Conditions for ocean acoustic matched field inversions to be unbiased and attain the Cramer–Rao bound

In recent years, a wide variety of acoustic techniques have been developed to probe the marine environment. These techniques typically require the nonlinear inversion of acoustic field data measured on a hydrophone array. The data, however, are often randomized by the addition of naturally occurring ambient noise or by fluctuations in the waveguide’s refractive index and boundaries. The nonlinear inversion of random data often leads to estimates with biases and variances that are difficult to quantify analytically. It has become popular in recent years to compute limiting bounds on the variance of these nonlinear estimators, since these bounds are usually much easier to obtain. A major problem, however, is that the bounds are only guaranteed to converge to the true variance of the estimator under sufficiently high signal-to-noise ratio (SNR). By applying higher order asymptotics, general expressions are obtained for the first-order bias and second-order variance of a general maximum-likelihood estimate. These expressions are applied in the matched field localization of a source in a shallow-water waveguide. The results quantitatively show that the Cramer–Rao bound provides an unrealistically optimistic estimate of the true variance in many practical scenarios, and the bias can be significant as the SNR passes below 0 dB.