Kevin R. James

A method for approximating acoustic-field-amplitude uncertainty caused by environmental uncertainties

In underwater acoustics, the accuracy of computational field predictions is commonly limited by uncertainty in environmental parameters. An approximate technique for determining the probability density function (PDF) of computed field amplitude, A, from known environmental uncertainties is presented here. The technique can be applied to several, N, uncertain parameters simultaneously, requires N+1 field calculations, and can be used with any acoustic field model. The technique implicitly assumes independent input parameters and is based on finding the optimum spatial shift between field calculations completed at two different values of each uncertain parameter. This shift information is used to convert uncertain-environmental-parameter distributions into PDF(A). The techniques accuracy is good when the shifted fields match well. Its accuracy is evaluated in range-independent underwater sound channels via an L(1) error-norm defined between approximate and numerically converged results for PDF(A). In 50-m- and 100-m-deep sound channels with 0.5% uncertainty in depth (N=1) at frequencies between 100 and 800 Hz, and for ranges from 1 to 8 km, 95% of the approximate field-amplitude distributions generated L(1) values less than 0.52 using only two field calculations. Obtaining comparable accuracy from traditional methods requires of order 10 field calculations and up to 10(N) when N>1.

A probability density function method for acoustic field uncertainty analysis

Acoustic field predictions, whether analytical or computational, rely on knowledge of the environmental, boundary, and initial conditions. When knowledge of these conditions is uncertain, acoustic field predictions will also be uncertain, even if the techniques for field prediction are perfect. Quantifying acoustic field uncertainty is important for applications that require accurate field amplitude and phase predictions, like matched-field techniques for sonar, nondestructive evaluation, bio-medical ultrasound, and atmospheric remote sensing. Drawing on prior turbulence research, this paper describes how an evolution equation for the probability density function (PDF) of the predicted acoustic field can be derived and used to quantify predicted-acoustic-field uncertainties arising from uncertain environmental, boundary, or initial conditions. Example calculations are presented in one and two spatial dimensions for the one-point PDF for the real and imaginary parts of a harmonic field, and show that predi...

Pekeris waveguide comparisons of methods for predicting acoustic field amplitude uncertainty caused by a spatially uniform environmental uncertainty (L)a)

Acoustic field calculations in underwater environments are often uncertain because the environmental parameters required for such calculations are uncertain. This letter compares the accuracy of direct simulations, the field shifting approximation, and polynomial chaos expansions for predicting acoustic amplitude uncertainty in 100-m-deep Pekeris waveguides having spatially uniform uncertain water-column sound speed. When this sound speed is Gaussian-distributed with a standard deviation of 1 m/s, direct simulations and polynomial chaos expansions, based on 21 field calculations, are more accurate than the field shifting approximation, based on two field calculations. This ranking reverses as the sound-speed standard deviation increases to 20 m/s.

Modeling uncertain source depth in range-dependent environments

Efficient and accurate estimation of the uncertainty in a transmission loss calculation is important for tactical applications of underwater acoustic propagation calculations. Uncertainty in source depth can contribute significantly to the overall transmission loss uncertainty. The unique relationship between source depth and transmission loss motivates a different approach to uncertainty estimation than that used for other environmental and sound-channel parameters. Prior research has shown that in a range-independent environment, source depth uncertainty can be efficiently modeled using the principles of reciprocity. This presentation describes a new approach to uncertainty estimation in range-dependent environments, based on the assumption that the relationship between source depth and transmission loss is approximately governed by the adiabatic approximation on a local scale. Transmission loss predictions are taken from RAMGEO results to solve for the unknowns in the resulting approximate formulation....

A comparison of acoustic uncertainty approximations in a Pekeris waveguide.

Uncertainty in environmental parameters is often the dominant source of error in underwater acoustic field predictions. This presentation provides comparisons of three techniques for assessing uncertainty in a predicted acoustic field caused by uncertainty in environmental parameters: field shifting, polynomial chaos expansion, and coarse uniformly sampled direct simulations. The uncertainty assessments are performed in a 100‐m deep Pekeris waveguide with an uncertain water‐column sound speed for frequencies of 100 Hz to 1 kHz at ranges of 1 to 10 km with a variety of common bottom types. The accuracy and computational efficiency of each approximation are quantified in terms of an absolute‐difference error norm for the probability density function (PDF) of acoustic field amplitude and the number of acoustic field calculations necessary to predict this PDF, respectively. In all cases, the true field‐amplitude PDF is determined from numerically converged direct numerical simulations. The strengths and limit...

A comparison of methods for approximating acoustic uncertainty in underwater sound channels

The accuracy and reliability of acoustic field calculations are often determined by uncertainty in environmental parameters. There are currently several approaches for predicting calculated‐acoustic‐field uncertainty arising from environmental uncertainties. This presentation outlines the current development of a method for predicting acoustic uncertainty based on correlations between variations in an uncertain parameter and spatial shifts within a calculated acoustic field. The results of this technique are compared with several other modern methods for predicting acoustic uncertainty such as direct sampling of environmental parameters, linearization and higher‐order finite difference approaches, the adjoint method for approximating derivatives of the acoustic field with respect to environmental parameters, and polynomial chaos methods. The advantages and limitations of each technique are presented for range‐independent shallow‐ocean sound channels at nominal ranges of 1 to 10 km and frequencies from 100...

A robust method for approximating acoustic field uncertainty in underwater sound channels

Environmental uncertainty in underwater sound channels is often the dominant source of error in acoustic field predictions. Standard Monte Carlo techniques to quantify the effect of this uncertainty are prohibitively time consuming for many applications. This presentation evaluates an alternative method for approximating the relationship between uncertain environmental parameters and uncertainty in the predicted field. Using FFT techniques, the effect of small changes in uncertain parameters is correlated to small spatial displacements of the original field. By appropriately extrapolating this relationship between uncertain parameters and the resulting field, an approximate probability distribution can be calculated for the field amplitude. A variety of test cases are shown to evaluate the accuracy of this method in underwater sound channels. Several uncertain parameters are addressed, including bottom characteristics, water column depth, source characteristics, and water column properties. Application of...

Approximating acoustic field uncertainty in underwater sound channels

Precise acoustic field prediction relies on accurate knowledge of many environmental parameters. Uncertainty in any of the inputs to a propagation routine leads to uncertainty in the predicted field. The most accurate but time‐consuming method of assessing the resulting field uncertainty is Monte Carlo simulation, which requires calculating the field with many possible combinations of inputs. This presentation covers a method to approximately determine the uncertainty in the predicted field, using only one additional field calculation for each uncertain parameter, when the input uncertainties are relatively small. The method relies on identifying how variations in uncertain input parameters can be mapped into spatial displacements in the original field calculation. Simultaneous uncertainty in multiple input parameters is handled using a net effective spatial displacement. In addition to expected values and standard deviations, the method allows for the generation of complete probability density functions ...

Probability density function predictions of acoustic uncertainty in an underwater sound channel

The utility of forward acoustic propagation models is often limited by incomplete knowledge of environmental parameters. For this reason, information about the resulting uncertainty of acoustic field calculations is a valuable addition to the solutions generated by current propagation routines. However, extensive Monte Carlo simulations are presently the only means for quantifying this uncertainty. This presentation explores alternative methods for obtaining a complete description of the uncertainty of a time‐harmonic acoustic field solution in the form of a probability density function (PDF). The isobaric contours of the field solution, including the uncertain parameters as independent variables, produce the relationship between acoustic pressure and the uncertain variables. In an ocean waveguide, the isobaric contours can be obtained approximately from a modal sum Green’s function using techniques similar to those that yield the waveguide invariant. Thus, information from a single field calculation can ...

Probability density function methods for uncertainty analysis in underwater acoustics

Forward modeling of underwater acoustic propagation is generally successful when environmental parameters and boundary conditions are known. Unfortunately, such information is seldom available at the requisite level of precision, and any imprecision introduces uncertainty into sound field predictions. Quantifying this sound‐field uncertainty is important for applications of acoustic propagation models such as matched‐field processing. This presentation describes a method for quantifying the underwater‐sound‐field uncertainty arising from imperfect knowledge of the environment and its boundaries. It is based on formulating and solving a transport equation for the joint probability density function (PDF) of the real and imaginary parts of a harmonic sound field. The appropriate equation is obtained by combining spatial derivatives of the PDF with physical laws drawn from guided wave mechanics. The inputs for solving the PDF‐transport equation are known or assumed distributions of the uncertain parameters. Solutions can be readily reduced to expected values, uncertainties, and confidence intervals for the predicted sound field. Results for simple test cases involving range‐independent isospeed underwater sound channels are considered, and compared to solutions obtained analytically or through Monte Carlo simulations. [Work sponsored by ONR.]