Thode Aaron, Zanolin Michele, Naftali Eran, Ingram Ian, Ratilal Purnima, Makris Nicholas C
Ocean Engineering Department, Massachusetts Institute of Technology, Cambridge 02139, USA.
J Acoust Soc Am. 2002 Nov;112(5 Pt 1):1890-910. doi: 10.1121/1.1496765.
Analytic expressions for the first order bias and second order covariance of a maximum-likelihood estimate (MLE) are applied to the problem of localizing an acoustic source in range and depth in a shallow water waveguide with a vertical hydrophone array. These expressions are then used to determine necessary conditions on sample size, or equivalently signal-to-noise ratio (SNR), for the localization MLE to become asymptotically unbiased and attain minimum variance as expressed by the Cramer-Rao lower bound (CRLB). These analytic expressions can be applied in a similar fashion to any ocean-acoustic inverse problem involving random data. Both deterministic and completely randomized signals embedded in independent and additive waveguide noise are investigated. As the energy ratio of received signal to additive noise (SANR) descends to the lower operational range of a typical passive localization system, source range and depth estimates exhibit significant biases and have variances that can exceed the CRLB by orders of magnitude. The spatial structure of the bias suggests that acoustic range and depth estimates tend to converge around particular range and depth cells for moderate SANR values.
将最大似然估计（MLE）的一阶偏差和二阶协方差的解析表达式应用于使用垂直水听器阵列在浅水波导中对声源进行距离和深度定位的问题。然后使用这些表达式来确定样本大小（或等效地，信噪比（SNR））的必要条件，以使定位MLE变得渐近无偏，并达到由克拉美罗下界（CRLB）表示的最小方差。这些解析表达式可以以类似的方式应用于任何涉及随机数据的海洋声学逆问题。研究了嵌入独立加性波导噪声中的确定性和完全随机信号。当接收信号与加性噪声的能量比（SANR）下降到典型被动定位系统的较低工作范围时，声源距离和深度估计会出现显著偏差，并且其方差可能比CRLB大几个数量级。偏差的空间结构表明，对于中等SANR值，声学距离和深度估计倾向于在特定的距离和深度单元周围收敛。
