Optimal scatterometer ambiguity removal using a successive correction method

Harald Schyberg and Lars-Anders Breivik

Norwegian Meteorological Institute,
P.O. Box 43 Blindern N-0313 Oslo, Norway
h.schyberg@dnmi.no

Abstract

For wind scatterometer data users not applying the data in variational data assimilation, it is of interest to have an ambiguity removal system or dealiasing resolving the directional ambiguity in the observations.

The "Pre-Scat" scheme developed for this purpose generally works well, but in a few cases it obviously picks the wrong ambiguity possibility. We here present an alternative scheme which is derived from Bayesian probability theory as finding the one of the wind possibilities which gives rise to the windfield with highest probability. This enables us to set up a cost function to be minimized which takes into account the same features as in the Pre-Scat scheme. In addition the new scheme is able to take advantage of the fact that the rotational part of the wind dominates over the divergent.

The minimization problem is solved using a modified version of the traditional data assimilation method of successive corrections. Our experience from a semi-operational comparison with Pre-Scat scheme is described. The new scheme is less dependent on a good first guess windfield than Pre-Scat, and its quality is comparable to Pre-Scat.

Keywords: Scatterometer, ambiguity removal, successive corrections

1. Introduction

Scatterometer data from the ERS satellites generally have a high degree of accuracy (see e.g. Stoffelen and Anderson, 1995), but its use is complicated by the fact that the retrieval algorithm results in two possible wind vector solutions, usually similar in windspeed and 180 degrees apart. For use of these observations in numerical models, modern assimilation methods like variational assimilation can best take advantage of the data without specifying in advance which of the two solutions corresponds to the correct one.

However, for most other types of application of the data, it is of interest to present only one single wind vector corresponding to the observation. At meteorological forecast centres this is of interest for the purpose of presenting the observations to the forecaster in near real time as they arrive. At that stage a first guess model windfield of poor quality may be the only guidance for the ambiguity removal, and other surface observations in the area are generally not available. For most ERS scatterometer users, such ambiguity removal has been done using the "Pre-Scat" scheme.

The ambiguity removal part of Pre-Scat has been developed mainly at UK Met Office, with contributions from IFREMER, ESA and ECMWF. The Pre-Scat scheme is a pragmatic iterative scheme based on (1) background information in the form of a first guess windfield from a numerical prognosis (which may be of low quality), (2) spatial coherence of observations and (3) consistency of the backscatter values with the scatterometer model function. The Norwegian Meteorological Institute (DNMI) is running operationally a version of this system.

The operational experience with Pre-Scat shows that it works well. However, for a small fraction of the observations the scheme seems to have picked the wrong wind alternative. In some cases this is evident because the real wind is usually close to be non-divergent, while the scatterometer observations seem to contain an unrealistic divergence.

This non-divergent nature of the wind is not taken explicitly into account by Pre-Scat. We therefore found it tempting to try to derive a new ambiguity removal system which takes into account the same three conditions as Pre-Scat, but in addition includes the constraint that the scatterometer swath should represent a field which is close to non-divergent.

The derivation of the scheme follows the basics of variational assimilation (see Lorenc, 1986). As opposed to Pre-Scat, the new scheme, using a variational approach, is optimal in a statistical sense. The formulation of a cost function to be minimized and its iterative solution is presented in section 2, while our operational experience with the scheme is discussed in section 3.

2. Method

The basic principle of variational analysis (see Lorenc, 1986), is to find an optimal estimate of the wind field, defined as that which corresponds to the most probable given the available observations. Our strategy is to use this most probable windfield to decide which of the two scatterometer ambiguity possibilities is the most probable one.

Using Bayes' theorem and assumptions on the error statistics, a corresponding cost function J to be minimized with respect to the field x can be found. In our case x is the field of wind vectors at the scatterometer observation points. The cost function consists of a term minimizing the distance to a first guess field, Jb, and a term minimizing the distance to the observations, Jo,

In our application the observations are the ambiguous wind pairs at the scatterometer observation points, which have gone through a processing from a backscatter s0 triplet. This pair of wind vectors is what is presented to the scheme, and it is assumed that one of them corresponds to the true wind. The scheme could also take into account other surface wind observation within the area, if available, which could give valuable information in the dealiasing.

The scheme is also based on having a background or first guess xb, which is found by interpolating a numerical model wind forecast to the observation points. The background cost term is given by

The background covariance matrix B is derived by assuming that all the errors in the background field are in the non-divergent part of the windfield (see Daley, 1991). In this way a constraint of minimizing the divergence over the scatterometer observation points is built in. A constraint of spatial coherence is also contained within this term, and the degree of coherence is determined by a decorrelation scale parameter. For more details on the derivation of the background covariance matrix, see Schyberg and Breivik (1996).

Assuming the observations are independent, the observation term splits up into contributions from each observation point, represented by a sum over the observation index k. We can derive an expression for each observation cost term using the relation with conditional probabilities outlined by Lorenc (1986). We denote the event of a given scatterometer observation "Ok", and this observation implies the two wind possibilities vOk1 and vOk2. Following Lorenc, we have

Eq. (4) is similar to an expression used for treating gross errors in Lorenc and Hammon (1988). Here "v and R1" is the event that the wind has the true value v and the correct wind alternative of the observed pair is vOk1.

It is reasonable to model P(O | v and R1) and P(O | v and R2) as two Gaussian distributions in vOk1 and vOk2 respectively. Then we obtain and , where we have defined, like done by Stoffelen and Anderson (1995),

jk1 and jk2 measure the distances to the two possible wind observation solutions which are approximately 180 degrees apart. m = 1,2 are the indices on the two winds in the ambiguous wind observation pair. (uk,vk) is the windfield at observation point and es is the standard deviation of scatterometer wind component observation error.

The probabilities P(R1|v) and P(R2|v) are a priori conditional probabilities for which of the aliases are true, with P(R1|v)+p(R2|v)=1 . We will assume that they are independent of the actual windfield v and omit the condition on v. As a simple option we can assume they are equally probable, that is setting P(R1)=P(R2)=1/2 , which will be used in the example to be presented here.

As another option, we can include additional information on the s0 triplet for a better estimate of P(R1) and P(R2). Such information exists e.g. in the form of the distance of the measured s0 triplet to each of the vOk1 and vOk2 points on the wind "cone" in the s0 triplet space. This cone is defined by the model function mapping two-dimensional wind vectors to the three-dimensional s0 triplet space (see Stoffelen and Anderson, 1995).

Because of errors and dependencies not taken into account by the model function, the s0 triplet will not lie exactly on this cone. Two wind solutions found in the pre-processing of the observation correspond to local minima in distance from the cone to the measured triplet. Statistics shows the so called "rank 1" wind solution, which is the one with smallest distance to the cone, to be slightly more probable. This is utilized within the Pre-Scat algorithm.

This extra information can be utilized for the new method within the above framework by letting vOk1 be the rank 1 solution and giving P(R1) a slightly larger value. Experiments showed, however, that for our applications, this did not give significantly different ambiguity removal results from assuming both solutions equally probable.

When applying the simplest option, we find for the cost contribution from each observation

It was shown by Schyberg and Breivik (1997) that the above variational problem can be solved by using a successive correction approach similar to that suggested by Bratseth (1986). The method iterates on a vector y and an analysis vector x, both containing v at each observation point. The iteration is initialized by setting x and y equal to the first guess field. The iteration value at each step is then found from

While P(R1) and P(R2) are prior probabilities for each of the two wind possibilities, w1 and w2, with the optimal most probable windfield inserted, can be regarded as posterior probabilities for the wind possibilities. The scheme assumes the ambiguity with the highest posterior probability to be the true one.

This ambiguity removal scheme contains several parameters which must be determined. Prior information on the probability of the "rank 1" and "rank 2" wind solutions enters into the choice of P(R1) and P(R2). The accuracy of the scatterometer observations is reflected in the value of es, and the quality of the first guess winds enters into a background error variance which scales the background covariance matrix B, as well as a decorrelation scale which determines the spatial coherence.

3. Operational use

Initial experiments showed the method to give better results than Pre-Scat for low-quality first guess fields (24 hour forecasts). The Pre-Scat operational routine run at DNMI, however, uses 6 hour forecasts as a first guess field, which gives a better quality of the dealiasing than with 24 hrs. forecasts. To compare the operational Pre- Scat with the new scheme, a parallel routine using the new ambiguity removal scheme with 6 hour forecasts as a background was performed.

Because of the high quality of the first guess fields, Pre- Scat is quite difficult to improve upon. However, we did detect a number of cases where the new scheme took advantage of coupling the spatial coherence to a requirement of minimizing divergence.

An example of such a case is given in Fig. 1. The figure shows the pressure field corresponding to the 6 hrs. forecast first guess field. A low north of Northern Norway appears in this field. The scatterometer winds processed using the new ambiguity removal system indicates that the low pressure centre in the first guess is misplaced and should be displaced towards northwest.

Figure 1. Ambiguity removal results. Blue arrows within the scatterometer pass area show results from the new ambiguity removal scheme, while green arrows shows points where Pre-Scat picked the other solution. The pink arrows elsewhere represent other wind observations. The contours show a 6 hrs. forecast pressure field corresponding to the first guess windfield.

The Pre-Scat scheme seems to rely more on the first guess field, and turn the winds opposite of the new scheme within an area, and does not seem to be able to produce information correcting the mispositioning of the low. This creates a divergent set of observations which seems unrealistic. This shows the new scheme to take advantage of the constraint of minimizing divergence.

There were, however, also a few cases where the new scheme produced questionable ambiguity removals with Pre-Scat seemingly superior. All in all the two schemes seem to be of comparable quality.

The new method shares the problem with traditional data assimilation methods of assuming a fixed background error covariance structure. This means that the background errors are assumed to have the same spatial distribution in all synoptic situations, which obviously is not an optimal method. (A similar problem also exists in the way Pre-Scat requires spatial coherence).

4. Conclusion

The new method seems to be of comparable quality to Pre-Scat when 6 hour forecasts are used as a first guess. The results from the new method is promising, but Pre- Scat is hard to improve upon when using high quality first guesses.

Acknowledgement

This work was supported by the Norwegian Space Centre.

References

Keywords: ESA European Space Agency - Agence spatiale europeenne, observation de la terre, earth observation, satellite remote sensing, teledetection, geophysique, altimetrie, radar, chimique atmospherique, geophysics, altimetry, radar, atmospheric chemistry