Data error estimation for matched-field geoacoustic inversion

Abstract

i n t r o d u c t i o nThe problem o f estimating seabed geoacoustic parameters by inverting m easured ocean acoustic fields has received considerable attention in recent years. M atched-field inversion (M FI) is based on searching for the set o f geoacoustic m odel parameters m that minimizes an objective function quantifying the misfit between m easured and m odelled acoustic fields. A num ber of approaches have been applied to this challenging nonlinear optim ization problem. In particular, adaptive simplex simulated annealing (A SSA) [1], a hybrid optim ization algorithm that combines local (gradientbased) downhill simplex moves within a fast simulated annealing global search, has proved highly effective for MFI.In a Bayesian formulation o f MFI, the objective function to be m inim ized is derived from the likelihood function corresponding to the assumed data uncertainty distribution. The likelihood depends on parameters describing data uncertainties (e.g., standard deviations) w hich are nuisance parameters in terms o f recovering seabed properties, but m ust be accounted for in a rigorous inversion. D ata uncertainties include both measurement errors (e.g., due to instrum entation and ambient noise) and theory errors (due to the simplified model parameterization and approximate acoustic propagation model). Theory errors in particular are generally not well known, and tend to increase w ith frequency due to the effects o f scattering, 3-D environmental variability, sensor location errors, etc. However, as m entioned above, data uncertainties are rarely w ell know n due to theory errors. The standard approach is to assume that the uncertainty weighting factor |df |2 /off is uniform over frequency, and minimize an objective functionHowever, this is often a poor assumption in practice [2]. A straightforward approach for unknown uncertainties is to explicitly estimate the standard deviations as part o f the inversion by minimizing the objective functionover m and o. The disadvantage to this approach is that it introduces F new unknown parameters O f, resulting in a more difficult inverse problem. A n alternative approach is to maximize the likelihood over Of, yielding the analytic solution THEORYFor acoustic data d f m easured at an N -sensor array at f= 1 ,F frequencies contam inated by independent, complex Gaussian-distributed errors w ith standard deviations Of, it can be shown that the likelihood function w hen source amplitude and phase are unknown is given by [2]whereSubstituting this back into the likelihood function leads (after some algebra) to an objective function E 4(m) = n F =1(1 -B f (m)).M inimizing this objective function treats the data standard deviations as implicit unknowns w ithout increasing the num ber o f parameters in the inversion. RESULTSThis section considers a synthetic study o f inversion performance for the various objective functions based on a shallow-water geoacoustic experiment carried out in the

Uncertainty analysis in matched-field geoacoustic inversions

“…A way to fix the defect of the likelihood function is to down sample the observations as adopted by Ref. 26. In the experimental data used here, due to the large separation of the array elements ͑5-m interelement spacing and 15 hydrophones in total͒, it was not necessary to down sample the data.…”