Application of a Doubly Stochastic Poisson Model to the Spatial Prediction of Unexploded Ordnance

Transcription

1 International Association of Mathematical Geology 2001 Annual Meeting, Cancun, Mexico, September 6-12 Application of a Doubly Stochastic Poisson Model to the Spatial Prediction of Unexploded Ordnance Sean A. McKenna, Geohydrology Department, Sandia National Laboratories, Albuquerque, New Mexico, USA Abstract The efficient characterization of sites contaminated with unexploded ordnance (UXO) is necessary prior to returning these sites to the public domain. Characterization plans must be based on an accurate prediction of the UXO distribution. A doubly stochastic Poisson process is proposed as the underlying model controlling the spatial distribution of UXO. This model allows the single parameter (intensity) of the Poisson distribution to vary spatially across a site. Historical information regarding site use is incorporated as a prior (soft data) estimate of the spatially variable UXO intensity. Limited sampling (hard data) of the site allows the prior estimate to be updated into a posterior estimate using geostatistical estimation techniques. This initial estimate of the UXO intensity can be compared to an intensity threshold specified by a regulatory body. Uncertainty in the initial estimation of the UXO intensity relative to the specified threshold can be quantified and additional sampling can be targeted in areas of greatest uncertainty. An example of this statistically based predictive approach is applied to an exhaustively known UXO data set. Geophysical surveys were used to locate over 15,000 buried objects with a high probability of discriminating scrap metal from UXO. This anomaly data set is used as a test case. Historical information defining the location of the target is used to develop a prior estimate of object intensity. A subset of the exhaustive data set is sampled along linear transects and used to create a variogram model of the intensity of the objects. The variogram model is used to estimate the spatial distribution of object intensity at the unsampled locations. The doubly stochastic Poisson model is assessed by testing the estimates of the Poisson intensity against the known intensity values at each location. Additionally, the actual count of object data at each location is compared to the corresponding value of the Poisson distribution as defined by the estimated intensity. McKenna 1

2 Introduction The characterization of UXO is an international problem at practice ranges and historical battlegrounds. Traditional field characterization techniques have proven to be inefficient and imprecise. More recently developed characterization technologies provide for rapid surveys with improved detection of UXO and discrimination of UXO from scrap. Improved characterization protocols are needed take advantage of these developments in characterization technology. Background Unexploded ordnance (UXO) is a problem across the globe at former test ranges and firing sites as well as at former battlefields. In general, UXO is found at these sites at relatively shallow depths (1-3 meters). Changes in land-use practices, erosion, frost heave and other environmental and anthropogenic processes may cause the UXO to become exposed at ground surface, or at least to make the burial depth sufficiently shallow such that the UXO is encountered during soil tillage (Young and Helms, 2000). UXO at the ground surface, or very shallow burial depths presents a major hazard to the public. For example in 1991 alone, 36 French farmers were killed from UXO left over at World War I and II battlefields (Young and Helms, 1999). More recently, UXO in Bosnia has become a hazard to the population there (Donovan, 1994). While the related problem of characterization of landmines has received the majority of the public focus, the problem of UXO disposition is critical and equally as challenging as landmine characterization. In the United States, the major UXO characterization problem is associated with current and former military testing ranges. The UXO characterization problem is particularly critical at Department of Defense (DOD) sites that are in the process of being turned back to the public sector. As pointed out by the DOD (SERDP, 2000), it is estimated that over 4.5 million hectares (11 million acres) of land potentially contain UXO. As is also pointed out by DOD, the actual UXO contamination may only occur in small portions of a site that is under investigation. The large amount of potentially contaminated land area and the observation that the actual UXO contamination may only occur in a fraction of this area make efficient characterization of the land for UXO extremely difficult. In particular, the potential for unnecessary expenditure of large amounts of characterization resources is high. Field Characterization Tools The traditional approach to collecting field data to characterize UXO sites is known as the magand-flag approach. This approach consists of sending a multiperson team into the field with hand-held magnetometers. These magnetometers are deployed on preselected gridpoints to search for UXO. Frequently, the grid is selected such that magnetometer readings are taken on spacings of only a few meters. The mag and flag approach has proven to be expensive, timeconsuming and the hand-held magnetometers alone are not able to distinguish between scrap metal and true UXO (see results in McDonald, et al. 2000). In response to the difficulties inherent in UXO characterization through the mag-and-flag approach, the DOD has made a large investment in the development of new approaches to sampling UXO sites. The majority of this investment has focussed on development of geophysical sensors that can be used to locate UXO in the subsurface. These sensors provide for the rapid, semi-automated McKenna 2

3 collection of large amounts of information. These sensors include magnetomoters, electromagnetic conductivity meters, seismic sources and receivers, synthetic aperture radar, and enhanced harmonic radar, and combinations of these sensors. The benefits that this investment in new UXO sensing technologies can provide, must be incorporated into the site characterization protocols. This work demonstrates an analysis technique that can be employed towards accomplishing this incorporation. Current Characterization Protocols The current UXO characterization guidelines rely on adaptations of traditional statistical techniques in an attempt to characterize what is inherently a spatial statistics problem the spatial estimation of the UXO intensity from a limited data set. Traditional statistical techniques do not account for correlation between samples. As an example, the U.S. Army Engineering and Support Center in Huntsville (USAESCH) has developed both the UXO calculator and the SiteStats/GridStats approaches to determining the number of samples that need to be obtained at a site (QuantiTech, 1995; USAESCH, 1999). Both of these approaches assume that the underlying mechanism responsible for the UXO distribution is the result of a Bernoulli process. In the case of the GridStats approach to site characterization, a grid is defined across the site and characterization of randomly chosen grid cells is conducted. The density of UXO within each grid is compared to a cleanup threshold and the grid is classified as passing (UXO density is below the cleanup threshold) or failing (the UXO density is above the cleanup threshold). The n sample grids are treated as independent trials where the probability of success (passing) is constant (sampling with replacement) (Walpole and Myers, 1989). USAESCH has also developed a variation on this technique in which the probability of success is not assumed to be constant (sampling without replacement) (USAESCH, 1999). A concise description of the statistical techniques underlying the current site characterization guidelines can be found in (Dohrman, 1997). As explained by Dohrman (1997), the number of samples necessary to characterize the site are determined by setting the acceptable error levels of false positives and false negatives and determining the number of grids to sample from the binomial distribution (sampling with replacement) or the hypergeometric distribution (sampling without replacement). The current characterization tools are based on sampling of a Bernoulli process. Inherent in the Bernoulli process is the assumption that the density of the UXO at each sample location is independent of the previous and the following sample locations. This assumption does not apply to the majority of UXO sites where the original deposition of ordnance was specifically clustered around target locations. Samples within the vicinity of a target area are likely to be similar to other samples in the same area. Samples obtained in areas away from the targets are likely to be similar to one another, but dissimilar to samples obtained near targets. The UXO intensity displays spatial correlation, not independence, and the assumption of a Bernoulli process cannot account for this correlation. This correlation between samples can decrease the number of samples necessary to achieve a certain level of confidence relative to the assumption of independence between samples (see Gilbert, 1987), especially if additional, more subjective information regarding the ordnance emplacement process can be incorporated into a sampling plan. Additionally, if applied in the correct situation, the Bernoulli process approach to site McKenna 3

4 characterization is still limited to answering the question of how many samples to obtain, but it cannot indicate where to collect those samples. The goal of this work is to introduce an alternative technique for characterizing UXO sites that takes into account the spatial correlation of a limited number of observed intensity values through the use of geostatistical techniques. This paper combines spatial Poisson processes and geostatistical techniques to develop an approach for the characterization of UXO sites. The protocol developed herein is inherently a Bayesian procedure that incorporates existing knowledge regarding the spatial distribution of UXO at the site into a prior estimate of the geophysical anomaly intensity (number of anomalies per area). This prior estimate is then updated with new sample information to produce a posterior estimate of the anomaly intensity. The spatial variability in the anomaly intensity is defined through spatial statistical estimation techniques. The probability of the anomaly intensity exceeding a cleanup goal for every location on the site is modeled. Decisions on where to sample and how many samples to obtain can be made by combining the approach demonstrated here with a data-worth procedure. Spatial Modeling The goals of the spatial modeling are two-fold: 1) To estimate the intensity of the Poisson process describing the UXO distribution at all locations within the domain 2) To assess the uncertainty in the intensity estimates with respect to a regulatory threshold. In order to achieve these two goals, it is necessary to use a spatial estimation method that can incorporate and honor both prior (soft) information and sample (hard) data in the estimation. Additionally, this method must produce a distribution of the estimates at each location that captures the uncertainty in the estimated intensity value. Incorporation of spatial correlation into the predictive modeling of UXO intensity is done through geostatistical techniques. These techniques rely on a conceptualization of the distribution of UXO objects as the result of a doubly-stochastic Poisson process. Doubly-Stochastic Poisson Processes The spatial distribution of UXO can be described as a realization of a spatial random function generated by a Poisson process. Poisson processes have traditionally been invoked to describe the number of independent events occurring within a specified unit of time or distance (e.g., number of customers per hour; number of defects per kilometer of fiber optic cable). k e P( k) (1) k! where P(k) is the probability of occurrence of k discrete events and is the mean occurrence rate, or intensity of the Poisson process. Traditional applications of the Poisson process model have assumed that the mean of the Poisson distribution is constant in time or space (see Cressie, 1993; Taylor and Karlin, 1998). Multidimensional, or spatial, Poisson processes with a constant mean are known as homogeneous Poisson processes. McKenna 4

5 There is precedent for using homogeneous Poisson process as model for the spatial distribution of ordnance. The following illustration is adapted from Harr (1987). In World War II, the Nazis launched flying bombs on London (mainly V1 and V2 rockets). The British were interested in determining whether or not these flying bombs were being guided towards certain targets within London. A grid of equally sized areas was sumperimposed over a map of London and the number of flying bomb impacts within each grid cell was compared to that predicted by a homogeneous Poisson distribution. The results showed that the spatial distribution of the bomb impacts was well predicted by the homogeneous Poisson distribution indicating that the impact locations were randomly scattered across London and the flying bombs were not being guided towards specific targets. This is an example of a homogeneous Poisson process. The expected value of the distribution (the total number of bomb hits in London divided by the total number of equally sized sample areas) was a constant across the entire area of London. Results showed that the same ordnance intensity applied to each grid cell. The homogeneous Poisson process applied to the spatial distribution of ordnance in London in World War II because the bombs were not directed at specific targets. This is not the case for the majority of UXO sites where test ranges often had specific targets designated on the ground. The intensity of the UXO will be considerably higher near these targets than across the remainder of the site. The case of a Poisson process with a spatially varying mean value is known as a doubly stochastic Poisson process (DSPP), or a Cox process, given as: k ( x) ( x) e P( x; k) (2) k! where x is a vector denoting the location within a multidimensional space. DSPP s describe the distribution of objects in a domain where the intensity of the objects within a given subdomain of constant is variable. These DSPP s have recently been used to describe the spatial occurrence of diamonds in sedimentary deposits (Caers, et al, 1997) and have also been applied to other types of mineral deposits. Spatial Prediction The spatial estimates of the UXO intensity will be used in a decision-making process that takes into account the uncertainty in the estimated intensity values. Therefore the end goal of the spatial estimation portion of this proposed work is to develop a map of the site that indicates the probability of exceeding, or not exceeding, a specified threshold for UXO density. This probability is best defined by a local conditional cumulative density function (ccdf) of the UXO intensity for each location within the site domain. As discussed in Goovaerts (1997), the ccdf at each location can be defined either through assumption of a parametric form for the ccdf (usually the multigaussian, MG, model is applied) or through a non-parametric technique (usually construction of the ccdf at a number of predefined thresholds through indicator geostatistical approaches). In this work, the MG model is applied to define the ccdf of the residuals about a trend surface at every location as explained below. Ordinary Kriging (OK) within a local search neighborhood is used to determine the mean and variance of the MG model at each location. For the MG model McKenna 5

6 with the mean and variance determined through OK, the standard normal Gaussian ccdf,g, at location x and for any value y is defined as. y est ( x n) OK G( x; y n) OK G OK ( x n) (3) where n is the number of surrounding data values, and est OK and OK are the OK estimates of the mean and standard deviation of the distribution given by: est OK n( x) OK ( x) y( x ) (4) i1 i i 2 OK ( x) C(0) n( x) i1 OK i C( x i x) OK ( x) (5) where x denotes the spatial location and n(x) is the number of data within the estimation neighborhood, OK are the OK kriging weights as determined from solution of the kriging system, C() is the spatial covariance function with the distance between any two points as the argument, and OK is the Lagrange parameter that accounts for the imposed unbiasedness constraint on the kriging weights: n( x) i1 1.0 (6) More details on the OK algorithms can be found in Goovaerts (1997), Deutsch and Journel (1998) and Olea (1999). OK i For the estimation of UXO intensity, the local ccdfs need to be conditioned to a prior estimate of the UXO intensity at each location as well as to the surrounding data. There are a number of variants of the kriging algorithm that can be used to incorporate the prior information. Most of these are based on the cokriging formulation (see Goovaerts, 1997, for more detail). For the application presented herein, the prior information is defined as a bivariate trend surface. The residuals are the difference between this trend surface and the measured data: resid(x) = trend(x)-data(x) (7) If the prior information as modeled by the trend surface is a perfect description of the underlying intensity values, then the residuals will describe noise (e.g., due to measurement error) about this trend surface, and by definition will be independent and normally distributed with a mean of zero. In the practical application of trend surfaces (see Davis, 1986), the trend surface is rarely an unbiased and precise estimate of the underlying process. In these situations, the distribution of residuals may exhibit spatial correlation and a non-zero mean. McKenna 6

7 The MG function used to define the local ccdfs of the residuals requires that the y data come from a standard normal distribution. In order to ensure that the residual data are normally distributed, the residuals are transformed to a standard normal distribution using the normalscore transform (Deutsch and Journel, 1998) prior to determination of est OK and OK with equations 4 and 5. The variogram describing the residual data is calculated and modeled in this normal-score space. The approach as outlined above makes use of the multigaussian assumption only to model the residual values and does not require the variogram modeling at multiple thresholds as is necessary in an indicator approach to estimating the ccdfs. Site Application The approach to the spatial modeling of UXO described above is applied and tested on a site for which an exhaustive geophysical survey exists. It is important to note that the characterization threshold and decisions made about this site are done for demonstration purposes only. Site Description The site chosen for a test application of the characterization approach developed in this report is the N-10 Target Area on the Pueblo of Laguna in west central New Mexico, USA. The N-10 site is arid grassland at an elevation of approximately 1750 meters above sea level. The site is a square that is roughly 560 meters on each side (area of 31.4 ha). The N-10 Target Area was used as a practice range for precision aerial bombing training, although no documentation on the actual ordnance used in the training is available. The site was in use for an indeterminate amount of time between the end of WW II and The data used in this example were collected in the summer of 1998 by the Naval Research Laboratory using the MTADS system (McDonald, et al., 2000). A set of magnetometers was towed across the site in a series of contiguous linear transects to exhaustively cover the site area. This exhaustive survey results in a map of the locations of over 15,000 geophysical anomalies (Figure 1). Not all of these anomalies are UXO - the vast majority are scrap material resulting from the explosion of the ordnance on impact. However, for this example application, the goal is the correct estimation of the intensity of the anomalies whether or not they are scrap or UXO. Incidentally, for the N-10 Target Area, the geophysical survey was capable of discriminating scrap from UXO with nearly 99 percent accuracy (McDonald, et al., 2000). Prior to the magnetometer survey the site was surveyed for scrap metal on the ground surface. Widely scattered bomb debris was found across the site with decreasing amounts of scrap with increasing distance away from a centrally located target (McDonald, et al., 2000). This information is used here to develop the prior estimate of object intensity. Prior Information and Initial Sampling Based on the surface survey of scrap metal and knowledge of the target location, a prior estimate of the UXO intensity across the site has been constructed. This prior estimate is a radial Gaussian function centered on the area with the greatest concentration of scrap metal. The trend surface defined by this Gaussian function is: McKenna 7

8 Northing (meters) Easting (meters) Figure 1. Map of surveyed object locations at the Pueblo of Laguna site (source: MacDonald et al., 2000) trend ( x x exp b 0 ) 2 y y c 0 2 (8) The center of the Gaussian model (x 0,y 0 ) also corresponds to the target location as determined from historical site use information. The standard deviation of the model in the x and y directions (b and c, in Equation 8) are estimated from the extent of scrap metal found in the surface survey. The Gaussian model of the anomaly intensity used as a prior estimate is compared to the actual, exhaustively known intensities as calculated on 20x20 meter grid cells in Figure 2. Note that the prior estimate does not capture the actual peak intensity values nor does it capture the SE-NW trend of higher anomaly intensities seen across the site. This type of imperfect prior knowledge of site conditions is thought to be typical of that encountered at characterization problems at many UXO sites. In this study, the prior information is used as a trend surface to define a non-stationary Poisson intensity value. The sample data are obtained and then subtracted from this trend defined by the prior information to produce a set of residuals. The residuals are transformed to standard-normal space and a variogram of the normal-score residuals is calculated and modeled. The ccdf of the McKenna 8

9 residuals is then estimated at all unsampled locations and the trend is added back to the estimated residuals to determine the final estimates of the actual anomaly intensity UXO Density (1/meter 2 ) Northing (meters) Easting (meters) Figure 2. Map of two-dimensional Gaussian function used as prior information on anomaly intensity calculated on 20x20 meter grid cells. The actual intensity values are shown as black squares for comparison and are calculated from the actual object data shown in Figure 1. A hypothetical intensity characterization threshold, c, of 0.10/m 2 is applied for this example. For the 20x20 meter panels onto which the intensity is mapped, this threshold intensity corresponds to 40 objects. The initial sampling plan is conducted by locating linear sampling transects such that they maximize the coverage of the intersection of the prior estimate with the characterization threshold. An additional pair of transects are placed at right angles over the center of the target. A contour map of the prior intensity estimate and the locations and values of the transect samples are shown in Figure 3, A and B respectively. Spatial Prediction For this example site, the spatial modeling approach outlined above is followed. The initial estimate is regarded as a trend surface of the estimated intensity (Figure 3, A). The geophysical sampling typically applied to UXO sites such as this one is done along 1-D transects (Figure 3, B). These initial transects are then compared to the estimated trend surface to determine residuals between the trend surface and the sample data (Figure 3, C and E). The residuals are converted to normal-score values (Figure 3, D and F) and variogram of the residuals is modeled (Figure 4). The histogram of residuals (Figure 3, E) shows a near normal distribution with a McKenna 9

10 mean of zero. The distribution is skewed slightly to the right by the positive residuals on either end of the northernmost east-west transect. The normal score transform creates a Gaussian distribution with a mean of exactly zero and a standard deviation of exactly 1.0 (Figure 3, F). An omnidirectional variogram of the normal-score values is fit with a Gaussian model (Figure 4). The parameters of the Gaussian variogram model are given in Figure 4. This variogram is used as input to a MG simulation routine (the program sgsim, Deutsch and Journel, 1998) to create multiple, equally probable realizations of the residual field in normal-score space. For each realization, the residual field is backtransformed to the raw residual space and then added back to the prior estimate to produce a final realization of Poisson intensity. A total of 100 realizations are created. The ensemble of realizations can be processed to assess the local uncertainty at each estimated location. These local uncertainty estimates can then be examined to determine the probability of exceeding the regulatory threshold and to guide future sampling and/or remediation. McKenna 10

11 Northing (meters) A 1e-4 1e-3 1e-3 1e e-2 1e-3 1e-2 1e e-1 1e-1 0 1e-2 1e e-1 1e-2 1e e-2 1e-3 1e-3 1e-4 1e-3 1e Easting (meters) B Transect Data C Transect Data Residuals D Normal-Score Transect Data Residuals E F Frequency Raw Residuals Number of Data 159 mean -1 std. dev. 4 coef. of var undefined maximum 0.16 upper quartile 1 median -1 lower quartile -3 minimum Frequency Normal Score Residuals Number of Data 159 mean 0 std. dev coef. of var undefined maximum 2.73 upper quartile 7 median 0 lower quartile -7 minimum Residual Normal Score Residual Figure 3. Contour map of the prior estimate of UXO intensity (A). UXO intensity sample locations (B). Residuals between prior estimate of UXO intensity and actual intensity values (C). Normal score transformed values of the residuals (D). Histogram of the residuals (E). Histogram of the normal-score transformed residual values (F). McKenna 11

12 N-S Residual Variogram Model Gaussian Range = 95 meters Sill = 0.90 Nugget = 0.10 Figure 4. Experimental and model variograms fit to normal score residual data. Results The results of the intensity modeling are summarized in Figures 5 and 6. Figure 5 shows the expectation map of the simulated residuals at each location in the left image and the conditional variance of the simulated residuals in the right image. The expectation and standard deviation of the final estimates, computed as residuals plus prior estimate, of the Poisson intensity are shown in Figure North North East East Figure 5.. Expectation map of the residual values (left image) and conditional standard deviation of residual values (right image), as created from 100 realizations. McKenna 12

13 North 50 North East East 28 Figure 6. Expectation map of the intensity values (left image) and conditional standard deviation of intensity values (right image), as created from 100 realizations. Local uncertainty in the estimates of the Poisson intensity is defined by each local ccdf. The uncertainty with respect to the particular characterization threshold can be determined by calculating the probability that the simulated Poisson intensity exceeds the characterization threshold. This probability value is the local probability of failure, P fail. For this work, the local reliability as determined through geostatistical simulation, R S, is defined as the probability of meeting the design criteria, or the complement of P fail : R ( x) P( ( x) ) 1.0 P ( x) (9) S c The decision regarding whether or not to remediate any particular location can be made by specifying an acceptable remediation design reliability, R D, and comparing the simulated reliability, R S, to the design reliability. If R S > R D, the decision panel is left as is. If R S R D, the decision panel is remediated. Specification of R D allows the decision-maker some control on the accuracy of the decision being made. It is noted that in this context, reliability is not directly equated with a level of confidence surrounding the estimation of a distribution parameter (e.g., mean or variance). Reliability, as defined in Equation 9, is based on the concept of engineering reliability as discussed in Harr (1987). For this example application, the probability of exceeding the characterization threshold of 0.10/m 2, P fail, is shown in Figure 7, A. There is a high probability of exceeding the threshold in the center of the site and in the northwest portion of the site. All transect sample values are honored in each simulation such that at any sample location P fail can only be or 1.0. The value of R S is compared to three different values of R D : 0.80, 0.90 and This comparison results in a remediation decision for each decision panel. The resulting decision maps are shown in Figure 7, B, C and D for the three different values of R D. Figure 7 demonstrates that as the specified design reliability increases, the area of the site classified for remediation also increases. It is possible that too high of an R D value may result in an overly conservative remediation design. fail McKenna 13

14 28 A 28 B Clean North North Leave East East C 28 D Clean Clean North North Leave Leave East East 28 Figure 7. The probability of failure map (A) and the remediation decision maps for design reliabilities of 0.80 (B), 0.90 (C) and 0.95 (D). The actions to be taken are clean (to be remediated) or leave (to leave as is). Model Assessment The adequacy of the DSPP as applied to the problem of estimating the true distribution of anomalies at the site is examined under the same characterization goal stated in two different ways. In the first case, the estimated intensities are compared directly to the specified intensity regulatory threshold. In the second case, the actual number of objects within an area is compared to the number of objects in that same area corresponding to a cumulative probability value of the Poisson distribution defined by the simulated intensity value. In the first case, the remediation McKenna 14

15 decisions across a range of remediation design reliabilities are compared to the decisions that would be made with complete knowledge of the object distribution at the site. In the second case, the same decisions are compared to the decisions made under complete knowledge of the object distributions for the same range of design reliabilities and also for a range of values of the Poisson cdf. The model assessment procedure consists of checking the decision made using the simulated values of the intensity against the decision that would be made if the exhaustive distribution of the objects was known to the decision maker. The decision made from the simulated values of the Poisson intensity is a function of both the simulated reliability, R S and the specified design reliability, R D. The four different results of decisions that can be made and their relationship to the actual intensity values are: Correct (A): R R ) and ( ) (10) ( s D true c Correct (B): R R ) and ( ) (11) ( s D true c False Positive: R R ) and ( ) (12) ( s D true c False Negative: R R ) and ( ) (13) ( s D true c The two types of correct decisions (A and B) occur when the location is correctly left as is (R S R D and true < c ) or when the site is correctly targeted for remediation (R S < R D and true c ). The two types of incorrect decisions arise when the location is unnecessarily remediated (False Positive) or when the location is not remediated, but in fact has a true that exceeds c (False Negative). It is the latter of the two incorrect decisions that can be of severe consequence in UXO remediation. The first assessment case consists of using the 100 simulated values of Poisson intensity at each location to determine whether or not the simulated reliability is less than or greater than the design reliability. This determination is then combined with the relationship of the true Poisson intensity to the threshold intensity and the decision result (Equations 10 through 13) is determined. The result of the decision is recorded for each location within the site and the final totals for each type of result are tabulated. It is recognized that the design reliability can be set to optimize different remediation objectives. For example, it may be desirable to select a value of R D that minimizes the number of false positive and false negative errors under a loss function that counts each false negative as being of equal importance to three false positives. As another example, the objective may be simply to maximize the number of correct decisions. In order to examine the changes in the decisions made as a function of R D, the model assessment exercise for Case 1 is conducted over a range of R D from to 1.0. For the second assessment case, it is recognized that the intensity threshold, c, can also be interpreted as a discrete number of objects. For a 20x20 meter decision area (400m 2 ) the McKenna 15

16 threshold intensity of 0.10/m 2 translates to 40 objects. This threshold, expressed as a number of objects, is denoted N c. Each simulated intensity value is the sole parameter necessary to fully define a univariate Poisson distribution for that location. Any specified value of the cumulative probability of this Poisson distribution (Poiss CDF ) corresponds to a discrete number of objects. For a given Poiss CDF value, the corresponding number of objects can be compared to N c and the proportion of realizations below or above N c can be determined. The higher the selected Poiss CDF value, the larger the corresponding number of objects will be and the higher the probability of exceeding N c. The proportion of realizations that exceed N c are then compared to the selected, R D, to determine whether or not to remediate the decision panel. This final portion of the second assessment case and the comparison to the true number of objects within each panel to determine the decision results are the same as for the first assessment case. It is emphasized, that under this second assessment case, each decision is a function of both the Poiss CDF and the R D. The results of the model assessment are presented in Figures 8 and 9 for assessment cases 1 and 2, respectively. Figure 8 shows the proportion of correct, false positive and false negative decisions made on the example site as a function of the design reliability. The total number of decisions is equal to the total number of cells in the model (784); however, at the 159 locations where the transect samples were obtained, the correct decision is always made. Figure 8 demonstrates that for this example application the proportion of correct decisions is greater than 0.90 across most of the design reliability values. This result verifies that the estimation of the Poisson intensity through geostatistical techniques is accurate. The highest proportion of correct decisions (0.964) occurs at a design reliability of between 0 and In this range, the proportion of false negatives decreases at a faster rate than the proportion of false positives increases. For design reliabilities greater than approximately 0.85, the number of false positives becomes very large causing expenditure of resources on unnecessary remediation. Given that the occurrence of false negatives is of greater consequence than the occurrence of false positives, it is interesting to note that for this example problem, the proportion of false negatives becomes zero at design reliabilities greater than The images in Figure 9 show the decision results as a function of both the design reliability, R D, and the cumulative probability of the Poisson distribution, Poiss CDF. In general, the higher the R D and the higher the value of the Poiss CDF, the more conservative the remediation decisions will be. The three images in Figure 9 show that the proportion of correct decisions is not a strong function of the Poiss CDF value the contour lines in the top image of Figure 9 are nearly vertical. Similarly, the Poiss CDF value does not exert a strong influence on the proportions of the False Positives (middle image of Figure 9). The contour maps in the top and middle images are nearly the complement of each other as the number of false positives increases, the number of false negatives decreases. The Poiss CDF value does have an influence on the proportion of false positives (bottom image of Figure 9); however, the maximum and minimum proportion of false negatives are only separated by 12 percent. The influence of Poiss CDF is strongest at lower design reliabilities and lower values of the Poisson CDF (lower left corner of the lower image in Figure 9). McKenna 16

17 Proportion of Decisions Correct Decisions False Positives False Negatives Design Reliability Figure 8. Proportions of different consequences (correct decision, false positive or false negative) for decisions made across the full range of design reliabilities. These results indicate that the simulated Poisson distributions are generally accurate and precise. The accuracy is demonstrated by the relatively high proportion of correct decisions across all design reliabilities. The precision of the simulated Poisson distributions is evidenced by the insensitivity of the decision consequences to the chosen value of the Poisson cdf. Together the images in Figure 9 show that the simulated Poisson distributions are centered in locations that provide correct decisions (accurate) and the distributions are narrow enough that the decision is consistent for all values of the simulated distribution (precise). A relationship can be drawn between the results in Figure 8 and the results in Figure 9. For Poisson distributions with an integer-valued intensity greater than or equal to 1.0, the median value of the distribution is equal to the intensity. Due to the discrete nature of the Poisson distribution, this does not necessarily imply that the distributions are symmetric. Therefore, the decision results shown in Figure 8 are roughly equivalent to taking a section through each of the contour plots in Figure 9 at the Y-axis value of 0. They are not exactly equivalent as the simulated Poisson distributions do not necessarily have an integer-valued intensity and some may have a Poisson intensity less than 1.0 for the 400m 2 decision panel. For Poisson distributions with non-integer intensity, the mode of a Poisson distribution is the largest integer less than the value of the intensity. For integer-valued intensities, there is a tied mode at both the integer one less than the intensity and at the intensity. Therefore, the median of the Poisson distribution (Results in Figure 8) at a location provides a good approximation of the maximum likelihood estimate of the number of objects at that location. McKenna 17

18 Correct Decisions Poisson CDF Design Reliability False Positives Poisson CDF Design Reliability False Negatives Poisson CDF Design Reliability Figure 9. Contour maps of the proportion of different consequences for decisions made across the full range of design reliabilities and Poisson CDF values. The proportion of correct decisions is shown in the upper image, the proportion of false positives is shown in the center image and the proportion of false negatives is shown in the lower image. McKenna 18

19 Conclusions The work has outlined and demonstrated an approach for the characterization and spatial modeling of UXO sites. This work considers the spatial distribution of UXO to be the result of a doubly stochastic Poisson process. In a DSPP, the individual objects are located independently of one another, but the Poisson intensity of the objects across the site is a spatially correlated property. The spatial correlation of the Poisson intensity can be quantified and defined through a variogram. The variogram can be used in geostatistical simulation techniques to model estimates of the Poisson intensity and the uncertainty about those estimates. The geostatistical simulation techniques are capable of acting as Bayesian operators to incorporate spatially exhaustive prior information and limited sample data into a posterior estimated of anomaly intensity. The spatial prediction techniques outlined herein are demonstrated on an exhaustively surveyed UXO site. Two different techniques were used to assess the spatial modeling approach. Both assessment techniques make use of comparing the simulated results to a specified design reliability in order to make decisions regarding cleanup. The first case assessment examined the ability of the modeling approach to estimate the intensity of the Poisson distribution at all locations relative to the actual intensity values. The second assessment case recognizes that the property being estimated is the sole parameter of the Poisson distribution. This allows for comparison of the number of discrete objects corresponding to any value of the cumulative Poisson distribution against the actual number of objects at all locations across a range of design reliabilities. For the example application studied here, the proposed technique for modeling the intensity of UXO provided accurate and precise estimates of the true anomaly intensity. Across the full spectrum of design reliabilities, the proportion of correct decisions remains above 90 percent with the exception of very high design reliabilities (>0.85) where the proportion of false positive decision results becomes large. These results show that for this example at the more commonly specified design reliabilities of 0.95 or 0.99, additional cleanup (false positives) will be undertaken without any increase in the post-remediation safety of the site. Total design cost was not considered in this study, but it is obvious from these results that the highest remediation design reliability option will not be the minimum cost option for any loss function. Future work on geostatistical modeling of UXO will focus on the necessary precision and accuracy (lack of spatial bias) in the prior estimate and on developing multivariate approaches (cokriging) to better discriminate between UXO and non-uxo objects in the subsurface. The incorporation of prior information will be extended to work with multiple target locations contained within site boundaries. Acknowledgements This work was funded under the Strategic Environmental Research and Development Program (SERDP) UXO Cleanup program. This paper was improved through the critical reviews of Chris Rautman and Erik Webb. The author acknowledges James McDonald of the Naval Research Laboratory for providing the MTADS data. Sandia is a multiprogram laboratory operated by the Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC-94-AL McKenna 19

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21 USAESCH, 1999, Ordnance and Explosives (OE) Sites Unexploded Ordnance (UXO) Statistical Estimation Standard Operating Procedure (SOP), CEHNC , U.S. Army Corps of Engineers, Engineering and Support Center, Huntsville, 10 pp. Walpole R.E. and R.H. Myers, 1989, Probability and Statistics for Engineers and Scientists, 4 th ed., MacMillan Publishing Company, New York, 765 pp. Young, R. and L. Helms, 1999, Applied Geophysics and the Detection of Buried Munitions, U.S. Army Corps of Engineers, McKenna 21