Mathematical Problems in Engineering

Volume 2016, Article ID 5856083, 14 pages

http://dx.doi.org/10.1155/2016/5856083

## Sensor Placement via Optimal Experiment Design in EMI Sensing of Metallic Objects

^{1}Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Vancouver, Canada^{2}Black Tusk Geophysics Inc., Vancouver, Canada

Received 12 July 2016; Revised 19 October 2016; Accepted 1 November 2016

Academic Editor: Sebastian Heidenreich

Copyright © 2016 Lin-Ping Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work, under the optimal experimental design framework, investigates the sensor placement problem that aims to guide electromagnetic induction (EMI) sensing of multiple objects. We use the linearized model covariance matrix as a measure of estimation error to present a sequential experimental design (SED) technique. The technique recursively minimizes data misfit to update model parameters and maximizes an information gain function for a future survey relative to previous surveys. The fundamental process of the SED seeks to increase weighted sensitivities to targets when placing sensors. The synthetic and field experiments demonstrate that SED can be used to guide the sensing process for an effective interrogation. It also can serve as a theoretic basis to improve empirical survey operation. We further study the sensitivity of the SED to the number of objects within the sensing range. The tests suggest that an appropriately overrepresented model about expected anomalies might be a feasible choice.

#### 1. Introduction

Unexploded ordnance (UXO) refers to malfunctioned, undetonated military munitions that are present at former war sites and military practice ranges. UXOs pose a serious problem worldwide on public and environmental safety due to their high-risk explosion threats to human life and potential leaching of toxic explosive chemicals into soil and groundwater system [1, 2]. In light of these concerns, the cleanup of UXO contaminated sites has drawn significant attention over the past two decades. The fundamental cleanup work requires to detect buried UXO at depth and to separate UXO from the vast amount of innocuous metallic debris. Geophysical sensing techniques based on electromagnetic induction (EMI) have been found to be the most effective for detecting and characterizing shallow UXOs [3].

An EMI sensing system consists of transmitter and receiver coils. The transmitter coil emits a primary magnetic field to illuminate the subsurface. A time varying primary field induces eddy currents in a nearby metallic object. By Faraday’s law [4], these induced currents produce a transient secondary magnetic field that can be measured by a receiver. Provided that the target dimension is small relative to the target-sensor distance and the primary fields around the target are approximately uniform, the transient scattering phenomena of a metal target can generally be well described by an equivalent induced dipole [4, 5]. Under this scattering model, the induced principal dipole polarizabilities are designated as the target signature for classification since they are intrinsic function of the geometry and material properties of an object. The principal polarizability characteristic of UXOs exhibits one strong axial polarizability and two equal weaker transverse polarizabilities [6–10]. That is the fundamental physical attribute used for differentiating potential UXOs from metallic debris in the EMI sensing classification approach. To reliably classify UXO requires accurate estimates of dipole polarizabilities from the geophysical inversion of EMI data [6–10].

EMI data are collected in either a dynamic mode or cued mode [3]. In a dynamic mode, the sensor is moved across the area of interest to map data anomalies that indicate metallic contamination. In a cued survey mode, the sensor is placed stationarily at locations of data anomalies identified in the dynamic survey. The cued survey helps to acquire high signal-to-noise ratio (SNR) data by increased stacking. In addition, instrument position errors are minimized. Therefore, a followup cued survey is often used to acquire data for improving polarizability estimates.

In this paper, we focus on the sensor placement problem in cued survey. For sites where anomalies are well separated, the current practice of cued sounding likely positions a sensor directly over a nearby target. For some circumstances the instrument might be repositioned above the target by inspecting the offset between the sensor location and the estimated location of an assumed single target. In this way, the SNR of cued data is maximized as possible. For sites with high anomaly density, we likely encounter instances where more targets are present in the field of view of the instrument. The objects near the sensor-edge might be poorly characterized due to the signal being dominated by objects closest to the center of the sensor. Additional soundings may be desired to better characterize the anomalies. Ideally, the addition of new data would reduce model uncertainty. The extent to which model uncertainty is reduced depends on the sensing locations. Therefore, we can treat adding soundings as an optimal experimental design (OED) problem [11, 12] in which optimal sensor positions are predicted in order to collect the most informative data for accurately estimating model parameters.

The theory of optimal experimental design has been well documented in statistics and extensively applied to various scientific and engineering fields [11, 12]. The OED theory contains design and model parameters. In the context of EMI sensing of UXO, the design parameter refers to the number and locations of sensors that control the survey quality. The model parameters refer to locations and polarizabilities of targets (see a brief review of the signal model in Section 2). An optimal survey design is implemented using statistical criterion [11, 12]. A common choice is the -criterion that minimizes the determinant of the model covariance matrix. The determinant in the -criterion is a function of both design and model parameters. When the true model parameters are known or specified, it can be relatively straightforward to find design parameters. For the EMI classification problem, we face a nonlinear optimization problem where optimizing design parameters depends on the model parameters that are unknown before the survey. To make the problem tractable, we consider a sequential design procedure [13, 14] that forecasts a new sensor placement using estimated model parameters given an initial/current survey. We then update model parameters using additional new data. This procedure is repeated until a stopping criterion is satisfied. A stopping criterion could be a predefined threshold of information gain or a maximum number of sensing.

The remaining parts of this paper are organized as follows. In Section 2, we describe the data model which briefly reviews the physics of the EMI sensing. In Section 3, the linearized model and the associated optimality design criteria are discussed. Then the sensor placement problem is formulated as a sequential experimental design (SED) process. In Section 4, the results from the synthetic and experimental data are presented to evaluate the SED technique. Section 5 summarizes the paper.

#### 2. Data Model

Consider a time domain electromagnetic system that is deployed near the surface to interrogate a buried metallic object. When the dimension of a target is small relative to the target-sensor distance, the low frequency EMI scattering of a metal target can be well represented by an equivalent induced dipole model [6, 7, 15, 16]. Mathematically, a target is characterized by the magnetic polarizability tensor at time :The elements of the tensor represent a dipole component in the th Cartesian direction due to a primary field in the th Cartesian direction. The polarizability tensor is symmetric and positive definite. The principal polarizabilities are obtained through an eigen decompositionwhere is the orthonormal eigenvector representing the th principal direction of dipolar polarization with respect to a reference system. The principal polarization are a function of the target’s size, shape, and material properties (e.g., magnetic susceptibility and electric conductivity).

Assuming a time domain EMI system consisting of transmitter-receiver coil pairs, we can express the corresponding measurements as [10]where is an measured data vector at time , is the number of targets in the field of view of the sensors, is an matrix denoting the sensitivities of the sensor system at to the th object located at , is a column vector whose components are the elements of the polarizability tensor of the th object, and is the additive noise vector. They are given bywhere is column vector representing spatial sensitivities of the th pair of transmitter-receiver to the object located at . and in (4) are the magnetic fields at the object location generated by that transmitter-receiver pair. Equation (3) describes the measurements of an EMI sensor system that are the function of target positions and their polarizabilities as well as a sensing location. The task of EMI sensing of UXO is to recover the principal polarizabilities from measurements for classification.

#### 3. Method

Our objective is to estimate model parameters that describe targets within view of the sensor and design parameters , in particular the location of the sensor. The model parameters are the target polarizations and locations. For objects within the sensor field of view, they might be collectively expressed aswhere contains the location vectors for the objects and contains their respective polarizabilities. For a single time channel, the dimension of is and the dimension of is . The dimension of is , where . The design parameters related to sensor location are contained in the modeling matrix of (4). We write (3) aswhere denotes the nonlinear forward function that maps the and on the data. is the data noise associated with a particular survey design . For simplicity, we have suppressed the variable in (5) and (6).

In standard data processing, the set of experiment design parameters are fixed, and a nonlinear inverse problem is solved for finding model parameters [17]. On the other hand, the optimal experiment design problem aims to find the design parameters that are expected to achieve the minimum model uncertainty [11, 12]. Experiment designs are compared using statistical criteria based on the information matrix. Calculation of the information matrix requires estimating the model covariance matrix. For the nonlinear optimal experiment design problem, we will linearize the forward model function (6) about a reference model. A design criterion based on the linear problem will then be utilized.

##### 3.1. Linearized Model Uncertainty and Optimal Design Criteria

To obtain the analytical expression of a model covariance matrix in the nonlinear inverse problem, we linearize the data functional about a prior estimated model [17]where and is the Jacobian matrix comprising the partial derivatives of the data functional evaluated at for the design ,A solution to the linearized equation of (7) can be achieved by minimizing an objective function [17]where are the data residuals between the measured data and the data predicted at the model . is generally treated as a data covariance matrix and is a prior model covariance matrix. The first term in (9) is the data misfit, measuring the discrepancies between the observed and theoretical quantities. The second term is a damping or regularization term that plays a role in controlling the size of a solution model relative to model . The formal linearized least square solution for (9) is given by [17]The corresponding posterior covariance matrix can be approximated as [17]As indicated in [13, 17], a model used for the linearization is often obtained as a local optimum point through an iterative optimization algorithm. The posterior of (11) estimated at convergence is viewed as a linearized measure of uncertainty in the estimated model parameters. This estimate forms the basis for various experimental design approaches [11, 12].

In the standard experimental design approaches, the model covariance matrix is converted into some scalar functions such as its determinant or trace, that is, or . Then an optimal design might be sought in terms of the - and -criteria [11, 12], given by and , respectively. Under assumption that the estimates are multivariate Gaussian, in the model space might be pictured as an ellipsoidal confidence region about the parameter estimates. Geometrically, or measures the size of this ellipsoid either in terms of its volume or its average semiaxes lengths. The smaller the ellipsoidal region, the smaller the uncertainty in parameter estimates. Posing or as a function of experimental parameters makes it possible to seek an optimal experiment over the experiment space and render the minimal postexperimental uncertainty in model estimates.

The optimization of experimental designs in our case is conditional on the assumed model . In principle, the optimal design can be achieved only when the prior estimate is close to the true parameters. In reality, such an assumption might not always hold. Thus for a nonlinear model, it is necessary to have a procedure in which the design can be sequentially updated upon the improved estimate of the model parameters after each experiment. In the following we present a* D*-criterion-based sequential experimental design.

##### 3.2. Sequential Experimental Design Optimization

In this section, we follow the approaches of augmenting experimental data [11, 13, 14] to obtain a sequential experimental design criterion and procedure in terms of the posterior covariance matrix.

Starting a current survey stage denoted as , we wish to find or predict the next sounding, denoted as future stage . The sensitivity matrix at stage can be expressed aswhere is a composite of the sensitivity matrices for experiments at stage and represents a sensitivity matrix that is to be determined via optimizing the criterion below. is formed by appending a future to the existing [11, 13, 14]. This is equivalent to adding data from a new sounding. Denote the model covariance matrix at stage as . With (11), its inverse can be expressed aswhere is the data covariance matrix at the th stage, given bywhere is a block data covariance matrix of size at the th sounding. For stage , the posterior covariance matrix is given byInserting (12) and using (13), the inverse posterior covariance matrix of (15) at step may be rewritten asUsing the matrix determinant lemma [18], we can havewhere is an identity matrix with the size of and . Therefore, we can introduce a new function asEquation (18) is a scalar measure of the model covariance reduction that can be obtained by adding a new sounding to the previous soundings. Conversely, may be called an expected information gain. The core part in the objective function is that is a model covariance matrix -weighted product of the scaled sensitivity matrix at the candidate sensing location. The way (18) works may be understood qualitatively. If the elements of model covariance matrix are larger at previous stages, will tend to be large by seeking a new experiment that can increase sensitivities in and thus reduce the model uncertainty expressed in (15). On the other hand, if the elements of model covariance matrix are already smaller at the previous stages at which large sensitivities were built upon, the value of would be small when adding a new experiment that likely resembles one of the previous ones.

The above presentation is general and can be reduced to some specific case. The form of (18) is similar to those in [13, 14], but its inclusion of the covariance matrices makes it convenient to incorporate available a priori knowledge about the model and accommodate various types of measurement noise into an optimal experimental design process. Specifying the prior covariance matrices requires statistical noise models that are representative of the experiment. However such statistical noise models are not always known or easily defined in reality. To simplify, one common practice is to ignore and assume a uniform uncorrelated data noise [13, 14]. In our current implementation, we also simply choose which represent no prior information about model in the beginning. However in the sequel, this choice is automatically updated as shown in (16) where from the previous survey locations can be viewed as prior model information for a next intended survey location and is embedded in (18). For a data covariance matrix, we consider an uncorrelated nonuniform error distribution in this study. That is, a standard error is assigned as a summation of the baseline error and a percentage of an observed datum [19]. Then the diagonal entry of is written as for th datum.

Having obtained the -optimality function (18), we can implement a sequential experimental design (SED) for sensor placement in the following manner.

*Step 1. *Given the survey parameters as at the th stage, finding the best model parameters is stated as the minimization of a nonlinear least squares function with some constraints (for algorithm details see [10, 20, 21]) where is a positive scalar used to provide a local trust region within which is allowed to change and represents that is constrained to be the elements of the set of symmetric positive semidefinite matrices.

*Step 2. *Given the current model and starting from the most recent sensor position in , we predict next sensing location represented as by maximizing The optimization can be carried out via the Matlab [22] function fmincon.

Repeat this survey augmenting process until a stopping criterion is attained. This stopping criterion may be a predefined threshold of information gain or a maximum number of sensing stages.

#### 4. Results

In this section, we evaluate the SED with two portable EMI sensor systems: one is the TEMTADS2x2 (Transient Electromagnetic Multisensor Towed Array Detection System) [23]; the other is the MPV (Man Portable Vector sensor) [24, 25]. Each sensor is described in the related subsections.

##### 4.1. Synthetic Experiments with TEMTADS2x2

First, we present results with synthetic TEMTADS2x2 time-domain data with a noise level of to illustrate the capability of the optimal experimental design for the sensor placement.

TEMTADS2x2 is a sensor system that consists of 4 coplanar small transmitters and 4 sets of tri-axial receiver cubes [23] (Figure 1(a)). The sizes of each transmitter and receiver are cm × 40 cm and cm × 8 cm. Two objects, 37 mm and 75 mm projectiles, are used in the numerical experiments. Their polarizabilities are shown in Figure 1(b).