D ata E rror E stimation in M atched - field G eoacoustic I nversion Stan E. Dosso, Michael J. Wilmut, and Jan Dettmer School of Earth and Ocean Sciences, University of Victoria, Victoria BC Canada V8W 3P6 1. introduction The problem of estimating seabed geoacoustic parameters by inverting measured ocean acoustic fields has received considerable attention in recent years. Matched-field inversion (MFI) is based on searching for the set of geoacoustic model parameters m that minimizes an objective function quantifying the misfit between measured and modelled acoustic fields. A number of approaches have been applied to this challenging nonlinear optimization problem. In particular, adaptive simplex simulated annealing (ASSA) [1], a hybrid optimization algorithm that combines local (gradient- based) downhill simplex moves within a fast simulated annealing global search, has proved highly effective for MFI. In a Bayesian formulation of MFI, the objective function to be minimized 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) which are nuisance parameters in terms of recovering seabed properties, but must be accounted for in a rigorous inversion. Data uncertainties include both measurement errors (e.g., due to instrumentation 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 with frequency due to the effects of scattering, 3-D environmental variability, sensor location errors, etc. [2]. This paper derives several likelihood-based objective functions and examines their performance in MFI of acoustic data with unknown, frequency-dependent uncertainties. is the normalized Bartlett (linear) correlator, df (m) are the data predicted for model m, and T denotes conjugate transpose. Maximum-likelihood parameter estimates are obtained by maximizing the likelihood over m. If the standard deviations Of are known, this is equivalent to minimizing the objective function E:(m) = 1^,(1 - Bf (m))|d f |2 la ) . However, as mentioned above, data uncertainties are rarely well known 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 function E 2(m) = X f =1[ - Bf (m)]' However, this is often a poor assumption in practice [2]. A straightforward approach for unknown uncertainties is to explicitly estimate the standard deviations as part of the inversion by minimizing the objective function E3(m = Z f =1[(1 - Bf (m))| df |2 laf + N lna/] over 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. An alternative approach is to maximize the likelihood over Of , yielding the analytic solution af = (1 - B f (m)) | d f |2 l N . 2. THEORY For acoustic data d f measured at an N -sensor array at f=1,F frequencies contaminated by independent, complex Gaussian-distributed errors with standard deviations Of, it can be shown that the likelihood function when source amplitude and phase are unknown is given by [2] F 1 L(m o) = n -— ^ exp[-(1 - Bf (m))| d f |2 1 af ], f =1( —a f ) where Bf (m) =|d f (m)Td f |2 l | d f (m) |2| d f |2 Substituting this back into the likelihood function leads (after some algebra) to an objective function E 4(m) = n F = 1(1 - Bf (m)). Minimizing this objective function treats the data standard deviations as implicit unknowns without increasing the number of parameters in the inversion. 3. RESULTS This section considers a synthetic study of inversion performance for the various objective functions based on a shallow-water geoacoustic experiment carried out in the Canadian Acoustics / Acoustique canadienne Vol. 32 No. 3 (2004) - 192