and the BF values, at grid points and station points, respectively:

o = obs - b,

a = s + b at station points ;

ag = g + bg at grid points ;

where obs corresponds to o, as defined in the previous section, ag and a are the final analysis estimates, bg and b are the vectors storing the BF values at grid points and at station points. The expressions shown above for the iterations still hold for the new variables.

2"4. S t a t i s t i c a l i n t e r p o l a t i o n . - The objective analysis algorithm commonly known as Optimum Interpolation (OI) or Statistical Interpolation assumes that a background field is available. OI is based on knowledge of error covariance matrices for both the observations and the background field. In practice, three matrices are required:

O S S = <(obs - t)(obs - t)T>, B S S = ((b - t ) (b - t)T>, B G S = <(bg - tg ) (b - t)T>,

where t and tg are vectors storing the (,true, values at station points and grid points, O S S and B S S are symmetric (M x M) matrices storing, respectively, observation and background field error covariances between couples of station points; B G S is a (L x M) matrix which stores background field error covariances between grid points and station points. The expectation values should be taken over some meaningful statistical ensemble.

The OI analysis at grid points is given by

g O I = B G S . ( O S S + B S S ) -1 o .

This is the estimate which minimizes the average of squared analysis errors [9, 7]

<(g - tg)T ( g -- tg)> = min, if g = gOi

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SUCCESSIVE-CORRECTION METHODS APPLIED TO ETC. 749

assuming a linear relation between estimate and observations (differences from the background field).

2"5. B r a t s e t h a lgor i thm. - Bratseth [9] noticed the formal similarity between the expression of gO1 and the limit g~ of the SC iteration, and suggested a modification to the SC algorithm which allows for a complete identification between weight matrices and covariance matrices.

Bratseth algorithm requires a ,,column normalization, instead of the usual ,,row normalization-:

g~+l=gn+WG.Q-l.(o-s~), 8n+i=sn§

where Q is the same (M • M) diagonal normalization matrix defined above. The last recursive formulas are equivalent to

g n + i - - W G ' Q -1" ~ ( I - W ' Q - I ) k ' o , 8 n + I - - W . Q -1. ~ ( I - W . Q - 1 ) k . o . (k = O . . . n ) (k = O . . . n )

In this way the normalization matrices (the Q's) can be simplified in the expression of the limit, which becomes:

g~ = W G . W - l o , s~ = o .

Using this kind of normalization requires the data to be detrended before the iteration. The normalized weight for each station depends on the local spatial density of observations. This has the effect of reducing, with respect to the Barnes algorithm, the error due to inhomogeneous distribution of data.

The comparison with OI is now straightforward:

or, more generally,

W G -- B G S , W -- O S S + B S S ,

W G . F -- B G S , W . F = O S S + B S S ,

where F is an arbitrary (M • M) matrix such that det (F) ~ 0. If covariance matrices are known and are used in place of the analitically

estimated weights, an SC iteration stopped at step N is an approximation to the OI analysis.

On the other hand, the Gaussian weights can be seen as an analytical estimation of covariances (covariance function depending only on distance).

The OI estimate at station points, in analogy with the estimate at grid points, is

8 oi - - B S S . ( O S S + B S S ) - 1 . o ,

while the limit of Bratseth algorithm is the observation vector itself. These limits are the same only when O S S = 0. In subsect. 2"7 a modification of Bratseth algorithm is described, allowing for O S S ~ O.

A better computational efficiency is obtained through a different formulation of the same algorithm, which is presented in the following.

The estimate at station points in Bratseth algorithm is

s~+l = Sn + W ' Q - I " ( o - Sn),

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750

from which (So = 0)

0 - - 8 n + 1 = (I - W ' Q - 1 ) ' ( o - 8 n ) "-= ( I - - W . Q - 1 ) n + I ' ( o - 8 0 ) : (I - W ' Q - 1 ) n + l

that is to say

which implies

o - s~ = (I - W ' Q - 1 ) n " o ,

(o - sk) = ~ ( I - W ' Q - 1 ) k "o. (k =O. . .n ) (k =O. . .n)

It can be noted that this series converges to the usual limit

(o - sk) = ~, (I - - W . Q - 1 ) k "o : Q ' W - 1 .

(k=O. . .oo) (k = 0 . . . oo)

The Bratseth algorithm can be written, at each step n, as

g n + l = W V ' Q -I" ~ ( o - s k ) , s ,~+l=W.Q -1. ~, (k = 0 ... n) (k = 0 ... n)

F. U B O L D I and A. B U Z Z I

( 0 - - 8 k ) .

" O ,

2"6. Daley algorithm. - An alternative algorithm approaches the same limit, gOi, taking into account explicitly the BF values in each iteration step, using a traditional row normalization [7]. For the analysis vectors

a g n + l = a g n + (QG + H) l"(WG'(obs - a~) + ~(bg - ag~)),

a n + l = a n + (Q + d ) - 1 .(W. (obs - a n ) + db - a n ) ) .

The algorithm, in this form, appears at each step n as a weighted mean which takes explicitly into account the BF value at the estimate point as if it were a (M + 1)-th station, with unnormalized weight represented by the scalar positive quantity s.

For the differences from the BF, we have

g n + I = ( Q G + H ) - I . ( Q G . g n - W G . s n + W G . o ) , sn+I=(Q+H)-~((Q - W) . Sn+W'o) .

This iteration can be studied more easily by considering the (L + M)-vector fi obtained by appending the station M-vector, s, to the grid L-vector, g:

f = (J~ 1), .-. f(L + M) )r = [ g T [ S T ]r = (g(1), ... g(L), S(1), --. S(M) )T,

where the indices in parentheses indicate vector components. The algorithm becomes

= p - 1 f n + l .(H" f~ + V 'o ) ,

gN= WG'Q -1" ~ (o - sk) , S N = W ' Q 1. ~ (o - sk) . (k = 0 . . . N - 1) (k = 0 . . . N - 1)

The computational efficiency is due to the fact that only the estimate at station points has to be performed at each step, while the estimate at grid points may be done only at the end of the iteration

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SUCCESSIVE-CORRECTION METHODS APPLIED TO ETC. 751

where the matrix H has dimension (L + M) • (L + M) and can be seen as a block matrix

- W G

The matrix V is obtained simply by appending the station weight matrix W to the grid weight matrix W G :

The matrix P is the diagonal normalization matrix obtained by appending (Q + d ) to ( Q G + d):

o

By defining

which implies (fo = 0, dl = f l )

the algorithm becomes

d n + l "~ f n + l - - f n ~

fn = ~ d k , (k = 1 ... n)

d n + l = P - I " H "dn �9

Let P H be the normalized matrix P - I " H

dn . 1 ---- P H " dn = P H n ' d l = p H n " f l ,

where P H n indicates the n-th power of P H .

In such a way the analysis at the (n + 1)-th step is

fn + 1 ~- E p H k - 1 . dl = ~ P H k" dl = ~, P H k" f l �9 (k= 1 . . .n + 1) (k= O...n) ( k = O . . . n )

Then, for each n

fn = ( I - p H n ) ' ( I - P H ) - l " f l .

The succession converges if all the eigenvalues of P H have modulus less than 1. The limit is

f ~ = ( I - P H ) - I . ~ .

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752 F. UBOLDI and A. BvzzI

The inverse matrix of (I - PH) can be calculated, in the case ~ ~ 0, t reat ing P H as a 2 • 2 block matrix:

( i _ P H ) _ l = [ ( I - Q G * ) - ( I - Q G * ) - I " W G * ' ( I - Q * + W * ) -1]

0 (I - Q* + W* ) - 1 "

Where the exponents * indicate normalization:

QG* = ( QG + H) 1. QG ,

Q* = (Q + H) -1. Q,

WG * = ( QG + H) -1" WG ,

W* = (Q + H) -1. w .

2"7. Observation error and data filtering with Bratseth algorithm. - A modifica- tion of Bratseth algorithm brings to the same limit as Daley algorithm, with smoothing of data [12].

Moreover, it does not have singularities for e = 0. In order to achieve Daley's (and OI) limit, the following iteration is considered, in

Then, using

f l = P - I " V ' o ,

the limit of the algorithm can be calculated:

g~ = W G . ( W + H) - 1 . o , s~ = W . ( W + H) 1 .o .

Since s ~ 0, the (limit) analysed field, at stations, is different from the observed values.

The comparison with the OI analysis suggests the following interpretation of the weight matrices:

2 2 = Oo/%, WG = B G S / ~ , W = B S S / ~ , I = OSS/~Zo.

That is to say the BF covariances carry the spatial covariance; s carries the 2 normalized with the BF error, ~ . The last observation mean square error, 00,

expression says that the observation error covarianee is estimated here by the scalar quantity zzo.

I t should be noted that this algorithm has a singularity for s = 0. In that case the algorithm is the same as the N-iteration Barnes SC algorithm previously shown (in subsect. 2"2), the limit being

g~ = Q G - 1 . W G . W - 1 . Q . o ,

which is different from the limit of Daley's algorithm (and OI) evaluated for ~ = 0:

g~ = W G . W - I . o .

Anyhow, the case ~ = 0 is not relevant, because it is assumed that the observations are always affected by errors.

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SUCCESSIVE-CORRECTION METHODS A P P L I E D TO ETC. 753

which zs is an M-vector (ZSo = 0):

ZSn + 1 : ZSn + ( W "~ H) �9 Q - 1 . (o - z sn) .

This is similar to the Bratseth iteration at station points (see subsect. 2"5), but an was added to the diagonal elements of matrix W, while the diagonal normalization matrix Q may remain the same.

The succession

E (o - zsk) = E ( I - ( W + H)" Q - 1 ) k ' o (k = O...n) (k = O...n)

has the following limit:

(o - zsk) = Q ' ( W + H) -1 "o. (k = 0. . . ~r

The estimate of the analysis at grid points and at station points may be performed at the final step of the iteration:

g N = W G ' Q -1 " ~ (o - zsk) , SN = W ' Q 1 . ~ (o - zsk) . (k = 0 . . . N - 1) (k = 0 . . . N - 1)

This succession has the same limit found for Daley algorithm. The matrix Q is actually arbitrary: (Q + eI), for instance, can be used in the same

way. It is important that in the final estimation at grid points and station points the

matrices W G and W are used, while the matrix (W+ H) is used during the iteration.

When covariance matrices are known, they can straightforwardly be inserted in this algorithm

ZSn+ 1 : ZSn "-[- ( O S S ~- B S S ) . F . ( o - ZSn) ,

where F is an arbitrary matrix (for instance the normalizing matrix of O S S + BSS) . The final estimation is

g N = B G S "F " ~ (o - zsk) , SN = B S S "F . ~ (o - zsk) . (k = 0 . . . N - 1) (k = 0 . . . N - 1)

3. - Application of SC algorithms to meteorological analysis.

In this section we present examples of the application of the different SC analysis algorithms described in sect. 2. We have chosen a particular date (21 February 1993) representing a case of meteorological interest, being characterized by incipient cyclogenesis in the lee of the Alps. We restrict our examples to the analysis of mean sea level pressure (m.s.l.p.) at 12 UTC on the Mediterranean region. The m.l.s.p. field is affected by mesoscale patterns, mainly induced by the orography, and at least part of this variability is revealed by the SYNOP reports. The available SYNOP station distribution for that particular time is shown in fig. 1. It can be seen that the station density is very inhomogeneous. This can represent a critical condition for analysis methods [8].

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754 F. UBOLDI and A. BUZZI

~

�9 �9 r �9 r r r

/

Fig. 1. - Distribution of SYNOP data in the analysis area, 21.02.1993 at 12.00 uTC.

The original data were subject to a preliminary quality control to discard values affected by large errors [13].

The analysis grid has a resolution of 0.5 • 0.5 degrees lat.-long. In all the examples below we use a background field which is provided by the ECMWF (European Centre for Medium-Range Weather Forecast, Reading, UK) analysis, interpolated on our grid from a grid of 1.5 • 1.5 degree resolution. Figure 2 shows such background field, describing a trough over Central I taly and a ridge north of the

H

Fig. 2. - The ECMWF analysis, 21.02.1993 at 12.00, used as a background field for the analysis algorithms.

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SUCCESSIVE-CORRECTION METHODS APPLIED TO ETC. 755

b) �9

Fig. 3. - a) Barnes two-step analysis, 21.02.1993 at 12.00, using background field, b) Barnes two-step analysis: increments from background field.

Alps embedded in a general north-westerly flow. At least some part of the orographically generated pressure pattern is captured in this field.

Figure 3a) shows the analysis field obtained by applying the Barnes two-step analysis method described in subsect. 2"1. The values of R are 336 km at the first step and 96 km at the second. The analysis is performed on the deviation from the BF, as suggested in subsect. 2"3. (a direct application of Barnes analysis to observations, without introducing the BF, produces very different results in data-void areas, and, therefore, it is not presented here). Figure 3b) shows the analysis increment with respect to the BF of fig. 2. The largest differences between this analysis and the BF

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756 F. UBOLDI and A. BUZZ1

H

I.

Fig. 4. - Bratseth analysis 21.02.1993, at 12.00, 20 iteration steps, R = 120km.

are seen in the orographic regions, in particular near the Alps, the Pyrenees and over the Mediterranean, east of Sardinia.

It is known that the Barnes method cannot cope well with inhomogeneous data distribution[8]. Better results, in this respect, are expected from more recent methods based on many iterations. Figure 4 shows the results of Bratseth method described in subsect. 2"5. In this example 20 iteration steps have been performed with R = 120 kin. Mesoscale features in this analysis are generally sharper than in fig. 3: note, for example, the low over the Ligurian Sea, extending to the east of Corsica, and the orographic high over the Pyrenees. In this case filtering of the observations depends only on the number of iterations.

An application of explicit data filtering using a parameter e is allowed by Daley algorithm described in subsect. 2"6. With ~ = 0.1 (other parameter values are the same as for Bratseth method) we obtain the analysis shown in fig. 5a). A comparison between fig. 5a) and fig. 4 shows that the differences between the two methods are relatively small. Figure 5b) shows the analysis increments with respect to the BF in this case. The largest deviations from the BF are again mainly associated with mesoscale topographic features, with highs/lows realistically located upstream/ downstream the mountains. Patterns are similar to those of fig. 3b), but in this case are of larger amplitude and better localized.

The results of the application of the method described in subsect. 2"7 are shown in fig. 6, which, as expected, is very similar to fig. 5a). The main difference between the two methods, as said above, is in the computational efficiency.

A reason for the small difference between fig. 4 and fig. 5a) and 6 is that a finite number of iterations means data filtering in any of the SC algorithms, even in Bratseth algorithm of fig. 4. Moreover, a larger value of the parameter e could help in distinguishing the results of algorithms which, in the limit, allow for data errors and algorithms which do not. Anyhow, the choice of the value of ~ was made for diagnostic use of operational analysis, intended to emphasize small-scale features.

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SUCCESSIVE-CORRECTION METHODS APPLIED TO ETC. 757

�9 IH o

Fig. 5. - a) Daley analysis 21.02.1993 at 12.00, 20 iteration steps, R = 120km. b) Daley analysis, 20 iteration steps, R = 120 km: increments from background field.

Finally, to allow for a bet ter evaluation of the above results, we show in fig. 7 the result of the application of the direct method discussed in subsect. 2"4, based on OI, when the covariance matrices are approximated by the weight matrices used in the SC algorithms, i.e. using the same values for the radius of influence, R = = 120kin and the pa ramete r s = 0.1. This indicates tha t the SC methods based on multiple iterations are apt to provide accurate results, even with a reasonable number of iteration steps.

A more detailed description and evaluation of SC methods and their application to meteorological objective analysis can be found in[14] and [13].

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758 F. UBOLDI and A. BUZZI

q

! . i L

Fig. 6. - Pedder analysis, 21.02,1993 at 12.00, 20 iteration steps, R = 120 km.

Fig. 7. - Optimum Interpolation, when the covariance is estimated by a Gaussian function of the distance, 21.02.1993 at 12.00.

4. - Successive-correction methods for unidimensional multivariate analysis.

W e p r e s e n t in this sect ion a s imple example of an appl icat ion of the SC a lgor i thm to mul t ivar ia te analysis . F o r the sake of s implici ty we l imit ourse lves to cons ider ing the unidimensional case.

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S U C C E S S I V E - C O R R E C T I O N M E T H O D S A P P L I E D T O E T C . 759

1.0

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+,

1 I I I I

2'0 4'0 6'0 8'0 100 Fig. 8. - One-dimensional, multivariate analysis; v = - d p / d x ; + = p; • = v; in the region with no data the curves show coherence with the relation.

Suppose that in the interval Xo. . .XL the variables v(x) and p(x ) a r e linked by the relation

v(x) = k . dp (x ) /dx .

The choice of this kind of relation was suggested by the geostrophic relation, which is usually taken into account in meteorological analysis. In this case v may represent the meridional component of the wind and p the pressure.

M observations are known in the same interval at the positions of M stations, X S l . . . XSM. The observed values are stored in the M-vectors o v for the variable v and op

for the variable p. The univariate analysis for the two variables, i.e. computed without taking into

account the relation, is (see subsect. 2"2):

vg~ + l = Vgn + Q G -1 . W G . ( ov - VSn ) , pgn + l = pgn + Q G 1. W G . ( op - ps~ ) .

vg and p g a re L-vectors, storing analysis estimates at grid points at step n. W G and QG, as usual, are the weight matrix and the diagonal orthogonalization

matrix, respectively. The SC algorithm minimizes the residuals between analysis estimates and

observations. The insertion of an appropriate term allows to include the relation between v and p

as a ~,weak, constraint. In this way the algorithm can be written as

Vgn + 1 --- V g n "~- ( Q G + f l Y ) - 1 " ( W G " (o r - v 8 n ) -~- f l (Fg~ - vg~) ) ,

Pgn + ~ = Pg~ + ( Q G + f l I ) - ~ ' ( W G ' ( o p - p s n ) + f l (Gg~ - P g n ) ) ,

where fl is a positive scalar, representing the weight for the condition. F and G are chosen so that the relation between v and p can be expressed in any of

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760 F. UBOLDI and A. BUZZI

the two equivalent ways (at grid points):

vgi = Fgi or Pgi = Ggi .

In the case of a unidimensional regular grid defined as xi = x0 + Ax, a possible choice of F and G can be found by discretizing the relation here considered between v and p:

vgi = ~(Pgi + 1 - Pgi - 1 )/2,

where a = k / A x .

We can obtain pg by evaluating the above expression at point i - 1 and i + 1 and solving for Pgi:

Pgi = ( v g i - i - v g i + l ) / a + ( P g i + 2 + P g i - 2 ) / 2 .

F g and Gg are the values of vg and pg which exactly satisfy the relation

F g i = a ( p g i + l - p g i - 1 ) / 2 , G g i = ( v g i - l - v g i + ~ ) / a + (pg i+e + p g i _ 2 ) / 2 .

The estimation of F and G at the station points is obtained as a linear combination of the conditions at the nearest-neighbour gridpoints. For the k-th station, lying in the interval between the i-th and (i + 1)-th gridpoints, at distance s h x from the i-th,

Fsk = (1 - s ) F g i + sFgi + 1 �9

The boundary condition requires the knowledge of the variables at two adjacent points at each boundary.

The numerical scheme was tested in a simple, idealized case in which we assume that the true values of the two variables are sinusoidal functions. We also assume k = - 1 , implying that v and p are represented by a sine-cosine couple.

The analytical functions were estimated on the station points, irregularly distributed in the considered interval, in order to generate the ,,observed,, values.

Two data-void subregions were chosen in the interval, in order to test the algorithm also in critical conditions.

Figure 8 shows an example of the results obtained in this case. The analysis is very close to the observed values in the regions where many ,,stations,, are located. In the regions without ,,observations,,, both p and v depart from the analytical value, but the relation between them still holds in this region.

Other case were tested in which the ,,station,, values of one of the two variables were affected by errors. The analysis partly corrected these errors as an effect of the prescribed relation.

The multivariate analysis algorithm presented here can be easily applied to more general relations between the analysis variables and can be extended to multidimen- sional cases.

A meteorological application of a different mulivariate analysis method based on an SC algorithm is described in[15].

5. - C o n c l u s i o n s .

In this paper we have reviewed and discussed some recently proposed methods of meteorological analysis based on successive corrections. Different formulations of such methods are available. Correct formulations of the iterative procedure not only lead to

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SUCCESSIVE-CORRECTION METHODS APPLIED TO ETC. 761

very accurate results, comparing favourably with direct methods (e.g. that of optimum interpolation) but they can also be computationally efficient. In the examples shown above the differences among the results of the multiple-iteration SC algorithms, applied to mesoscale analysis based on SYNOP observations and ECMWF analyses used as background fields, are very small. However, analysis accuracy is very important in dynamical meteorology, particularly when the analysis is used to initialize numerical forecast models in which initial errors may quickly grow to large amplitudes.

We thank Dr. P. Bonelli and Mr. W. Fossa of ENEL-DSR-CRAM for their continuous support. This work has been supported under project METACO, jointly conducted by ENEL-DSR-CRAM, Milano and CISE, Segrate MI.

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