Stephen D. Billings

Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods

In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small- to medium-sized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. The usual formulation of such a problem can suffer from an unfortunate scale dependence not intrinsic in the problem itself. This scale dependence occurs, for instance, when fitting polyharmonic splines in even dimensions. We present and analyze an alternative formulation, available for all strictly conditionally positive definite basic functions, which does not suffer from this drawback, at least for the very important example previously mentioned. This formulation changes the problem into one involving a strictly positive definite symmetric system, which can be easily and efficiently solved by Cholesky factorization. The second section considers a natural domain decomposition method for the interpolation equations and views it as an instance of von Neumanns alternating projection algorithm. Here the underlying Hilbert space is the reproducing kernel Hilbert space induced by the strictly conditionally positive definite basic function. We show that the domain decomposition method presented converges linearly under very weak nondegeneracy conditions on the possibly overlapping subdomains. The last section presents some algorithmic details and numerical results of a domain decomposition interpolatory code for polyharmonic splines in 2 and 3 dimensions. This code has solved problems with 5 million centers and can fit splines with 10,000 centers in approximately 7 seconds on very modest hardware.

Discrimination and classification of buried unexploded ordnance using magnetometry

Magnetic surveys are quite successful at locating buried steel objects, including unexploded ordnance (UXO). However, many of the anomalies apparent in a magnetic image of a contaminated area are from metallic debris, shrapnel, and geological variations in ferromagnetic concentration. Observations are usually made in the far-field of the object so that, in most cases, we can only recover the dipole moment of a buried item. Due to self-demagnetization effects, the magnitude and direction of induced magnetism varies significantly with ordnance orientation. This results in an infinite number of ordnance-like objects that can reproduce a given dipole moment. To discriminate, we define a library of ordnance items expected to occur in the area and find how closely each recovered moment matches one of the UXOs in this library. We define the size of this mismatch as the remanent magnetization and produce a prioritized dig-list on the assumption that items with lower remanence are more likely to be UXO. Such a ranking scheme proves to be very effective when implemented at two sites in Montana. The analysis reveals that live-site and emplaced UXO have significantly different remanence and implies that previous tests of magnetic discrimination performance on seeded sites have been overly pessimistic.

Interpolation of geophysical data using continuous global surfaces

A wide class of interpolation methods, including thin‐plate and tension splines, kriging, sinc functions, equivalent‐source, and radial basis functions, can be encompassed in a common mathematical framework involving continuous global surfaces (CGSs). The difficulty in applying these techniques to geophysical data sets has been the computational and memory requirements involved in solving the large, dense matrix equations that arise. We outline a three‐step process for reducing the computational requirements: (1) replace the direct inversion techniques with iterative methods such as conjugate gradients; (2) use preconditioning to cluster the eigenvalues of the interpolation matrix and hence speed convergence; and (3) compute the matrix–vector product required at each iteration with a fast multipole or fast moment method.We apply the new methodology to a regional gravity compilation with a highly heterogeneous sampling density. The industry standard minimum‐curvature algorithms and several scale‐dependent ...

Magnetic models of unexploded ordnance

Magnetometry is widely used for the characterization of areas contaminated by unexploded ordnance (UXO). To successfully discriminate hazardous UXO from nonhazardous clutter requires accurate models of the ordnance response. This paper develops an ordnance library with 15 different items using total-field magnetic data collected over a test stand. The induced and remanent magnetizations were obtained by varying the three-dimensional orientation of each item and measuring the magnetic field on a horizontal plane in the dipolar regime. Replicate measurements using multiple specimens of the same ordnance returned very similar induced magnetizations. The fitted moments were used to estimate the detection depths for different sensor noise floors. A prolate spheroid with a 3.5 aspect ratio was used to provide a good approximation to the detection depths for many of the ordnance items. Assuming a 1-nT noise floor, these orientation-dependent detection depths ranged from 10 to 17 times of the objects diameter

Smooth fitting of geophysical data using continuous global surfaces

Continuous global surfaces (CGS) are a general framework for interpolation and smoothing of geophysical data. The first of two smoothing techniques we consider in this paper is generalized cross validation (GCV), which is a bootstrap measure of the predictive error of a surface that requires no prior knowledge of noise levels. The second smoothing technique is to define the CGS surface with fewer centers than data points, and compute the fit by least squares (LSQR); the noise levels are implicitly estimated by the number and placement of the centers relative to the data points. We show that both smoothing methods can be implemented using extensions to the existing fast framework for interpolation, so that it is now possible to construct realistic smooth fits to the very large data sets typically collected in geophysics.Thin‐plate spline and kriging surfaces with GCV smoothing appear to produce realistic fits to noisy radiometric data. The resulting surfaces are similar, yet the thin‐plate spline required ...

Evaluating the effects of magnetic soils on TEM measurements for UXO detection

The task of discriminating UXO from non-UXO items are more difficult when sensor data are contaminated with geological noise originating from magnetic soils. In particular, magnetic material affects the decay curve characteristics in electromagnetic surveys and this can adversely affect inversion calculations that try to estimate parameters of the UXO or determine whether the item is UXO or scrap. This talk focuses on the effects magnetic soils and magnetic viscosity have on time domain and frequency domain electromagnetic sensor data. By forward modelling of a 1-D layered earth model we investigate the sensitivity of surveys to magnetic soils, and attempt to determine the relative strengths of magnetic soils and buried metallic objects. In particular, we consider soil susceptibility models appropriate for Kaho’olawe Island, Hawaii, where the UXO clearance project has been hampered by the presence of highly magnetic basaltic soils. Finally, we investigate how perturbations in signal due to magnetic soil affect the ability to recover parameters of the dipole model presented in Pasion and Oldenburg (2001a).

Unexploded ordnance discrimination using magnetic and electromagnetic sensors: Case study from a former military site

In a study at a military range with the objective to discriminate potentially hazardous 4.2-inch mortars from nonhazardous shrapnel, range, and cultural debris, six different discrimination techniques were tested using data from an array of magnetometers, a time-domain electromagnetic induction (EMI) cart, an array of time-domain sensors, and a time-domain EMI cart with a wider measurement bandwidth. Discrimination was achieved using rule-based or statistical classification of feature vectors extracted from dipole or polarization tensor models fit to detected anomalies. For magnetics, the ranking by moment yielded better discrimination results than that of apparent remanence from relatively large remanent magnetizations of several of the seeded items. The magnetometer results produced very accurate depths and fewer failed fits attributable to noisy data or model insuffi-ciency. The EMI-based methods were more effective than the magnetometer for intrinsic discrimination ability. The higher signal-to-noise ...

Joint and Cooperative Inversion of Magnetic and Time Domain Electromagnetic Data for the Characterization of UXO

Magnetics and electromagnetic surveys are the primary techniques used for UXO remediation projects. Magnetometry is a valuable geophysical tool for UXO detection due to ease of data acquisition and its ability to detect relatively deep targets. However, magnetics data can have large false alarm rates due to geological noise, and there is an inherent non-uniqueness when trying to determine the orientation, size and shape of a target. Electromagnetic surveys, on the other hand, are relatively immune to geologic noise and are more diagnostic for target shape and size but have a reduced depth of investigation. In this paper we aim to improve discrimination ability by developing an interpretation method that takes advantage of the strengths of both techniques. We consider two different approaches to the problem: (1) Interpreting the data sets cooperatively, and (2) Interpreting the data sets jointly. For cooperative inversion information from the inversion of one data set is used as a constraint for inverting another data set. In joint inversion, target model parameters common to the forward solution of both types of data are identified and the model parameters from all the survey data are recovered simultaneously. We compare the confidence with which we can discriminate UXO from non-UXO targets when applying these different approaches to results from individual inversions. In this paper we focus on the details of the joint and cooperative inversion methodologies. Examples of the application of the methodology to TEM and magnetics data sets collected at the former Fort Ord in California are presented. This work is funded in part by the U.S. Army Engineer Research and Development Center and the Army Research Office.

UXO Discrimination and Identification Using Magnetometry

Magnetic surveys are quite successful at locating buried steel objects, including unexploded ordnances. However, many of the anomalies apparent in a magnetic image of a contaminated area are from metallic debris, shrapnel and geological variations in ferromagnetic concentration. Methods to discriminate between UXO’s and clutter are required to expedite clearance efforts. Anomalies from buried ordnance are essentially dipolar due to the rapid full-off with distance of higher order poles. This means that, in most cases, we can only recover the dipole moment of a buried item. Consequently, we have developed a three stage procedure for ordnance discrimination and identification. The first stage consists of a local gradient based inversion for the dipole location and magnitude. The second stage uses the recovered dipole moment and relies on partial or complete shock demagnetization when the ordnance strikes the ground. Any remnant magnetism is erased and the direction of induced magnetization is constrained to lie within about 60 of the Earth’s field. We apply the method to 822 anomalies excavated in the Helena Valley, Montana and find that we can identify all ordnance items and leave almost 50% of non-UXO’s in the ground; a significant cost reduction. The third stage is to classify items identified as UXO into different ordnance types. Due to self-demagnetization effects the magnitude and direction of induced magnetism varies significantly with ordnance orientation relative to the Earth’s field. This results in an infinite number of ordnance like objects that can reproduce a given dipole moment. However, given a finite number of possibilities we can identify which item requires the least remnant magnetism to produce the recovered dipole moment. Assuming that 60 and 81 mm mortars, and 76, 90, 105 and 155 mm projectiles can be present in the Helena Valley, we can assign the correct ordnance type about 60% of the time.

EXAMPLES OF THE EFFECT OF MAGNETIC SOIL ENVIRONMENTS ON TIME DOMAIN ELECTROMAGNETIC DATA

During September 2004 a field study was carried out on the Hawa iian Island of Kaho’olawe to explore various aspects of the effect of magnetic soils on time domain electromagnetic (TEM) measurements. This field work was in support of two Strategic Environmental Research and Development Program funded research projects (UX1355 and UX1414) whose goals are to inv estigate the source and spatial variability of magnetic soil anomalies, to create a methodology for mode lling the response of magnetic soils, and to develop TEM data collection techniques that can better di scriminate between the response of magnetic soils and unexploded ordnance (UXO). Detailed electromagn etic surveys were carried out at a test site on the island. The data from the surveys verify the commonly hel d b lief that magnetic variations in the soil can complicate the identification of UXO. However, the data a lso show that short wavelength variations in the TEM response due to micro-topographic variations and co il orientation effects can generate responses that could mask a UXO and/or result in a false positive. An ove rview of the surveys, a discussion of preliminary results and some practical recommendations fo r urveying in magnetic soil environments will be presented.