Assimilation of ERS SAR Wave Spectra in a Wave Prediction Model

Abstract

Inverted wave spectra from ERS Wave Mode products have been applied in the ocean wave forecast service at the Norwegian Meteorological Institute. Elements of the operational system are explained briefly, and examples of the impact of including the SAR wave data in the operational wave model runs are showed. It turns out that due to the intermittency of the data, the average improvement is minor for the areas covered by the wave model. However, it is shown that in those cases where there is impact, this is mainly positive.

Keywords: SAR, assimilation, wave model

Introduction

This paper describes the value of inverted SAR wave spectra in the wave forecasting service for the North and Norwegian Sea carried out by the Norwegian Meteorological Institute (DNMI) supported by the Norwegian Space Centre. In order to use the data operationally, the information needs to be interpolated in space and time from a very limited number of observation points to the larger area of interest. This is done by assimilating the data in the operational wave model at DNMI. By the assimilation the model wave fields are updated to produce new wave fields as initial condition for the next model forecast. In later years wave observations have become available from satellites, and work has been done to utilize this information to improve wave model results. At present altimeter wave heights from the European Remote Sensing satellite (ERS) are operationally assimilated at several centres, among them at DNMI (Breivik and Reistad, 1994). Advanced methods, including wave model dynamics in the assimilation scheme attempting consistent analysis of wave field and wind history, have also been developed (e.g. Heras et al. 1994, Bauer et al. 1996). These are computationally expensive and so far not in operational use. A further development in assimilation of satellite wave data is to utilize the full spectral information from the Synthetic Aperture Radar (SAR). A method for assimilation of two dimensional wave spectra by subdivision of the full spectra into separate partitions suitable for optimal interpolation has been developed by Hasselmann et al. (1994). In the present study an other sequential method for assimilation of the full two dimensional SAR wave spectra, is presented. Unfortunately, the currently inverted SAR wave spectra are not independent from the wave model spectra since the SAR inversion algorithm depends on a priori information from the wave model. Previous studies (Breivik et al. 1995) have shown that there is nearly a full correlation between model and inverted SAR wave spectrum parameters in the wind sea part. This is actually expected since the SAR information fades out at about 0.12-0.15Hz. However in the swell part of the spectrum the SAR data contain new information. In special cases, e.g. with poor wind history and correspondingly poorly defined swell, significant improvement from the assimilation might be expected.

SAR Observations and the Wave Model

DNMI receives the AMI (Active Microwave Instrument) Wave Mode product in near real time on the GTS. The product is described by Krogstad et al. (1994). A second wave product has also been received in near real time on request from Tromsø Satellite Station (TSS) directly based on the high resolution SAR images. This paper will however concentrate on the use of Wave Mode. The results in assimilation proved to very similar for Wave Mode data and TSS Image data (Breivik et al. 1996). The Wave Mode product is a SAR image spectrum obtained from a 3 look, amplitude averaged 5 times 8 km imagette recorded every 200 km along the track. The resolution of the imagette is similar to the regular full resolution images (20x16m). About every second recording is distributed daily on GTS. The spectrum consists of spectral values given in 12 directional and 12 wavenumber bins ranging approximately from 100 to 1000m in wavelength and covering a 180 degrees sector. The inversion procedure is based on a quasi-linear version of the Hasselmann forward ocean-SAR spectral transform expressing the SAR image modulation spectrum, (see Krogstad et al., 1994) for details. The ocean-SAR inversion amounts to finding an ocean wave spectrum such that when the forward ocean-SAR transform is applied, the result conforms to the observed SAR spectrum. Unfortunately, it has been realised that the ERS-1 Wave Mode spectra suffer from an inferior processing which complicates its applicability. The ocean-SAR inversion theory, has been developed for the spectrum of the relative wave modulation seen in the SAR image. However, the received spectrum is a linear combination of the wave modulation spectrum and the speckle spectrum with unknown scaling and relative weight between the two. Moreover, the imagettes are look-summed in amplitude instead of in intensity as required by the ocean-SAR theory. The speckle spectrum has been determined by analysis of a set of a separate set of imagettes and the scaling procedure is described in Krogstad (1994). An automatic control procedure is run operationally prior to the inversions in order to remove excessive low wavenumber noise and spectra with no definite wave features. As a result of this data control, approximately 50 per cent of the spectra are rejected. In September, October and November 1995 a total of 217 ERS-2 wave spectra and 583 ERS-1 wave spectra passed the data control. This gave approximately 4.5 observations per assimilation cycle within the wave model area. The wave model WINCH runs on a rectangular grid with a fine mesh (~75 km) covering the North Sea, Norwegian Sea, and the Barents Sea, and a coarse mesh (~150 km) covering the western parts of the North Atlantic. WINCH is a second generation, deep water, spectral discrete model. The wave spectrum is divided into 24 directional bands with 15 degrees bandwidth and 15 frequency bands ranging from 0.04 Hz to 0.24 Hz. Each spectral bin is propagated individually with the energy input proportional to the existing energy multiplied by a function of the friction velocity and the angle between the wind and wave directions. The wind input is obtained from the operational weather forecast model HIRLAM. The a priori wave information for inversion procedure is a wave spectrum taken from a neighbouring grid point of WINCH. The 180 degrees directional ambiguity in the SAR image spectrum is removed by reading probable directions from the a priori spectrum and selecting the weight functions accordingly. Outside the azimuth pass band and for high range wave numbers in the SAR spectrum there is no wave-related information and it is reasonable to keep the a priori wave spectrum unchanged. Inside this domain it is possible to give the a priori wave spectrum some weight away from the peaks in the SAR spectrum, but in the present study the SAR spectrum determines the wave spectrum completely within this domain. The wave model spectra are defined for the frequency range from 0.04-0.3Hz, whereas the inverted AMI Wave Mode spectra only cover 0.04-0.125Hz. A final merging of the inverted and a priori spectra therefore has to be carried out. The inverted spectrum is first converted to the wave model format and extended above 0.125 Hz using a f -4 frequency spectrum and a constant directional distribution equal to that at 0.125Hz. In the final spectrum, the inverted spectrum is used below 0.125Hz, the a priori wave model spectrum above 0.153Hz, and a gradual transition in between. The operational SAR-to- wave spectrum inversion procedure is run prior to the analysis. A previous wave model forecast (+2 to +12 hours) is used as a priori information. For the period from 1 September to 30 November 1995, inverted wave spectra based on AMI Wave Mode both from ERS-1 and -2 have been compared with collocated wave spectra from WINCH. Statistics are presented for total sea, wind sea and swell, separately. Spectral energy below the wind sustained frequencies in the wave model is treated as swell. The wave parameters presented in Tables 1 for ERS-1 are significant wave height, SWH (m) and peak period, PP (sec.).

Since the SAR inversion only affects frequencies below ~.13Hz, the correlation between inverted and a priori spectra is quite high in the wind sea part. Even the correlation for swell SWH is above 0.9. The standard deviation of difference, Sde, for wind sea SWH is 0.22m as opposed to 0.48m for swell SWH. There is only small negative mean differences. The Peak Period is longer for ERS swell compared to WINCH, which is realistic since operational experience with the WINCH model indicate that the model tends to underestimate the swell period. A scatter plot for the swell part of peak direction, PD is shown on Fig. 1. Somewhat surprisingly, the diagram shows very little scatter. Since the peak direction is fully determined by the SAR spectrum, there seems to be little additional directional information in the SAR data not already available in the a priori model spectrum. The results for ERS-2 Wave Mode are very similar as for ERS-1 (Breivik et al. 1996).

The analysis

The assimilation scheme is based on a modification of the successive correction method as proposed by Bratseth (1986). The basic idea is similar to Statistical Interpolation. At DNMI a similar scheme is in operational use in the atmospheric analysis/forecast model system. A simplified version of the scheme was developed for the assimilation of altimeter wave heights (Breivik and Reistad, 1994). Since wave height depends only on the total wave energy, some pre-assumptions about the shape of the wave spectrum have to be made, and the wave height assimilation therefore works as a scaling of the wave energy while preserving the ratio between wind sea and swell. With the inverted SAR spectra full directional wave spectra are available, and the assimilation may be carried out without predefined assumptions about the spectral shape, and the assimilation scheme has thus been further developed for assimilation of full wave spectra observations. The analysis starts with a first guess field of wave spectra derived from a wave model short forecast. This field is then corrected by the available observations. The method is based on the following two iterative equations:

Here is wave energy as a function of frequency and direction. Subscripts i and j refer to observation points while subscript x refers to a grid point. Superscripts O and A refer to observed and analysed values, N is the number of observations in use, and k is an iteration counter. The iterations are initiated by a spatial interpolation of the first guess value to the observation points. The analysis weights, aij and axj, are functions of model and observation error co-variances. We assume that the errors in different frequency/direction bins are uncorrelated, which means that the analysis treats each (n, q)-bin independently. We also assume that model and observation errors are uncorrelated. The weighting functions are dependent on frequency, n,

Here m(n)ij is the model error co-variance function, sP(n) the standard deviation of the model error, and rij is the distance between observations. b(n) is the radius of influence which is a typical distance for which the model errors become decorrelated. Furthermore, d(n)ij is the observation error covariance function. Since there is no spatial overlap between the SAR observations, we assume that the errors in different observation points are uncorrelated, i.e. that d(n)ij = 0 for i not equal j. Observation error covariance then reduces to the observation variance, d(n)ii = (so(n) )2, where so(n) is the standard deviation of the observation errors. M(n)j is calculated for each observation as a function of m(n)ij and d(n)ij in a way that makes Eq. (2) converge. It can be shown that this method makes ExA converge towards the results of Statistical Interpolation (Bratseth, 1986). In the formulation of the weight functions it is assumed that the model and observation errors are uncorrelated. This might be wrong since the model background has been used in the SAR wave spectra retrieval for high frequencies as described in the previous sections. To deal with this, the analysis weights a(n) are reduced for high frequencies where the background has been used in the inversion. The influence radius b(n) vary with frequency so that long-wave information influence a larger area of the analysed field than shorter waves. b(n) is ranging from 300km for n=0.04Hz, to 60km at 0.24Hz. The size of the expected standard deviation of the errors, sP(n) and so(n) determine the relative influence of model and observation at the observation point. In the present study sP(n) = 3so(n) where sP(n) varies smoothly such that sP(n)=1 for n= 0.04Hz, increasing to 5.2 for n=0.1Hz, and dropping again to 1 for n=0.24Hz. An alternative way to perform the analysis is to divide the spectrum into a small number of distinct partitions characterized by bulk wave parameters like the total energy, mean frequency and direction, and then perform the analysis on each of the partitions using Statistical Interpolation. This method is described by Hasselmann et al. (1994) and implemented and tested on several cases with spectral buoy observations by Voorrips et al. (1996). The method drastically reduces the number of variables to be analysed. The partitions of the spectrum are considered to represent separate wave systems originating from independent meteorological systems, and the model errors of different partitions are assumed to be uncorrelated. The main problem with this partition method is how to assign each partition of the observed spectrum to the correct partition of the first guess spectrum especially when there are large differences between observations and the first guess, cases which will be of major interest for the analysis. The modified successive correction scheme used in the present work is computationally less demanding, and there is no big need to reduce the number of variables. However, in the present scheme there is no attempt to include the error correlations between each spectral bin, and the cross correlations are set to zero. Our experience with the system, evaluating a lot of analysed spectra proves that this works satisfactorily.

Case studies

Several cases have been studied in more detail in order to investigate the impact of assimilating SAR wave spectra. Figures are presented in Breivik et al, 1996, and in the proceeding following these symposium. Here we just give some general comments. The wave model were run forward in time using wind prognoses from the weather forecast model HIRLAM. SAR data were assimilated every second hour when available. In most cases the period and the relative distribution of energy in the spectra were quite similar in WINCH and AMI Wave Mode. The total energy however may vary. In several cases the wave energy was significantly reduced by the assimilation. As foreseen the impact were mainly in the swell part of the spectra. Studying the impact on the forecasts 2 - 24 hours after assimilation it is seen that the impact was propagated along with the group velocity of the longer waves and gradually reduced. The radius of the influence area was seen to be in accordance with the values specified for b above. In an example from 22 November 1995 significant wave height was reduced about 1.3 m in an area outside Northern Norway. The area influenced by the assimilation propagated towards ice covered areas south-east of Svalbard. Even 16 hours after the assimilation there was more than 0.7m difference in wave height. This case, and similar cases chosen because of significant differences between model first guess and observations, shows that the information is well assimilated with impact in an area around the observations as given by the influence radius in the analysis. The information is further propagated by the model up to 24 hours after assimilation. The impact gradually decreases as new wind input enters the model, or the wave systems reach the coast or areas covered by sea ice.

Results from paralell run

In order to assess the impact of the SAR data, an additional operational wave model routine including assimilation of the inverted SAR spectra was run parallel to the regular operational wave model forecasts from October 1995 to March 1996. The results from the assimilation run, ass, and the operational run, noass, was compared to in situ wave measurements from two platforms in the North Sea, Gullfaks and Ekofisk, and with the ERS-2 altimeter wave height measurements. The in situ wave measurements are three-hourly and are given in half metres. At Ekofisk waves are measured by a wave buoy.

Table 2 shows the mean significant wave height, SWH, from the measurements and the model data, the RMS difference, and the correlation coefficient for a validation period 1.11.1995 to 29.1.1996. The statistical parameters are given at analysis time and for four different forecast intervals (3-12 hours, 15-24 hours, 27-36 hours, and 39- 48 hours). On average there is only a very small positive impact of the assimilation of SAR data. The limited impact is certainly due to the relatively small number of SAR observations, approximately 9 observations per day in the whole integration area for the model. Moreover the influence of the observations is roughly limited in space by the size of the influence radius. To get a significant impact on specific locations as the North Sea platforms one actually needs nearby SAR observations which indeed are very few.

In Table 3 only situations from Ekofisk North Sea platform where the difference between ass and noass exceeds 0.3 m are included in the statistics. There are few cases for forecasts times longer than 24 hours only 2. The improvements in RMS and correlation, achieved by the assimilation, are slightly larger in these cases compared to the previous. The two runs have also been compared to SWH measurements from the ERS-2 altimeter. Results from 1 October 1995 to 9 January 1996 are presented in Table 4. Only cases where the absolute difference between ass and noass exceed 0.5 m is included in the statistics which means that only a very small part of the total number of co-locations between altimeter observations and model results are used. As seen from the table, the assimilation in these cases gives a small, but positive impact. For the short prognosis the standard deviation of errors is reduced by 0.13 m, and the correlation coefficient is increased by 0.08. For the long prognosis the bias in these cases is reduced by 0.43 m which is quite significant. This shows that at the few places where SAR-data are available they have significant potential for improving the wave forecast.

Conclusions

In the present paper we have considered the impact of including inverted SAR spectra in the operational wave model at the Norwegian Meteorological Institute. It appears to be useful information about wave height and wave period in the swell part of the inverted spectra, but, somewhat surprisingly, relatively little new directional wave information compared to the wave model WINCH. Operational use of inverted SAR spectra needs a careful data control, and the objective data control implemented in the system works satisfactory and eliminates spectra that are un-physical and contains a lot of noise. The relatively simple assimilation method, where the energy in every frequency and directional bin is analysed separately, works well in an operational environment. The new information gained from the SAR spectra are in some cases kept in the model for more than 24 hours as shown in Section 4 . However, usually the differences from the background analyses without SAR are small. Evaluated against independent wave measurements in the North Sea and ERS-2 altimeter wave heights, the assimilation is only to a limited extent capable of improving the average results at given locations. However the in the cases where there is impact this is mainly positive. There are several reasons for the low averaged impact. First, there are very few observations, typically 2-3 observations used in each assimilation cycle. Secondly, too much information from the wave model is used in the derivation of SAR wave spectra, and the amount of new information included in a wave model spectrum by assimilation of a SAR spectrum is limited. A third important reason for limited impact is the fact that the model (and real world) wave fields are strongly controlled by the input wind field. A short time after the start of the wave forecast, the input wind field has much more influence on the sea state than the initial state itself has. The effect of correcting the initial state therefore gradually dies out. It is only for swell, propagating undisturbed by the wind, that new information introduced by the assimilation of observed spectra will be kept in the model for a long time. Improved results might be expected when assimilation methods taking the time development of the total wind and wave field into account in a consistent manner are operationally available. The somewhat negative conclusion is that assimilation of SAR sea state data within the present analysis system and with the present ERS data coverage, is not sufficiently useful to warrant an operational setup for the Norwegian waters. This conclusion may however also change if the ongoing work with deriving wave spectra from SAR without any prior model wave information, the cross spectrum method, is successful (Engen et al., 1995). It can not be excluded that such new SAR processing methods together with increased data coverage can enable an assimilation system for sea state with a significant positive impact from SAR data. The more positive impact from assimilation of radar altimeter wave heights (Breivik and Reistad, 1994), for which the data coverage is much higher, does show a potential for observations to improve wave analysis and forecasts. This potential is presently not fully realized by the ERS SAR observations.

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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