Conference Agenda

Forest SAR tomography is a key tool for monitoring biomass. It consists of processing a stack of coherent SAR images to retrieve several observables related to the biomass. One of the challenges of forest SAR tomography is that the shape of the forest’s reflectivity profile is unknown. Usually, some prior is introduced, e.g., the reflectivity profile is assumed to have a Gaussian or an exponential shape. However, the choice of this prior impacts the estimation performance, as an incorrect model induces estimation biases.

This communication focuses on estimating the mean height, the spread, and the total backscattered power of the forest reflectivity profile. It presents a new estimation method that does not require, nor assume, any shape for the reflectivity profile of the forest. Instead, the unknown reflectivity distribution is characterized by its central moments.

First, a discussion on the observability of the forest SAR tomography problem justifies the introduction of the moments to characterize the distribution of scatterers. The key idea is that the spread of the scatterers is small with respect to the system ambiguity, typically 5 meters vs. 90 meters. Therefore, the spectrum of the reflectivity density is smooth and can be approximated by the first terms of its Taylor expansion, which involves the moments of the distribution.

Based on this observation, a new model for the covariance matrix is obtained by cutting the Taylor expansion at some order D, which is a hyperparameter to set. In this model, the first D moments are (new) parameters to estimate, along with the usual parameters. Then, a new estimation algorithm is proposed by applying Covariance Matching Estimation Techniques (COMET) to fit the empirical covariance of the measurements with the proposed model using a least-squares minimization. The main advantage of this model is its computational complexity. All the parameters but one, the mean height of the forest, enter linearly in the model of the covariance matrix. Consequently, the estimation algorithm requires only the optimization of a single parameter. Moreover, as the model relies on the moments and does not assume any prior on the shape of the reflectivity profile, it is robust to incorrect modeling. However, the cut-off induces a misspecification and potentially estimation biases.

The performance of the algorithm is assessed in realistic SAR tomography simulations. Scenarios are proposed to estimate the forest profile with and without ground cancellation. The estimation algorithm is compared with the maximal-likelihood estimator (MLE), which assumes a shaping distribution. As expected, when well-specified, the MLE obtained the best performance, but it is outperformed by the proposed algorithm when the assumed shape is incorrect. Finally, the choice of the hyperparameter is discussed and a tradeoff is highlighted. Large cutting orders allow better reconstructions with smaller biases but are slower to converge, while small cutting orders are more robust but present larger biases.

Due to its robustness and its low complexity, the proposed algorithm is particularly suited for forest SAR tomography, where it can efficiently process wide areas. The next step will be to apply it to Biomass data.