SAR Ocean Wave Inversion - A Hierarchical Approach

Retrieval of two dimensional ocean wave spectra from synthetic aperture radar data has been performed for several years using a variety of di erent approaches. Due to the complexity of the problem potential users of SAR ocean wave retrieval schemes like weather centers have found it diÆcult to judge about the quality and the bene t of SAR retrieved wave spectra. The present study gives insight into the potential as well as the shortcomings of SAR wave retrieval. The study is based on a hierarchical approach, which starts with the most simple inversion schemes, based on strong simpli cations of the imaging mechanism. Linear as well as quasilinear approximations are used for the inversion process. In a rst step no a priori information is used in order to investigate the information coming solely from the SAR. SAR retrieved wave heights are compared with ECMWF wave height data. The comparison demonstrates di erent e ects caused by the simpli cations made in the retrieval scheme. Based on these observations the inversion scheme is extended step by step using more sophisticated forward models as well as a priori information. The a priori information is introduced using a maximum a posteriori approach. The bene t of using cross spectra instead of conventional image power spectra is analyzed. The study is based on a global dataset of ERS-2 complex wave mode data, which were processed with the DFD processor BSAR. The presented work is meant to be a preparation for the coming ENVISAT ASAR ocean wave products. INTRODUCTION The retrieval of ocean wave spectra from synthetic aperture radar (SAR) data has been studied for several years now. New interest in the subject is coming up with the near launch of the ENVISAT satellite. The ENVISAT ASAR will provide SAR data, which will lead to improved ocean wave measurements compared to the former ERS SAR data. The ASAR wave mode data will have the following advanced features: acquisition every 100 km along the track SAR cross spectra with low noise level and resolved wave propagation direction available as standard product original SAR images (imagettes) are available SAR ocean wave imaging is a strongly non-linear mechanim which involves loss of information in particular about short ocean waves propagating in the satellite ight direction (azimuth). Several sophisticated algorithms have been developed which provide 2d wave spectra using di erent kinds of a priori information [1], [2], [3]. Although the proposed algorithms provide reasonable results, there is still discussion about the question of how much ocean wave information is contained in the SAR data alone. The present study analyzes this question using a step by step approach. Ocean wave information is retrieved from complex SAR data using inversion techniques of growing complexity. The study is based on complex ERS-2 SAR wave mode data which were processed with the DLR BSAR processor [4]. The wave mode raw data were kindly provided by ESA. Hierachy of Forward models SAR ocean wave retrieval schemes proposed so far are based on the modulation Î of SAR intensity images. Î = I < I > < I > (1) Neglecting motion e ects (e.g. assuming a real aperture radar (RAR)) the relation between the ocean wave spectrum F and the variance spectrum P of Î is to rst order be given by: P k = 0:5 jT RAR k j 2 Fk + 0:5 jT RAR k j 2 F k (2) The RAR transfer function T is composed of three parts T = T tilt + T hydro + T rb (3) with tilt modulation transfer function (MTF) T , hydrodynamic MTF T , and range bunching MTF T rb [5]. A contour plot of jTj is shown in Fig. 1 (A). Taking into account the impact of sea surface motion the SAR image variance spectrum P can to rst order be expressed in terms of a SAR transfer function T P k 0:5jT SAR k j 2 exp(i! t)) Fk + 0:5jT SAR k j 2 exp( i! t)) (4) with the SAR MTF T given by T = T i kx T v k (5) where T v is the orbital velocity transfer function [5] and is the slant range to platform velocity ratio. However, it turned out that this linear expression (eq. 4) is a poor approximation in many cases [6]. Taking into account the nonlinear imaging mechanims Hasselmann [7] derived an integral equation for the mapping of an ocean wave spectrum F into the SAR image variance spectrum P . This relation was later extended to cross spectra Pk of two looks separated by a short time t [8]. Pk( t) = 1 4 2 exp( k x < 2 >) Z dx exp( i k x) exp(k x 2 f(x)) (1 + f(x) + i kx (f (x) f( x)) +(kx ) (f(x) f(0))(f( x) f(0)) (6) The autocorrelation and cross correlation functions f(x),f and f are given as follows: f(x) = 0:5 Z dx (F (k) jT k j 2 exp(i! t) +F ( k) jT k j 2 exp( i! t)) exp(i k x) (7) f(x) = 0:5 Z dx (F (k) T k T v k exp(i! t) +F ( k) T k T v k exp( i! t)) exp(i k x) (8) f(x) = 0:5 Z dx (F (k) jT v k j 2 exp(i! t) +F ( k) jT v kj 2 exp( i! t)) exp(i k x) (9) The variance < 2 > of the azimuthal image point shift is given by < 2 >= Z jT v k j 2 Fk dk (10) with orbital velocity MTF T . A contour plot of jT j is given in Fig. 1 C) assuming the ERS-2 SAR imaging con guration. SAR ocean wave measurements are based on the inversion of the transform given by eq. 6. Due to the complexity of the transform it is desirable to seek for simpli cations. Several approximations of the transform have been proposed. Expanding eq. 6 to rst order with respect to the wave spectrum F yields the linear transform eq. 4. Expanding only the integral part of eq. 6 to rst order with respect to F yields the quasi linear transform P quasi k = exp( k 2 x < 2 >) (0:5jT k j 2 exp(i! t)) Fk + 0:5jT SAR k j 2 exp( i! t))F k) (11) A contourplot of the lter function exp( k x < 2 >)jT k j 2 with p < 2 > = 90 m is shown in Fig. 1 B). A comparison of the three di erent approximations for image power spectra (e.g. t = 0) is given in Fig. 2. The input wave spectrum F with a signi cant waveheight of 5.5 m is shown in A). A wind sea system of about 150 m and a swell system of about 400 m can be seen travelling in diagonal direction with respect to the azimuth range reference system of the SAR sensor. The waveheight is 5.5 m and the rms azimuthal -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 azimuth wavenumber [rad/m] -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 ra ng e w av en um be r [r ad / m ]