the german bight: preparing for sentinel-3 with a cross

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three geophysical parameters: sea surface height (SSH), significant wave height (SWH) and wind speed at 10 meters height (U10) from CryoSat-2 data in our.
THE GERMAN BIGHT: PREPARING FOR SENTINEL-3 WITH A CROSS-VALIDATION OF SAR AND PLRM CRYOSAT-2 ALTIMETER DATA Fenoglio-Marc L. (1), Buchhaupt C. (1), Dinardo S. (2), , Scharroo R. (3), Benveniste J. (4), Becker M. (1) (1)

Technische Universität Darmstadt, Germany, [email protected] (2) SERCO/ESRIN, Frascati, Italy, [email protected] (3) EUMETSAT, Darmstadt, Germany, [email protected] (4) ESA/ESRIN Frascati, Italy, [email protected]

ABSTRACT As preparatory work for Sentinel-3, we retrieve the three geophysical parameters: sea surface height (SSH), significant wave height (SWH) and wind speed at 10 meters height (U10) from CryoSat-2 data in our validation region in North Sea. The CryoSat-2 SAR echoes are processed with a coherent and an incoherent processing to generate SAR and PLRM data respectively. We derive precision and accuracy at 1 Hz in open ocean, at distances larger than 10 kilometres from the coast. A cross-validation of the SAR and PLRM altimeter data is performed to investigate the differences between the products. Look Up Tables (LUT) are applied in both schemes to correct for approximations applied in both retracking procedures. Additionally a numerical retracker is used in PLRM. The results are validated against in-situ and model data. The analysis is performed for a period of four years, from July 2010 to May 2014. The regional cross-validation analysis confirms the good consistency between PLRM and SAR data. Using LUT the agreement for the sea wave heights increases by 10%. 1. INTRODUCTION Synthetic Aperture Radar (SAR) altimetry is expected to provide improved precision and along-track resolution compared to the conventional low-resolution mode (LRM) radar altimetry. CryoSat-2 enables a quantitative comparison of SAR and Pseudo-LRM (PLRM) data derived from a coherent and an incoherent processing of the same SAR echoes respectively. In this respect, CryoSat-2 is very similar to Sentinel-3. Main novelty in this paper with respect to the preliminary work in [1] is the building of Loop Up Tables (LUT) to correct for approximations in the retracking using the Brown and SAMOSA2 models. LUT have been considered in [2, 3, 4]. As alternative a numerical retracker is used here for PLRM, which uses the real Point Target Response (PTR) in the Brown

model instead of a Gaussian approximation for the waveform retracking. No instrumental LUT correction is required in this case because the real PTR is included. Level 2 data are constructed using LUT and the numerical retracker and validated. In Section 2, the methodology to build PLRM waveforms, LUT and finally both PLRM and SAR Level 2 data is detailed. In Section 3 the results obtained during cross-validation and in-situ validation are described.

2. METHOD 2.1 Computation of PLRM waveforms The procedure described in [5,6] has been applied to generate the PLRM data without LUT table. Each 20Hz waveform is computed by incoherently averaging 256 echoes from four bursts of 64 echoes each. Each burst has its own time tag and geo-location. In the first step four bursts are gathered and the time between the second and the third is computed as a reference for the 20Hz waveform. The 20Hz timing is corrected for the time tag bias of 0.520795 msec. The geo-location is linearly interpolated, the height is corrected for the time tag bias by adding the time bias multiplied with the height rate to the altitude. 2.2 Computation of LUT for PLRM For the case of pulse-limited (LRM) altimeter Brown expressed the statistical expectation, i.e. the average return power received from a rough scattering surface, in terms of a convolution of three terms which are a function of the delay t:

𝑊 𝑡 = 𝐹𝑆𝑆𝑅 𝑡 ∗ 𝑃𝐷𝐹 𝑡 ∗ 𝑃𝑇𝑅(𝑡)

(1)

where: FSSR is the flat sea surface response, PTR is the radar point target response, and PDF is the surface elevation probability density function of specular points In the Brown model several approximations have been made. In particular it is assumed that:

(1) the PDF function is a Gaussian function. (2) the PTR, which is a sinc2 function, is approximated by a Gaussian, PTR(t) = exp(-t2/σp), with   σp   =   αp   τp,   with τp, being the time resolution and αp=  0.513   (3) the modified Bessel function within the FSSR term is approximated by exponentials (this approximation is good enough for mispointing angles smaller than 0.2°). In general in open sea the Brown’s model and the Maximum Likelihood Estimator with three free parameters (MLE-3) are used to estimate three parameters from the average return signal contained in the waveforms. The minimizer used here to solve the estimation problem is a Trust Region algorithm [7,8]. The estimated parameters are epoch t0, amplitude Pu and σc. The σc is a function of significant wave height (SWH) and time resolution 𝜏!  according to Eq. 2: 𝜎! =   (

!"# ! ) !!

+ (𝛼! 𝜏! )! =

𝜎! ! + 𝜎! !

(2)

!"#

with 𝜎! ! = ( )! , 𝜎! ! = (𝛼! 𝜏! )! . The PTR !! approximation leads to errors in the parameters estimated by MLE-3 [3]: whereas the epoch and the amplitude are estimated quite well, the SWH needs a correction. Therefore a correction term for the SWH needs to be estimated and stored in a Look up table. This LUT would give, for example, the LUT correction (LUTcorr) to the estimated SWH as function of the SWH estimated by MLE-3 (Fig. 1 bottom).

In our processing we do not directly compute the LUT as function of SWH, instead we evaluate a LUT giving the αp parameter as function of the SWH. The method is as follows: 1. Numerical pulse-limited waveforms encompassing as

PTR a squared sinc-function, and therefore not usng the PTR approximation, are simulated from known parameters. We select a Pu amplitude of one, an epoch of bin 68 and SWH starting from zero to 20 m with a step size of 0.1m. The pointing error angles, the skewness coefficient and the thermal noise are set to zero. The waveforms are generated in the frequency domain using a fast circular convolution algorithm. 2. For each SWH in input, the numerical waveforms are fitted with a scheme employing the Brown model and the MLE method [2]. Three different methods for estimating the LUT have been considered. The first LUT is computed using a MLE where only the parameter αp (Eq. 2) is estimated and the other two parameters amplitude and epoch are held fixed. In the second LUT the approach used in the first LUT is followed and, in addition, the waveform is (area equal to the Brown model with the same parameters). Finally the third LUT is computed using a MLE-3 where the three parameters amplitude, epoch and σs are estimated. We use here LUT2 because the same procedure is used for the SAR data in GPOD [9]. 3. The SWH and corresponding αp are stored in a LUT giving αp as function of SWH (Fig. 1 top). 4. The real echoes are hence retracked using the Brown model and an MLE-3 method. This time, the third estimated parameter is the SWH. For each iteration in MLE-3 the αp corresponding to the SWH is computed from the LUT αp table, then σc is computed from both αp and SWH (Eq. 2, Fig. 2 top). We can also evaluate the corrected SWH using the LUTcorr (Fig. 2 bottom). This method is in analogy with the method used to compute the LUT in SAR mode as done in the ESRIN SAR prototype [9] and described in the next section. 2.3 Computation of LUT for SAR As for the PLRM processing, also in the SAR processing in GPOD a table giving the αp parameter as function of the SWH is computed. The method is as follows:

Figure 1. LUT for PLRM as αp (LUTαp, top) and as correction to SWH (LUTcorr bottom)

1. Numerical SAR waveforms (encompassing as PTR a squared sinc) are simulated selecting an amplitude of one, an epoch t0 of zero and significant wave heights starting from zero to 20 m with a step size of 0.1 m. The

pointing error angles, the skewness coefficient and the thermal noise are set to zero. The simulation is made in the spectral domain.

SWH as free fitting parameter and with a fixed αp value of 0.513. The PLRM data that we process including the LUT correction are called TUDaL hereafter.

2. For each SWH in input the simulated waveforms have been normalized and fitted against the SAMOSA2 model and the free parameter αp is estimated for each input SWH by minimizing the rms error between the complete numerical solution and the SAMOSA2 model.

2.4  Computation  of  PLRM  without  LUT    

3. The input SWH and corresponding estimated αp are stored in a Look Up Table giving αp as function of SWH. Figure 2a shows the LUT estimated. 4. The real SAR echoes are hence retracked using the SAMOSA2 model and a Least Square method wherein, for each fitting iteration, the αp (corresponding to the iterative trial SWH) is extracted from the LUT. The minimizer used to solve the estimation problem in GPOD is a Levenberg-Marquardt algorithm [10].  

We develop a numerical retracker for PLRM, called SINC2, which uses the real Point Target Response (PTR) in the Brown model instead of a Gaussian approximation for the waveform retracking. No instrumental LUT correction is required because the real PTR is included. The numerical retracker is based on a fast circular convolution algorithm, estimating the three parameters amplitude A, epoch t0 and σs. The PLRM data that we process using the numerical retracker are called TUDaN hereafter.

RESULTS 3.1 Precision As indicator for the precision of each parameter at 1 Hz, we consider the standard deviation of the parameter itself (SSH, SWH or U10) over 20 consecutive 20 Hz measurements and evaluate those as a function of SWH. We then compute the median of the standard deviations for each SWH to obtain the performance curve and we particularly consider the value for a SWH of 2 m.   As  in  [1]  both  SAR  SSH  and  SWH  measurements  have   a  higher  precision  than  PLRM.  At 2 m wave height, the SSH have precision of 0.9 cm in SAR and 1.9 in PLRM. Instead wind   speed   measurements   have   higher   precision   in   SAR,   For   wind   speed   instead,   the   precision  is  higher  in  PLRM  than  in  SAR.  Precision  is   4  cm/s  and  5.7  cm/s  respectively  for  SAR  and  PLRM.       Table 1. Precision of SWH (cm) depending on LUT SWH 0.5 1.0 2.0

Figure 2. LUT for SAR as αp (LUTαp, top) and as correction to SWH (LUTcorr bottom) We built as well a tabulated SWH correction (LUTcorr.) against SWH (Fig. 2 bottom) by fitting the simulated waveforms against the SAMOSA model using this time

TUDa 16.8 14.8 10.8

TUDaL 10.6 10.5 10.1

TUDaN 10.7 10.8 10.1

SAR 10.11 8.5 7.2

  Fig.  3  shows  the  results  for  SWH.  The  LUT  correction   increases  significantly  the  precision  of  SWH  for  small   SWHs,  as  from  Table  1.  For  SWH  of  2  m,  the  precision   of   SWH   is   7.2   cm   in   SAR,   10.8   cm   in   PLRM   without   LUT   and   10.1   cm   in   PLRM   with   LUT   and   in   the   numerical   retracker.   For   SWH   of   2   metres   the   precision  improves  in  SAR  by  a  factor  1.4.  For  lower   SWH   the   ratio   increases   and   is   1.8   for   SWH   of   0.5   metres.      

 

Figure 3. Precision of SWH without LUT (above), with LUT (middle) and in SINC2 (below) 3.2 Cross-comparison of PLRM and SAR The   regional   cross-­‐validation   of   PLRM   and   SAR   shows   a   good   agreement   between   the   uncorrected   SSH   obtained   by   both   types   of   processing,   with   a   standard   deviation   of   the   differences   of   3   cm   and   a   bias   of   1   cm   in   agreement   with   [1].   Similarly   good   agreement   is   found   for   the   wind   speed   U10,   with   a   standard   deviation   of   the   differences   between   SAR   and  PLRM  of  26  cm/s,  bias  of    2  cm/s,    slope  0.992).       Results   are   very   similar   to   those   presented   in   [1]   over   a   short   interval   and   are   independent   from   the   LUT  applied.     Instead, the effect of the LUT is relevant for SWH, as it causes a reduction of 10% on the RMS difference between SAR and PLRM. ESRIN SAR [7] and PLRM data including the LUT correction (TUDaL) are in very good agreement. The standard deviation decreases from 21 to 19 cm when the LUT is applied . Differences are higher at small SWHs and reduce significantly with applied LUTcorr (Fig. 3).

Figure 4. SWH Scatterplot of SAR and PLRM without (above) and with LUT (middle) and in SINC2 (below) The altimeter wind speeds agree at best with the ECMWF wind data and the altimeter wave heights with the DWD/LSM data as in [1].

Table 2. Statistics of 1 Hz CryoSat-2 minus in-situ HELG sea level (16 pts) in 10-20 Km distance of open sea uncorrected height (m) PLRM-FIN1 TUDa-FIN1 TUDaL-FIN1 TUDaN-FIN1 SAR-FIN1

mean 0.051 0.052 0.051 0.045 0.024

std 0.090 0.089 0.091 0.089 0.110

cor 0.987 0.987 0.987 0.987 0.980

slope 1.027 1.028 1.026 1.018 1.010

SI 0.002 0.002 0.002 0.002 0.003

Figure 6. Scatterplot of SWH at FINO-1 from PLRM without (square) and with LUT (circle orange), numerical (circle black) and SAR (triangle)

Figure 5. SWH difference of SAR and PLRM without (above) and with LUT (middle) and in SINC2 (below) 3.3 In-situ validation Altimeter data with a maximum distance of 20 km from in-situ data have been considered. The sea state bias correction was computed as 5% of the SWH. In-situ comparison of PLRM and SAR data shows a similar accuracy in SAR and PLRM for SSH and U10 (Tables 1, 3) and higher accuracy in SAR than in PLRM for SWH (Table 2). The standard deviation of the differences between the three parameters from altimetry and in-situ data are 11 cm, 18 cm, and 1.9 m/s for SAR and 9 cm, 29 cm, and 1.8 m/s for PLRM. The agreement increases for SWH when the LUT is applied and with the PLRM numerical retracking (TUDaL and TUDaN). In those cases the standard deviation of the residuals decreases to 19 and 21 cm respectively (Table 2). The best agreement between SAR and PLRM is obtained using the numerical retrackers.

Figure 7. Scatterplot of SSH near the Helgoland tide gauge from PLRM without/with LUT (square/circle orange), numerical (circle black) and SAR (triangle) Table 3. Statistics of 1Hz SWH (m) of CryoSat2 and insitu FINO-1 AWAC (26 pt) in 10-20 km distance PLRM-FIN1 TUDa-FIN1 TUDaL-FIN1 TUDaN-FIN1 SAR-FIN1

mean 0.02 0.02 0.004 -0.009 0.03

std 0.295 0.290 0.195 0.210 0.183

cor 0.955 0.956 0.965 0.963 0.980

slope 1.16 1.17 0.95 1.02 1.11

SI 0.17 0.17 0.13 0.14 0.11

Table 4. Statistics of 1Hz U10 (m/s) of CryoSat2 and insitu FINO-1 (38 pt) in 10-20 km distance PLRM-FIN1 TUDa-FIN1 TUDaL-FIN1 TUDaN-FIN1 SAR-FIN1

mean -0.650 -0.673 -0.788 -0.869 -0.747

std 1.836 1.829 1.884 1.867 1.954

cor 0.889 0.890 0.881 0.883 0.868

slope 0.761 0.761 0.790 0.784 0.738

SI 0.19 0.18 0.20 0.20 0.19

References 1. Fenoglio-Marc L., Dinardo S., Scharroo R., Roland A., Sikiric M., Lucas B., Becker, M., Benveniste, J., Weiss, R. (2015). The German Bight: a validation of CryoSat-2 altimeter data in SAR mode. Adv. Space Res. doi: http://dx.doi.org/10.1016/j.asr.2015.02.014, 2015. 2. Amarouche, L., Thibaut, P., Zanife, O. Z., and Dumont, J.-P. (2004). Improving the Jason-1 Ground Retracking to Better Account for Attitude Effects. Marine Geodesy, 27:171–197. 3. Amarouche L., Zanife O.Z., Steunou N., Vincent P. and Raizonville P. (2014). Jason-1 Altimeter Ground Processing Look-up Correction Tables, Marine Geodesy, 27:3-4, 409-431, doi:10.1080/01490410490902133. 4. Thibaut P., Amarouche L., Zanife O.Z., Steunou N., Vincent P. and Raizonville P. (2014). Jason-1 Altimeter Ground Processing Look-up Correction Tables, Marine Geodesy, 27:3-4, 409-431, doi:10.1080/01490410490902133. 5. Scharroo, R. (2014). Algorithm theoretical basis document 0.3

Figure 8. Scatterplot of SWH at FINO-1 from PLRM without (square) and with LUT (circle orange), numerical (circle black) and SAR (triangle) CONCLUSIONS The main result of this work is the quantification of the effects of the Look Up Table corrections when applied to PLRM. The regional cross-validation analysis over the four years confirms the previous results corresponding to a time interval of two years [1]. The effect of the LUT is relevant in the PLRM SWH, and leads to a reduction of 10% of the RMS difference between SAR and PLRM when is applied. Precision and accuracy of SWH increase using LUT. The best agreement with SAR SWH (standard deviation 18 cm) is obtained with the numerical retracker. The SAR data agree at best with the in-situ data. No relevant changes are seen in the range and wind speed parameters. Differences are higher at small SWHs and reduced significantly with applied LUTcorr .

6. Smith, W. H. F. and Scharroo, R. (2014). Waveform aliasing in satellite radar altimetry. IEEE Transactions on GEOSCIENCE and REMOTE SENSING, 53:1671–1682. 7. Coleman, T.F. and Y. Li (1996), "An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds," SIAM Journal on Optimization, 6, 418-445 8. Coleman, T.F. and Y. Li (1994). On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds, Mathematical Programming, 67, 2, 189-224 9. Dinardo S., Lucas B., Benveniste J. (2014). Cryosat-2 SAR processing on demand service at ESA G-POD,” in Proc. Conf. on Big Data from Space 10. Marquardt, D. (1963). An Algorithm for LeastSquares, Estimation of Nonlinear Parameters. SIAM J. Appl. Math. 11, 431-441, 1963.