A MODEL FOR WIND SPEED AND WAVE HEIGHT RETRIEVAL FROM RADAR ALTIMETER MEASUREMENTS

A numerical algorithm is here proposed to simulate the measurement of ocean wave height and wind speed by radar altimeter. The algorithm takes into account a realistic ocean wave spectrum and employs a parametric sub-model to simulate the electromagnetic waves scattering from the surface, thus providing greater flexibility and realism than the analytical models currently used. The model is shown to be consistent even when different backscatter models are employed, and numerical results prove that the measured value of the significant wave height does not depend upon the wave period while on the opposite wind speed measurements are significantly affected by it. The aim of this work is two-fold: on the one hand to provide a tool to improve the understanding and thus the accurate calibration of altimeter data, and on the other hand to develop a model which could eventually be used to simulate the sea state response as seen by the altimeter and by other microwave instruments. INTRODUCTION The measure of the heavy sea by means of altimeter radar is a widely consolidated practice: the technique is based on the emission of a short impulse of radio waves and on the analysis of the temporal behaviour of the reflected signal. Together with the measurement of the wave motion, wind speed on the surface of the sea is currently evaluated on the basis of the percentage of reflected energy; estimating this latter parameter is much harder task and – as it will be shown in the following it is also influenced by wave motion itself. The direct action of wind causes ripples to appear on the sea surface; such ripples are caused by the balance between surface tension and gravity and their height and length are of the order of a few centimetres, thus substantially smaller than the proper sea waves, generated by the balance between gravity and momentum. Ripples cause the backscattering of electromagnetic waves through the Bragg interference, a very complex problem to simulate. The algorithms for the extraction of wind data to leave from the pulse response (wave form) have been implemented and developed in the early eighties: a classic paper by Brown [1], described the general principles of the model. A new formulation of these procedures is given in [2] and in [3]; the measures of sea state are systematically collected and verified [4],[5]. In order to critically review these algorithms and possibly to improve their performances and to point out possible errors especially in the wind speed evaluation, a numerical model of simulation of the coupled sea surface/radar altimeter system is being developed by the Authors and is here described in its basic form. The same model, after proper testing and calibration, could eventually be developed and employed to simulate the sea surface data provide by other sensors such as the scatterometer and the SAR. METHODOLOGY The conceptual simulation model described here has been implemented through two separate computer codes: the first one simulates the sea surface by synthetically generating a space and time succession of sea surface heights, while the second simulates the altimeter response. The first program is based on widely experienced and now universally accepted techniques [6 ] [7],[8] It makes use of a JONSWAP sea spectrum (Fig.1), expressed by: ( ) { } ( ) [ ] ) ( / 25 . 1 exp ) 2 ( 2 2 / 1 / 5 . 0 exp 4 5 4 2 θ γ π π d f f f Ag S p f f p ∗ ∗ ∗ − = − − − − (1) d(θ) being the spreading function, where θ is the angle measured from the main wave propagation direction. Fig. 1. Wave spectrum The actual implementation of the procedure presented here is based on the large experience of wave synthesis techniques which have been developed in the past. In carrying out the numerical experiment shown in this paper, the spreading function has been taken as a constant, thus representing a one –dimensional (and thus long crested) sea The formula used for the simulations is therefore: The ai terms represents the spectral amplitude values, ; ωi and ki represent the terms of angular frequency and wave number which are connected by the linear dispersion relation, while the terms φi represent the pseudo-random phase angles through which the pseudo-random realisations of a given of sea state are produced. [9] The consistency of results to the number of realisations, and the number of spectral components was checked [10] in order to limit the computational effort and also in order to validate the procedure. In order to evaluate the performance of the data retrieval algorithm for different sea states, tests were performed for various values of the characteristic wave height Hs, which is related to the total spectral energy and therefore to the A value in (1), ( ) ( ) ) 2 ( cos ∑ + + = i i i i i x k t a x φ ω η 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 fp A i 2 Hs = 3m, fp = 1/6 sec and the spectrum peak period Tp (Tp=1/fp). The other parameters such as: ω (shape parameter), γ (amplification parameter), α (function of the parameter of the fetch) have been kept constant. Given the very short altimeter pulse length (of the order of 3 ns) as compared to even the shortest time periods in the sea spectrum, all the realisations are stationary, i.e. based on a fixed time t. A single realisation is therefore an array of η values at fixed intervals dx, which represents a stretch of sea of length L (Fig. 2). Fig. 2. Instantaneous water height The second software module simulates the sea surface altimeter response in two steps: first the pulse response Ir(t). is calculated, and then the intensity of the reflected signal Io (t) (wave form) is obtained by simply convoluting the time history of the signal emitted by the antenna Ifs(t) with Ir(t). The pulse response is calculated by dividing the signal energy Io into segments of intensity Io/L x dx, L being the footprint length; for each segment the backscattering coefficient σo is then calculated and the reflected signal is delayed it according to the distance from the single element to the altimeter. In our model the backscattered energy by the elementary surface of length dx is numerically calculated as a function of the incidence angle θ and therefore of the local sea slope dη/dx. On of the main reason for implementing a numerical – rather than an analytical – model is to allow a wide flexibility in the choice of the backscattering sub model; in this paper the following equation is assumed for σo : Varying the alpha (α) parameter (Fig. 3) the shape of the diagram varies so as to represent a large variety of behaviours of the surface. ( ) [ ] ) 3 ( 90 sin ° − = θ α σ K o -200 -150 -100 -50 0 50 100 150 200 0 50 100 150 200 250 300 d (m) et a (c m ) Hs = 3m, fp = 1/6 sec