Cramér-Rao bounds for radar altimeter waveforms

The pseudo maximum likelihood estimator allows one to estimate the unknown parameters of Brown’s model for altimeter waveforms. However, the optimality of this estimator, for instance in terms of minimizing the mean square errors of the unknown parameters is not guarantied. Thus it is not clear whether there is some space for developing new estimators for the unknown parameters of altimetric signals. This paper derives the Cramér-Rao lower bounds of the parameters associated to Brown’s model. These bounds provide the minimum variances of any unbiased estimator of these parameters, i.e. a reference in terms of estimation error. A comparison between the mean square errors of the standard estimators and Cramér-Rao bounds allows one to evaluate the potential gain (in terms of estimation variance) that could be achieved with new estimation strategies. 1. PROBLEM STATEMENT AND DATA MODEL Altimeters such as Poseidon-2 on Jason-1 and Poseidon-3 on Jason2 provide useful information regarding the sea surface around the Earth. Altimeters send pulses which are frequency linear modulated and transmitted toward the ocean surface at a given pulse repetition frequency. After reflection on the sea surface, these pulses are backscattered and received by the altimeter. The formation of the resulting altimeter echoes (also called return powers) is illustrated in figure 1 extracted from [1]. Three distinct regions can be highlighted in the received altimeter signal: the first region (“thermal noise only” region) corresponds to the thermal noise generated by the altimeter before any return of the transmitted signal from the ocean surface (figure 1-a). The second part called the “leading edge” region contains all information about the ocean surface parameter and the altimeter height (figure 1-b and -c). Finally, the last part of the received signal referred to as a ”trailing edge” region (figure 1-d) is due to return power from points outside the pulselimited circle. Altimeter signals can be used to estimate many interesting ocean parameters, such as the significant wave height or the range, using a retracking algorithm [2]. This estimation assumes the received altimeter waveform can be modeled accurately by Brown’s model [3], [4]. A simplified formulation of Brown’s model assumes that the received altimeter waveform is parameterized by three parameters: the amplitude Pu, the epoch τ and the significant wave height H . The resulting altimeter waveform denoted as x(t) can be written x(t) = Pu 2 1 + erf t− τ − ασ c √ 2σc e −α t−τ− ασ 2 c 2 + Pn (1) Fig. 1. Construction of a radar altimeter waveform. where σ c = H 2c 2 + σ p, erf (t) = 2 √ π R t 0 e 2 dz stands for the Gaussian error function, c denotes the light speed, α and σ p are two known parameters depending on the satellite and Pn is the instrument thermal noise. The retracking algorithm estimates the thermal noise from the first data samples and subtracts the estimate from (1). As a consequence, the additive noise Pn can be removed from the model (1) with very good approximation. The received signal x(t) is sampled with the sampling period Ts, yielding xk = Pu 2 " 1 + erf u− τ − ρH √ 2 p μH + σ p !# e 2 , (2) where xk = x(kTs) and the following notations have been used u = kTs − ασ p, v = −αkTs + ασ p 2 , ρ = α 4c , μ = 1 4c , δ = α 8c . Figure 2 shows the waveform model and the influence of the three parameters Pu, τ and H . The epoch, τ corresponds to the central point of the “leading edge”. The amplitude, Pu represents the amplitude of the waveform, while the significant wave height H is related to the slope of the “leading edge”. Altimeter data are corrupted by multiplicative speckle noise. In order to reduce the influence of this noise affecting each individual echo, a sequence 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100