ESTIMATION OF DIRECTIONAL SEA SPECTRA FROM ERS/SAR IMAGES OF MEDITERRANEAN AREAS: A CASE STUDY

ABSTRACT

An inversion technique for estimating sea wave directional spectra from Synthetic Aperture Radar (SAR) images is applied to a set of ERS-1 data relevant to selected Mediterranean areas. The approach followed is based on an analytical definition of the transform which maps the sea wave spectrum into the corresponding SAR image spectrum (SAR modulation transfer function). The solution of the inverse problem is determined through a numerical procedure which minimises a proper functional. A suitable iterative scheme is adopted, involving the use of the SAR modulation transfer function. Although widely applied to ocean environments, the method has not been yet extensively tested in smaller scale basins, as for instance the Mediterranean sea. The main purpose of this paper is providing new experimental data relevant to the Mediterranean Sea, to discuss the results obtained by the above inversion technique and to compare them with buoy derived sea truth measurements.

INTRODUCTION

The analysis and the interpretation of SAR images of the sea surface strongly relies on the comprehension of the interaction between electromagnetic waves and the sea waves. Although many interpretative models have been proposed by several authors [Ulaby F.T. et al, 1986], [Kasilingam D.P. and Shemdin O.H., 1990], a model that exhaustively explains the several and complex phenomena which characterise the formation of SAR images is not yet available. In this context, the estimation of the sea spectrum from SAR images represents a particularly difficult task. One of the greatest difficulties arises from the time-dependent nature of the sea surface [Alpers, W.R., Rufenach, C.L., 1979]. However, spectral analysis as applied to SAR images has demonstrated, under particular hypotheses, the possibility of evaluating some typical parameters of the sea surface, as for instance the two-dimensional wave power spectrum. Some authors have proposed functional relationships (SAR modulation transfer function) between the spectrum of SAR images and the spectrum of the sea surface. In particular, although in some specific conditions the SAR modulation transfer function can be approximated by a linear expression [Monaldo, F.M., Lyzenga, D.R., 1986], in the most general case a non-linear transform is needed [Hasselmann, K., Hasselmann, S., 1991]. The relationship between the sea wave and the SAR image spectrum must be inverted to provide an estimate of the sea wave directional spectrum from the corresponding SAR image. The inversion is accomplished by an iterative algorithm, which is based on the minimisation of a proper cost functional [Hasselmann, K., Hasselmann, S., 1991], [Engen, G., et al., 1994].

This paper is concerned with the application of the above inversion procedure to ERS-1 SAR PRI images of selected Mediterranean Sea areas. These areas have been chosen for the possibility of comparing the results obtained by the inversion algorithm with sea truth measurements provided by buoys. We underline that the method used has not been yet extensively applied in small scale basins, as for instance the Mediterranean Sea.

Finally, it is worth noting that, due to the complexity of the SAR image formation process, numerical simulation may provide a very important tool for interpreting and validating the analytical expressions for the SAR modulation function. In the analysis, we made partial use of a SAR ocean simulator [Corsini, G., Manara, G., Monorchio, A., 1995], based on a realistic description of the interaction between the incident electromagnetic waves and the sea surface. In particular, it was usefully employed for a more detailed interpretation of the effects produced on the corresponding backscattered signal by the main physical phenomena at the sea surface, as for instance tilt modulation and azimuthal cut-off. We identify as a further step in the analysis the substitution of the analytic representation of the SAR modulation transfer function with a more accurate description based on the above numerical simulator. This problem will be subject of future work.

INVERSION ALGORITHM

The sea state can be statistically characterised by its wave spectrum [Apel, J.R., 1987], which describes the spectral power density as a function of the propagation vector . The first step in the definition of a numerical procedure for retrieving the sea wave directional spectrum from SAR images is to determine the relationship between the variance of the SAR image intensity and the variance of the wave height with respect to the reference plane. In the most general case, this relationship can be expressed as follows [Hasselmann, K., Hasselmann, S., 1991], [Krogstad, H.E., 1992]:

(1)

where and denote the sea surface directional power spectrum and the SAR image power spectrum, respectively. In (1), is a mapping function, generally dependent on the wave vector and on the sea wave spectrum .

An orthogonal reference frame (x,y) will be assumed in the following, where x denotes the sensor flight direction (azimuth), while y, perpendicular to the flight line, is the SAR antenna bearing direction (range), projected onto the reference surface plane. In some particular cases [Monaldo, F.M., Lyzenga, D.R., 1986], eq. (1) reduces to the following quasi-linear approximate expression:

(2)

We note that the previous relationship can be directly inverted to obtain an estimate of the quasi-linear approximation of the SAR transfer function. However, in the most general instances the SAR transfer function is non-linear. In this latter case the inversion procedure can be performed by resorting to an iterative scheme [Hasselmann, K., Hasselmann, S., 1991]. The optimal estimate of the sea wave spectrum is obtained by minimizing a suitable cost functional [Engen, G., et al., 1994]:

(3)

In (3), and represent the first guess estimation of the sea wave spectrum and the oserved SAR image spectrum, respectively. The choice of the weighting functions and is usually made with the objective of optimising the convergence of the numerical procedure. The same observation is valid for the numerical parameter . is defined in [Engen, G., et al., 1994].

The first guess wave spectrum is used to initialise the iterative process. It is worth noting that the convergence of this process strongly depends on the similarity between the first guess wave spectrum and the actual wave spectrum. Moreover, is the estimated SAR image spectrum at the generic iteration. The minimisation of the above functional, i.e. the condition , can be easily reached within an iterative scheme that makes use of the quasi-linear approximation between the sea wave spectrum and the SAR image spectrum. Assuming at the first step , we evaluate the corresponding estimation of the SAR image spectrum at the generic n-th step of the iterative scheme from the best fitted wave spectrum through the non-linear transform. The solution at the -th step is obtained by that at the previous step, by means of the following relationships:

,

(4)

.

(5)

The variation , used to correct the SAR image spectrum, is calculated from the variation of the sea wave spectrum by using the quasi-linear approximation of the SAR modulation transfer function. Finally, the term can be obtained by replacing expressions (4) and (5) in the cost functional (3) [Hasselmann, K., Hasselmann, S., 1991].

NUMERICAL RESULTS

The algorithms previously described have been applied for estimating sea wave spectra from a set of ERS PRI images relevant to specific areas of the Mediterranean Sea. Samples of the numerical results obtained are reported in this Section. In order to estimate the effectiveness of the inversion algorithm, a suitable correlation coefficient between the best fitted SAR image spectrum and the ERS SAR spectrum has been introduced. It is defined as:

.

(6)

The first area under consideration contains the Ponza Isle. In the same zone a buoy of the Italian National Hydrographic Service periodically collects sea spectrum data. The buoy coordinates are: 40o 52' N, 12o 57' E; it measures the height of the sea surface together with rolling and pitching with respect to the magnetic north with a sampling frequency of 1.28 Hz.

Fig. 1 shows the numerical results obtained by applying the iterative algorithm. In particular, the estimated ERS-1 SAR image spectrum is reported in Fig. 1.a. The first guess sea wave spectrum (Fig.1.b) is obtained by modifying the omnidirectional Pierson sea wave spectrum by a spreading function of the following kind: , with . The wind friction velocity used in the Pierson model (40 cm/s) has been determined through heuristic considerations. The predicted SAR image spectrum at the final step of the iterative algorithm is shown in Fig. 1.c. The correlation coefficient between the images in Fig. 1.a and Fig. 1.c is equal to 0.6. The weighting function in eq. (3) has been defined as . Finally, the estimated sea wave spectrum is reported in Fig. 1.d. The significant wave height results equal to 4.18 m in good agreement with the buoy derived estimation (3.9 m).

The second testing zone is a Sardinian Sea area in front of Alghero. Again, a buoy of the Italian National Hydrographic Service is present in the scene, with coordinates 40o 32' N, 8o 06' E. In particular, Fig. 2.a shows the ERS-1 SAR image spectrum. In this case, the first guess sea wave spectrum has been obtained from the omnidirectional spectrum measured by the buoy, using the same spreading function as in the first example (Fig. 2.b). The optimization procedure employs the cost function in eq. (3), where

(7)

is the gaussian weighting window proposed in [Engen, G., et al., 1994] and is the weighting window proposed in [Hasselmann, K., Hasselmann, S., 1991]:

.

(8)

In expression (8), b is a small real positive constant, which is introduced to avoid a divergence in those points where . The values of parameters in the weighting functions are reported in Table I.

Table I

The correlation coefficient between the ERS SAR image spectrum and the best fitted one at the final step of the algorithm (Fig. 2.c) is equal to 0.863. The image of the estimated sea wave spectrum (Fig. 2.d) shows a sharp peak in a direction which is rotated of about 127° with respect to the azimuthal direction. It is characterized by a wavelength of about 160 m; this is in agreement with the buoy recorded data. The significant wave height obtained by integrating this spectrum is equal to 6.46 m, a value which is consistent with that measured by the buoy (6 m).

CONCLUSIONS

The inversion procedure for retrieving the sea wave power spectrum from SAR images proposed in [Hasselmann, K., Hasselmann, S., 1991], [Engen, G., et al., 1994] has been applied to selected areas of the Mediterranean Sea. The method has been tested on a set of actual SAR PRI images, recorded by the European satellite ERS-1. The iterative inversion algorithm has been checked for different inizialisation procedures and the results obtained have been compared in terms of the correlation coefficient between the ERS SAR image spectrum and the SAR spectrum predicted by the algorithm at the final step. High values of this correlation coefficient can be achieved by estimating the first guess sea wave spectrum from buoy derived measurements. A good agreement is obtained between the estimated value of the significant wave height and that measured by the buoy system.

Acknowledgements

This work was performed in the framework of the ESA experiment ERS 1-2 n. A02.I111. The authors would like to thank the "Servizio Idrografico e Mareografico Nazionale" of the Italian "Presidenza del Consiglio dei Ministri" for providing the sea truth measurements.

REFERENCES

Ulaby, F.T., Moore, R.K., Fung, A.K., 1986:

Microwave Remote Sensing, Artech House Inc., Norwood.

Kasilingam, D.P., Shemdin, O.H., 1990:

Models for Synthetic Aperture Radar imaging of the ocean: a comparison, Journal of Geoph. Research, Vol. 95, No. C9, pp. 16263-16276.

Monaldo, F.M., Lyzenga, D.R., 1986:

On the estimation of wave slope- and height-variance spectra from SAR imagery, IEEE Trans. on Geosci. and Remote Sensing, Vol. 24, pp. 543-551.

Hasselmann, K., Hasselmann, S., 1991:

On the nonlinear mapping of an ocean wave spectrum into a Synthetic Aperture Radar image spectrum and its inversion, Journal of Geoph. Research, Vol. 96, No. C6, pp. 10713-10729.

Engen, G., Johnsen, H., Krogstad, H.E., Barstow, S.F., 1994:

Directional wave spectra by inversion of ERS-1 Synthetic Aperture Radar ocean imagery, IEEE Trans. on Geosci. and Remote Sensing, Vol. 32, No. 2, pp. 340-352.

Alpers, W.R., Rufenach, C.L., 1979:

The effect of orbital motions on Synthetic Aperture Radar imagery of ocean waves, IEEE Trans. on Antennas and Propagat., Vol. 27, No. 5, pp. 685-689.

Corsini, G., Manara, G., Monorchio, A., 1995:

Sea wave spectrum estimation from SAR images: a simulation based approach, Proc. IGARSS ‘95, Florence, Italy, July 10-14, pp. 936-938.

Apel, J.R., 1987:

Principles of Ocean Physics, International Geophysics Series, Vol. 38, Academic Press, London.

Krogstad, H.E., 1992:

A simple derivation of Hasselmann's nonlinear ocean Synthetic Aperture Radar transform, Journal of Geoph. Research, Vol. 97, No. C2, pp. 2421-2425.

Fig. 1 - Thyrrenian Sea area close to the Ponza Isle: a) ERS-1 SAR image spectrum; b) first guess sea spectrum; c) best fitted SAR image spectrum; d) estimated sea wave spectrum. (Orbit: 8970, Frame: 2781; Date: 3/04/93, Image center: 40°56’24"N, 13°20’24"E).

Fig. 2 - Sardinian Sea area: a) ERS-1 SAR image spectrum; b) first guess sea spectrum; c) best fitted SAR image spectrum; d) estimated sea wave spectrum. (Orbit: 20007, Frame: 2781; Date: 13/5/95, Image center: 40°54’36"N, 8°18’36"E).

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