Page 52 - Spring 2015

sound Propagation in the Atmospheric boundary layer
3. Errors reducible through better statistics and probabil- ity models.
4. Errors reducible through better numerical methods and grids.
Regarding the physics models (item 1), the basic physics un- derlying sound propagation outdoors are now well under- stood. Regarding data (item 2), as discussed in the previous section, it remains challenging if not impossible to charac- terize the environment at a sufficiently high spatial and tem- poral resolution so as to accurately predict outdoor sound propagation in a deterministic sense. The statistics and probability models (item 3) relate to our previous discus- sion of the spatial spectrum and simulation of atmospheric turbulence. However, these models, even though quite com- plicated mathematically, still neglect commonplace features such as ground property variations, sloping of the terrain, interactions of the flow with objects such as trees and build- ings, gravity waves, and intermittency of turbulence.
Numerical methods and grids (item 4) were not discussed much in this article beyond the Introduction. Much prog- ress has been made in numerical methods for sound propa- gation in recent decades, and this will likely continue to be an active and productive area of research. A key to utilizing numerical methods effectively will be to employ them in an overall predictive framework that addresses uncertainties in data and statistics for the environment (items 2 and 3). This is where stochastic methods such as Monte Carlo integration come into play. Conventional methods for propagating er- rors and uncertainties, based on linear expansions and small perturbations, usually do not apply because the relationship between the parameters and the predictions is complex and highly nonlinear. Monte Carlo methods are now frequently employed in other related fields such as numerical weather forecasting, where ensemble forecasting is regularly used to assess the uncertainties in the forecasts.
Considering the prediction problem broadly, the received sound level will depend on a set of stochastic variables, which can include many different inexactly known param- eters such as the source and receiver heights, the horizontal separation (range), the source level, wind speed and direc- tion, thermal stratification, ground impedance, vegetation, topography, building positions, and material properties. The gist of the Monte Carlo approach is to specify (based on either empirical data or theoretical considerations) prob- ability distributions for these parameters, randomly sample from the distributions, and perform a propagation calcula- tion for each random sample. Quantities of interest, such as
50 | Acoustics Today | Spring 2015
Rarely, if ever, are measurements of the atmosphere and terrain available with the sub-wavelength resolution necessary to accurately predict the sound field, in a deterministic sense.
the mean square sound pressure and its variance, can then be estimated. The more direct approach of numerically in- tegrating over the distribution for each of the parameters would require discretizing integrals into a finite number of intervals along each of the variable axes. Unfortunately, the number of evaluations of the integrand would then increase exponentially with the number of parameters and thus quickly become computationally impractical. Suppose, for example, there are 8 parameters to be sampled, and the inte- gration for each is partitioned into 64 discrete intervals. That would amount to 648 = 2.8x1014 evaluations of the integrand (e.g., solutions of a parabolic equation). If each evaluation takes 1 s, almost 9 million years would be needed to finish! A further disadvantage of this approach is that if only a few of the parameters dominate the end result, most of the evalua- tions of the integrand become unnecessary.
Monte Carlo methods are particularly valuable in such situ- ations (Evans and Swartz, 2000). In ordinary Monte Carlo sampling (MCS), each random sample of the parameter space (from which the integrand is then evaluated) is drawn independently of the other samples. A disadvantage of or- dinary MCS is that it tends to randomly undersample cer- tain parts of the input parameter space while oversampling others. Stratified sampling is commonly employed to ensure that important regions of the parameter space are adequate- ly sampled. The overall parameter space is partitioned into strata (subvolumes) and then each stratum is sampled in- dependently. The strata should be mutually exclusive and exhaustive. The overall statistics are then determined by ap- propriately weighting statistics as calculated from the indi- vidual strata. In the context of outdoor sound propagation, atmospheric conditions have often been categorized into acoustical classes based on the wind speed, propagation di- rection relative to the wind, and atmospheric stability. Such schemes exemplify stratified sampling.
Latin hypercube sampling (LHS) is a general and widely used scheme for stratified sampling. In LHS, the domain for each variable is first partitioned into N equal-probability in-