Michael D. Collins

The parabolic equation method is the most effective approach for solving wave propagation problems in environments with strong vertical variations and gradual horizontal variations. This approach efficiently provides accurate solutions for problems involving sloping fluid-fluid interfaces, sloping solid-solid interfaces, and sloping solid boundaries. There remains a need for improvement for the case of sloping fluid-solid interfaces. The most promising approach to date is the combination of a single-scattering approximation and the treatment of part of the fluid layer as a solid layer with low shear speed. This is a singular perturbation, which results in rapid oscillations near the interface between the low-speed layer and the true solid layer. It is demonstrated here that the rapid oscillations may be eliminated by using slip conditions at the interface between the low-speed layer and the true solid layer. Stability problems were encountered when attempting to directly implement the slip conditions, but this problem was resolved by using an equivalent condition involving the normal derivative of the tangential displacement. This progress does not fully resolve the case of the sloping fluid-solid interface. There remains a need to find a vertical interface condition that is compatible with the slip conditions and that provides accurate results for problems involving sloping fluid-solid interfaces.

The Ivory-billed Woodpecker (Campephilus principalis) is an elusive bird that has repeatedly been feared extinct only to be rediscovered during the past hundred years. The most recent rediscovery, which took place in Arkansas, was announced in an article that was featured on the cover of Science in 2005. Despite published reports of sightings in Florida and Louisiana in the years that followed, the issue became controversial when nobody managed to obtain a clear photo and critics attacked relatively weak video evidence that was presented in the original article. The issue began to develop into a science scandal when the critics used specious arguments to delay the publication of the strongest evidence, which consists of three videos that were obtained during encounters with birds that were identified in the field as Ivory-billed Woodpeckers and that show field marks, body proportions, flights, and other behaviors that are consistent with that species but no other species of the region. The scandal culminated in a decision by the U.S. Fish & Wildlife Service to declare the species extinct in 2021, which was made without addressing the strongest evidence. An update is given here on recent developments. The decision to declare the species extinct was based on a five-year review of evidence for persistence. During an interview, it came to light that the person who performed the review was unaware of basic facts about the strongest evidence. Evidence that was obtained at another site in Louisiana [Latta et al., “Multiple lines of evidence suggest the persistence of the Ivory-billed Woodpecker (Campephilus principalis) in Louisiana,” Ecology and Evolution (2023)] is discussed and compared with the strongest evidence. Videos obtained with a drone on a sunny day often show white markings that do not correspond to actual field marks. It is demonstrated that a video that was purported to show an Ivory-billed Woodpecker on the basis of apparent white markings is apparently a Pileated Woodpecker (Dryocopus pileatus). The bird in that video appears to have black trailing edges on the dorsal surfaces of the wings, which are consistent with the Pileated Woodpecker, not the Ivory-billed Woodpecker. The wingbeat frequency and an upward swooping landing also seem to be consistent with the Pileated Woodpecker but are not consistent with historical accounts of the Ivory-billed Woodpecker or flights appearing in videos that contain the strongest evidence.

The parabolic equation method is an accurate and efficient approach for solving nonseparable problems in ocean acoustics in which there are horizontal variations in the environmental parameters. Many range-dependent problems may be solved using 2-D parabolic equation models that ignore coupling of energy between planes of constant azimuth. When azimuthal coupling must be taken into account, the splitting method may be used to efficiently solve a 3-D parabolic equation that handles the depth operator to higher order but handles the azimuth operator only to leading order. Despite the fact that this approximation provides a favorable combination of accuracy and efficiency for 3-D problems, run times have generally been regarded as prohibitive for the long-range problems that are often of interest in ocean acoustics. It is demonstrated here that, when propagation paths from source to receiver are confined to a relatively narrow neighborhood of the vertical plane containing the source and receiver, it is practical to solve 3-D problems out to long ranges by using nonuniform azimuthal sampling, with fine sampling near the vertical plane and extremely coarse sampling elsewhere.

This paper reviews various approaches for localizing submerged acoustic sources under adverse conditions. It is essential to obtain data of the highest possible quality when there are adverse conditions, such as uncertainties in the environment, source motion, and low signal-to-noise ratio. Focalization is an approach in which the source location and environmental parameters are treated as unknowns. Due to a parameter hierarchy in which source location outranks environmental parameters, there may be many realizations of the environment that bring the source into focus; the ambiguity in the environment can be an advantage if the primary objective is to localize the source. Environmental uncertainty is often associated with environmental complexity, which can be an advantage by reducing the ambiguity of the source location. Obtaining a high-quality estimate of the covariance matrix may be difficult when there is source motion, but the complexity of the received field from a moving source is another factor that can reduce ambiguity. It may be possible to localize a source that is buried in noise when an estimate of the noise covariance matrix is available. During the development of approaches for localizing submerged sources, much of the focus has been on one-dimensional vertical arrays. An extension of the multi-valued Bartlett processor to the case of a rectangular array was designed to take advantage of the extra dimension of the array and appears to have the potential to be the most powerful combination of hardware and signal processing to date.

The parabolic equation method provides an unrivaled combination of computational efficiency and accuracy for wave propagation problems in the geosciences in which the environment has strong vertical variations and gradual horizontal variations. The development of this approach has been an active area of research for several decades in ocean acoustics, which includes problems involving sediment layers and ice cover that support shear waves. It is demonstrated here that this progress has culminated in a parabolic equation model for fast and accurate seismic computations in laterally varying environments. The model is tested for problems involving sloping boundaries and interfaces, variable layer thickness, continuous variations of the elastic parameters within layers, and a Rayleigh wave propagating along variable topography. The seismic parabolic equation model is based on an outgoing wave equation and rational approximations of operators for generating initial conditions, propagating the solution through stratified regions that approximate a laterally varying environment, and estimating transmitted fields across the vertical interfaces between regions.