An iso-deviant strategy for efficient acoustic grids with EAGLE

Some Navy operations require extensive acoustic calculations. The standard computational approach is to perform the calculations on a regular grid of points and radial directions. Unfortunately, the time required to achieve the necessary accuracy often renders them tactically useless. The Efficient Acoustic Gridder for Littoral Environments (EAGLE) [J. Acoust Soc. Am., 2003] demonstrated that irregular gridding can provide an order of magnitude efficiency improvement (over regular grids), and that the efficiency of irregular gridding increases with the number of grid dimensions. EAGLE\´s point selection mechanism is guided by the local features of the grid itself, such as the local slope, rather than by analysis of the environmental inputs to the acoustic calculations. On each iteration, EAGLE produces an acoustic field with ever-increasing resolution and accuracy, and the sampling density of sub-regions is proportional to their acoustic complexity without resort to complex (and computationally expensive) environmental analyses or a priori assumptions. OGRES was able to reduce the number of acoustic calculations by a factor of 5-9 for a given level of grid uncertainty, or to reduce uncertainty by 2-8 dB for a given average grid density. As desired, it concentrated effort where acoustic complexity was high and analysis of the density variation across the grid increased physical insight into the relationship between oceanographic environments and local acoustic complexity. The present work presents a new concept, a point-selection mechanism based on local predictive error. Predictive error is defined as the difference between a known value at a location and an estimate of its value based on all other points gathered to date, so it resembles the statistical process known as "jack-knifing". Tests show that a grid of the predictive error is at least as smooth as, and usually much smoother than, the underlying dataset, so it can be interpolated (using the same interpolat- r used for the predictions) and probed for extreme values. These extrema are then used to steer the next round of point selections. Since predictive error at a given point is a measure of the uncertainty removed from the grid by addition of that point, it can be used to estimate uncertainty variation across the grid. Consistent removal of extremal uncertainty rapidly forces the irregular grid toward a narrow band of low uncertainties, a property we refer to as "iso-deviance" (ie., the grid may be irregular when viewed in Cartesian space, but would appear as regularly gridded if viewed in uncertainty-space). Most successive gridding schemes are terminated subjectively, based on "experience" with a particular type of data; iso-deviance objective termination. Also, the point selection mechanism is independent of the interpolator in use, other than its requirement that the predictive error grid and the dataset both use the same interpolator (e.g., linear, optimal, etc.). Any consistent differential efficiencies can be used to objectively measure a given interpolator\´s fitness vis-a-vis that data type.