Spatial Inference of Nitrate Concentrations in Groundwater Dawn Woodard Operations Research & Information Engineering Cornell University joint work with Robert Wolpert, Duke Univ. Dept. of Statistical Science and School of the Environment and Michael O Connell, Waratah Corporation, Durham, NC 1
Outline 1 Nitrates in Groundwater 2 The Data 3 Existing Approaches for Pollutant Estimation 4 Bayesian Moving-Average Models 5 Results 6 Conclusions and Future Work 2
Nitrates in Groundwater High levels of nitrates in groundwater can cause health and environmental problems Nitrate contamination in groundwater can be due to: agricultural fertilization, septic systems, etc. 4
Nitrates in Groundwater Measurements of nitrates in groundwater have been obtained over the mid-atlantic states [Ator 1998]: > 8.3 mg/l mid range < 0.75 mg/l 5
Nitrate Estimation Desire geographic interpolation of nitrate levels Distinct regulatory goals require inference at distinct geographic scales... fine-scale, regulatory units (e.g. counties), hydrologic units (e.g. watersheds)....as well as distinct risk measures average nitrate concentration, probability of exceeding a threshold, averaged by region, maximum nitrate concentration occurring in each region. 6
Nitrate Estimation 1. Wish to perform inference for multiple scales and risk measures without refitting the model or ad-hoc aggregation 2. Need the uncertainty associated with all estimates of risk measures 3. Desire a nonparametric approach 7
Model Summary We utilize a nonparametric spatial statistical model for nitrate concentrations at all locations Bayesian approach: uncertainty about the nitrate concentration and its average over various regions are all random variables......for which we can compute expected values (best overall estimates) and probabilities of exceeding specified thresholds 8
Data Summary Nitrate measurements from 929 wells in the mid-atlantic states Taken between the years of 1985 and 1996 10
Existing Approaches for Pollutant Estimation Pollutant concentrations can be estimated separately for each region. This leads to unreliable estimates for regions with few measurements When inference is desired at a single spatial partition (e.g. counties), lattice models can be used 12
Existing Approaches for Pollutant Estimation Kriging allows smooth spatial interpolation. It models the pollutant concentration Λ(x) at location x X as: log Λ(x) = J X j (x)β j + Z (x) j=1 where Z (x) is a mean-zero Gaussian process. 13
Existing Approaches for Pollutant Estimation A kriged surface with only an intercept term β 0 : > 8.3 mid range < 0.7525 Latitude 34 36 38 40 42 0 6 12 82 80 78 76 74 Longitude 14
Existing Approaches for Pollutant Estimation The interpolated surface is reasonable. > 8.3 mid range < 0.7525 Latitude 34 36 38 40 42 0 6 12 82 80 78 76 74 Longitude 15
Existing Approaches for Pollutant Estimation However, the confidence intervals are very wide in many locations, even where there is much data. Lower Bound: Upper Bound: > 8.3 mid range < 0.7525 > 8.3 mid range < 0.7525 Latitude 34 36 38 40 42 0 6 12 Latitude 34 36 38 40 42 0 6 12 + 82 80 78 76 74 82 80 78 76 74 Longitude Longitude 16
Existing Approaches for Pollutant Estimation The Gaussian process model makes strong assumptions about the distribution of the nitrate concentration The wide confidence intervals may be due to the data violating these assumptions Let s look at a nonparametric alternative 17
Moving-Average Models Ickstadt and Wolpert (1997) and Wolpert and Ickstadt (1998) introduced methods for interpolating intensities of spatial point processes by modeling the intensity Λ(x) as a moving average of an underlying stochastic process The approach has been used in non-point-process applications: identifying proteins in mass spectroscopy [House, Clyde, and Wolpert 2006] inferring temporal fluctuations in sulfur dioxide pollution [Tu 2006] 19
Moving-Average Models The concentration Λ(x) at location x X is modeled as: Λ(x) = J X j (x)β j + j=1 M m=1 k(x, s m )γ m for k(x, s) a kernel function on X S. The number M, locations s m, and magnitudes γ m of the mixture components are uncertain; so are the coefficients β j. 20
Moving-Average Models Interpretation of the spatial portion of the model, m k(x, s m)γ m, for pollutant level estimation: the pollutant surface is the sum of an unknown number of point sources with unknown locations and magnitudes......where the pollutant spreads out from each source in a manner consistent with the kernel k(, ) 21
Moving-Average Models The ith measurement Y i is assumed to have a log-normal distribution centered at Λ(x i ): log Y i Normal( log Λ(x i ),σ 2 ) The measurement variance σ 2 is unknown 22
Moving-Average Models The kernel is specified as: k(x, s) =exp where d > 0 is a constant { 1 } x s 2 2d 2 This decreases smoothly as a function of distance from the center s 23
Prior Specification The parameters are σ, β,m, {s m }, {γ m } For the nitrates analysis we do not include covariates, so β is not in the model Following common practice we assign for fixed α σ > 0 and ρ σ > 0 σ 2 Gamma( α σ,ρ σ ) 24
Prior Specification The spatial term in the model can be rewritten where M m=1 k(x, s m )γ m = Γ(ds) = is a discrete measure on S. M m=1 S k(x, s)γ(ds) γ m δ sm (ds) Under some reasonable assumptions (e.g. Γ(A) and Γ(B) are indep. for disjoint sets A, B S), our prior on Γ must be a Lévy random field We use the well-known gamma random field on a bounded set S R d 25
Prior Specification The gamma random field prior for Γ(ds) implies that for fixed α, ρ, ɛ > 0: The number of mixture components M has a Poisson distribution: M Poisson(α S E 1 (ρɛ)) where E 1 is the exponential integral function Conditional on M, the locations s m are independently uniformly distributed on S The magnitudes γ m are independently distributed with a truncated gamma distribution, having density: f (γ) γ 1 e ργ 1(γ >ɛ) 26
Prior Specification The constants d, α, ρ, ɛ, α σ, and ρ σ are given reasonable values using expert knowledge and information in the data. 27
Prior Specification These choices lead to prior surfaces Λ(x) like this one: Latitude 34 36 38 40 42 0 6 12 82 80 78 76 74 Longitude 28
Prior Specification There are some areas with high nitrate concentrations; these areas have random (unknown) locations a priori: Latitude 34 36 38 40 42 0 6 12 82 80 78 76 74 Longitude 29
Nitrate Inferences The posterior mean surface for the nitrate concentration is: Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 6 12 82 80 78 76 74 Longitude 31
Nitrate Inferences There are hot spots in the Chesapeake region, southeast Pennsylvania, etc. Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 6 12 82 80 78 76 74 Longitude 32
Nitrate Inferences There are low-nitrate areas in West Virginia, eastern North Carolina, etc. Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 6 12 82 80 78 76 74 Longitude 33
Nitrate Inferences Areas with sparse / no data have expected concentration close to the prior mean of 4.4 mg/l. Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 6 12 82 80 78 76 74 Longitude 34
Nitrate Inferences The posterior standard deviation of the nitrate concentration is: Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 2 4 82 80 78 76 74 Longitude 35
Nitrate Inferences This is a measure of estimation uncertainty. Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 2 4 82 80 78 76 74 Longitude 36
Nitrate Inferences Areas with sparse / no data have high uncertainty. Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 2 4 82 80 78 76 74 Longitude 37
Nitrate Inferences Most areas with numerous measurements have low uncertainty. Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 2 4 82 80 78 76 74 Longitude 38
Nitrate Inferences The spotty quality of the figure is due to noise from the computational method, and could be eliminated with, e.g., parallelization. Latitude 34 36 38 40 42 > 8.3 mid range < 0.7525 0 2 4 82 80 78 76 74 Longitude 39
Nitrate Inferences Average nitrate concentrations over counties can be obtained: Latitude 34 36 38 40 42 > 5 mg/l mid range < 1 mg/l 82 80 78 76 74 Longitude 40
Nitrate Inferences So can the probability that the nitrate concentration exceeds the regulatory limit, averaged by county: Latitude 34 36 38 40 42 > 8 % mid range < 2 % 82 80 78 76 74 Longitude 41
Nitrate Inferences This probability is low in most regions with a lot of data: Latitude 34 36 38 40 42 > 8 % mid range < 2 % 82 80 78 76 74 Longitude 42
Nitrate Inferences This probability is equal to its prior value of 5% in regions with sparse / no data: Latitude 34 36 38 40 42 > 8 % mid range < 2 % 82 80 78 76 74 Longitude 43
Nitrate Inferences Again there is a hot spot in the Chesapeake region: Latitude 34 36 38 40 42 > 8 % mid range < 2 % 82 80 78 76 74 Longitude 44
Nitrate Inferences The green counties around the edge are due to edge effects of the model. Latitude 34 36 38 40 42 > 8 % mid range < 2 % 82 80 78 76 74 Longitude 45
Conclusions The Bayesian moving-average model allows inference of a variety of risk measures at a variety of spatial scales. Uncertainty measures are available for all these estimates. The model is nonparametric. It has a desirable interpretation in the context of pollutant level estimation. 47
Conclusions The moving-average model has a computational advantage over kriging for large data sets Likelihood evaluation for the moving-average model is O(NM), where N is the number of data points and M is the number of mixture components. Likelihood evaluation is O(N 3 ) for kriging. 48
Future Work Covariates such as climatic, geologic, and land use factors could be added to the nitrates analysis. The fixed kernels could be replaced with kernels that have priors on the scale and eccentricity. This would allow the model to capture, e.g., pollutant point sources that spread out more in one direction than another due to flow patterns. Additional risk measures, e.g. the % of population exposed to nitrate levels above 10 mg/l 49
References Ator, S. W. (1998). Nitrate and pesticide data for waters of the mid- Atlantic region. USGS Open File Report 98-158, Reston, VA: U.S. Geological Survey. House, L. L., Clyde, M. A., and Wolpert, R. L. (2006). Nonparametric models for peak identification and quantification in mass spectroscopy, with application to MALDI-TOF. Discussion Paper 2006-24, Duke Univ. Dept. of Statistical Science. Ickstadt, K. and Wolpert, R. L. (1997). Multiresolution assessment of forest inhomogeneity, in Case Studies in Bayesian Statistics, Vol. III, NY: Springer-Verlag, pp. 371-386. Tu, C. (2006). Bayesian nonparametric modeling using Levy process priors with applications for function estimation, time series modeling, and spatio-temporal modeling. PhD thesis, Duke Univ. Dept. of Statistical Science. Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika, 85, 251-267. 50