6.2 CRAMER–RAO BOUND

In all approaches, the mean square error induced by the estimator is

$MSE=\langle {\Vert \hat{\mathbf{\theta }}-\mathbf{\theta }\Vert }^2\rangle =\mathrm{cov}[\hat{\mathbf{\theta }}]+\mathbf{b}(\hat{\mathbf{\theta }})\cdot {\mathbf{b}}^{\mathrm{{\rm H}}}(\hat{\mathbf{\theta }})$(6.16)

$\mathrm{cov}(\hat{\mathbf{\theta }})\ge {\mathrm{C}}_{CRB}={\mathbf{J}}^{-1}(\mathbf{\theta }(\mathbf{{\rm A}}))$(6.17)

$\mathbf{J}(\mathbf{\theta })=-\langle \frac{\partial }{\partial \mathbf{\theta }}{(\frac{\partial }{\partial \mathbf{\theta }}\ln p(\mathbf{A}|\mathbf{\theta }))}^{\mathrm{T}}\rangle$(6.18)

$\langle {({\hat{\theta }}_p-{\theta }_p)}^2\rangle \ge J_{pp}^{-1}$(6.19)

6.3 LEAST SQUARE ESTIMATION

There are applications where it is necessary to estimate the values of a set of parameters based on the generation model of the deterministic part of the data but without a knowledge of the involved pdfs. In this case one can estimate the parameter values by minimizing the error between the data and the model, according to some metric. A common metric is the sum of the square of the error, and the estimation method is the least-squares (LS) technique. The LS is used in a large variety of applications due to its simplicity in the absence of any statistical model, and for this reason it can be considered an optimization technique for data-fitting rather than an estimation approach. Furthermore, LS has no proof of optimality, except for some special cases where LS coincides with MLE as for additive Gaussian noise, and it is the first pass approach in complex problems.

In radar remote sensing of parameters vector x, it is through a geophysical model function GMF, $g$;

$\mathbf{y}=g(\mathbf{x})$(6.20)

In general, $g$ is a complex and highly nonlinear function that relates the input vector (observation space) and the output vector (parameter space) includes an electromagnetic scattering model, dielectric model (e.g., soil and ocean surface), and wind-wave model (ocean), where the electromagnetic scattering model describes the relationship between the radar echo and the target parameters (both geometrical and electrical), while the dielectric model or the wind-wave model converts the electrical parameters to the physical parameters of interest (e.g., soil moisture content, ocean winds). Reminding from the uncertainties of the roughness parameters, and RCS measurements in previous chapters, what we discussed about (6.1) is an ideal case. In practice, uncertainties must, more or less, be attached to $\mathbf{y}$, $g$, and $\mathbf{x}$. Physically, the noise sources come from different aspects. For roughness parameters, please refer to Equations 2.40 and 2.41 in Chapter 2.

A data model, in a proper way, is to enable a mapping of measurement space to parameter space, which we are of interest. Here, following, we detail the procedures. To be more explicitly and realistic, Equation 6.20 is re-expressed by

$\mathbf{y}=\mathbf{W}\mathbf{x}+\mathbf{u}$(6.21)

From previous discussions, we may summarize the estimation error sources in terms of the remote sensing of surfaces as (1) the surface parameter uncertainty and (2) the measurement uncertainty, as described in Equation 6.21. Fur one often faces the situation where one measurement set must be mapped to more than one parameter set; i.e., an ill-posed problem. As shown in Figure 6.3 the parameter sets (object domain) correspond to the same measurement result $\mathbf{y}$ (data domain). If we want to determine one representative value $\hat{x}$, in the minimum squared error sense, a sample average of the known parameter sets will be used as the retrieval result. As in mapping problem, there always come with issues of uniqueness, existence, and stability. Regularization theory and neural networks with deep learning are often applied to resolve these problematic issues.

6.4 KALMAN FILTER-BASED ESTIMATION

It has been illustrated from previous chapters, the inverse mapping from radar measurements to surface parameters will be highly nonlinear and an analytic function does not usually exist. Indeed, the retrieval of surface parameters can be viewed as a problem of mappings from the measured signal domain onto the range of surface characteristics that quantify the observation [7–9]. To solve the problems of multidimensional retrieval, the application of a neural network to microwave remote sensing has been carried out by many investigators, because of its ability to adapt to geophysical multidimensionality and its noise robustness in realistic remote sensing. It becomes apparent that the parameter estimation problem invokes two issues that must be simultaneously solved. One is the need for an inversion scheme that provides the mapping, between the parameters to be inferred and the measurements, in a least-squares error sense. Because parameter estimation must come from a finite set of radar observations and measurements, an ill-posed must be solved. Neural networks seem to be well suited for these purposes. Another is the need for GMF models that relate the radar echoes and the geophysical parameters. Thus, the GMF includes two parts: the scattering model relating to the dependence of radar returns to surface roughness and permittivity; the dielectric model relating the permittivity to the moisture content under various conditions.

The Kalman filter (KF) is the optimal MMSE estimate of the state for time-varying linear systems and Gaussian processes in which the evolution of the state is described by a linear dynamic system. In a statistical sense, $\mathbf{x}$ constitutes a random variable due to spatially and temporally varying properties, such that

$\mathbf{x}={\mathbf{x}}_t+{\mathbf{x}}_n$(6.22)

$\mathrm{E}(\mathbf{x})=\mathrm{E}({\mathbf{x}}_t),\forall {\mathbf{x}}_n~\mathit{N}(0,{\sigma }_{{\mathbf{x}}_n}^2)$(6.23)

Note that the radar response is formed by, in general, the scattering matrix. It has been argued that [10] non-quadratic regularization is practically effective in minimizing the clutter while enhancing the target features via:

$\hat{\mathbf{x}}=\text{arg}\min \{{\Vert \mathbf{y}-\mathbf{W}\mathbf{x}\Vert }_2^2+{\gamma }^2{\Vert \mathbf{x}\Vert }_p^p\}$(6.24)

Direct solving of Equation 6.24 perhaps is possible but demands intensive computation resources. From preceding chapters, we also see that the scattering behavior, both pattern and strength is complicatedly determined, in a stochastic sense, by three surface parameters. Hence, in search of the cost function minima in Equation 6.24, we may seek a neural network approach. Perhaps one disadvantage of the neural network is that it constitutes a black box for most users. Extensive studies show that it is a powerful tool for handling complex problems involving bulky volume data in high dimensional feature space. The inverting parameters vector from measurements, a neural network offers an effective and efficient approach and will be detailed in what follows.

Modified from a multilayer perceptron (MLP), a dynamic learning neural network (DLNN) was proposed [11] and is adopted. Figure 6.4 schematically depicts the configuration of a dynamic learning neural network (DLNN). It features: (1) every node at the input layer and all hidden layers are fully connected to the output layer; (2) the activation function is removed from each output node; the output of the modified network can be characterized as the weighted sum of the polynomial basis vectors. Such modifications form a condensed model of the MLP in which the output is a weighted sum of compositions of polynomials. Hence, with measurement error matrix $\mathbf{u}$, as in Equation 6.1, $\mathbf{W}$ is the network weight matrix.

The network training or learning scheme, based on the Kalman filter technique [12] that lends itself to a highly dynamic and adaptive merit during the learning stage, is described below. To begin, the basic concept of Kalman filtering is briefly described, with notation given in Figure 6.5.

The measurement equation takes the form for one-step n:

${\mathbf{y}}_n=\mathbf{W}{\mathbf{x}}_n+{\mathbf{u}}_n$(6.25)

${\mathbf{x}}_{n+1}={\mathrm{\Phi }}_n{\mathbf{x}}_n+{\mathbf{B}}_n{\mathbf{v}}_n$(6.26)

$\mathrm{E}[{\mathbf{u}}_m{\mathbf{v}}_n^{\mathrm{t}}]=0,\forall m,n.$(6.27)

${\hat{\mathbf{x}}}_n={\tilde{\mathbf{x}}}_n+K_n({\mathbf{y}}_n-\mathbf{W}{\tilde{\mathbf{x}}}_n)$(6.28a)

${\tilde{\mathbf{x}}}_{n+1}={\mathrm{\Phi }}_n{\hat{\mathbf{x}}}_n$(6.28b)

$K_n={\tilde{\mathbf{P}}}_n{\mathbf{W}}^{\mathrm{t}}(\mathbf{W}{\tilde{\mathbf{P}}}_n{\mathbf{W}}^{\mathrm{t}}+{\mathbf{R}}_n)$(6.29a)

${\hat{\mathbf{P}}}_n={\tilde{\mathbf{P}}}_n-K_n{({\tilde{\mathbf{P}}}_n{\mathbf{W}}^{\mathrm{t}})}^{\mathrm{t}}$(6.29b)

${\tilde{\mathbf{P}}}_{n+1}={\mathrm{\Phi }}_n{\hat{\mathbf{P}}}_n{\mathrm{\Phi }}_n^{\mathrm{t}}+{\mathbf{B}}_n{\mathbf{Q}}_n{\mathbf{B}}_n^{\mathrm{t}}$(6.29c)

$\mathrm{E}[{\mathbf{u}}_m{\mathbf{u}}_n^{\mathrm{t}}]=\{\begin{array}{l}{\mathbf{R}}_n,m=n\hfill \\ 0,m\ne n\hfill \end{array}$(6.30a)

$\mathrm{E}[{\mathbf{v}}_m{\mathbf{v}}_n^{\mathrm{t}}]=\{\begin{array}{l}{\mathbf{Q}}_n,m=n\hfill \\ 0,m\ne n\hfill \end{array}$(6.30b)

${\mathbf{y}}_{\kappa }={\mathbf{w}}_{\kappa }\mathbf{x}(\kappa =1,2,\dots ,L)$(6.31)

$y_{\kappa }^i={\mathbf{w}}_{\kappa }^i\mathbf{x}+v_{\kappa }^i$(6.32)

${\mathbf{w}}_{\kappa }^{i+1}={\mathbf{w}}_{\kappa }^i{\mathbf{A}}^i+{\mathit{u}}_{\kappa }^i{\mathbf{B}}^i$(6.33)

${\hat{\mathbf{w}}}_{\kappa }^i={\tilde{\mathbf{w}}}_{\kappa }^i+{\mathbf{g}}_{\kappa }^i[d_{\kappa }^i-{\tilde{\mathbf{w}}}_{\kappa }^i\mathbf{x}](i=1,2,\dots ,N)$(6.34)

${\tilde{\mathbf{w}}}_{\kappa }^{i+1}={\hat{\mathbf{w}}}_{\kappa }^i{\mathbf{A}}^i$(6.35)

where $d_{\kappa }^i$ is the desired output, ${\tilde{\mathbf{w}}}_{\kappa }^i$ is the one-step predicted estimate, and ${\hat{\mathbf{w}}}_{\kappa }^i$ is the filter estimate of ${\mathbf{w}}_{\kappa }^i$, respectively, and ${\mathbf{g}}_{\kappa }^i$ is the computed Kalman gain, which is viewed as an adaptive learning rate and its computation is according to Equations 6.31–6.35.

From the previous chapter, the sensitivity analysis in response to radar scattering is essential. Knowing the parameter bounds is important to the network performance. A neural-based estimation seems powerful and yet efficient as long as training data is effective. The MLP together with learning by Kalman filter is expected to resolve the highly nonlinear, and complex decision boundary problems, as will be demonstrated in the following chapter.