Geometry Design for DOA Estimation in Seismic 2D-Arrays: Simulation Study

Sensor array geometry has a direct impact on the direction-of-arrival (DOA) estimation of a seismic signal. In this paper, we design a planar array that aims to optimize the DOA estimation of a narrowband signal in the sense of the minimum mean-squared-periodic-error (MSPE) obtained by the maximum a-posteriori (MAP) estimator of the DOA. We investigate the MSPE of the MAP estimator as a main design criterion and compare it with the criteria: 1) the cyclic Bayesian Cramér-Rao bound (CBCRB); and 2) the expected log-likelihood (ELL). The theoretical properties of these criteria are discussed. We show that minimizing the CBCRB is equivalent to maximizing the expected Fisher information matrix. Additionally, maximizing the ELL under a uniform prior is equivalent to minimizing the Kullback-Leibler divergence between the posterior PDF and its estimation. The criteria are compared across three different array geometries, specifically: small arrays, uniform circular arrays (UCAs), and concentric circular arrays (CCAs). Simulation results show that 1) direct MAP-MSPE optimization notably exceeds CBCRB- and ELL-based designs, especially in small arrays; 2) UCAs have suboptimal performance compared to non-circular arrays in many scenarios; 3) under the MAP-MSPE criterion, CCAs match unconstrained design performance with lower computational complexity, making them preferable for smaller arrays; 4) for CCAs and larger UCAs, CBCRB and MAP-MSPE designs yield similar results, while the ELL design excels in the case of small UCAs. Our results highlight the need for selecting suitable array geometries and design criteria in accordance with the scenario and array size in order to achieve the best DOA estimation results.