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v Geostatistical Analysis of Spatial Data Noel Cressie Iowa State University 5.1 Introcluction AH data have (more or less) precise spatial and temporal labels associated with them. That is, a measurement is obtained from a particular location at a particular time, although that information may be lost by omission or made less precise by aggregation. For most of this chapter, it is assumed that only the data's spatial labels are important hence the term spatial data. As a discipline, spatial statistics has components of ah the classical areas of statistics, such as design, statistical methods (including data analysis and diagnostics), stochastic modeling, en cl statistical inference. Importantly, the spatial labels form an integral part of a spatial statistical analysis. Geostatis- tics is the area of spatial statistics that is concerned mostly with prediction of unknown values at given locations (or of aggregations over given regions). Typically, the prediction is based on univariate and bivariate distributions of the spatial values, and these distributions (or appropriate moments of them) are estimated from an initial analysis of the data. The prefix "geo" in geostatistics originally implied statistics pertaining to the earth (Matheron, 1963; see also Hart, 1954, who used the term difl:er- ently from Matheron, in a geographical context). However, more recently, geostatistics has been used to solve problems in agricultural engineering, atmospheric science, ecology, forestry, hydrology, meteorology, remote sens- ing, etc. Although it is Matheron's development of the area within mining 87

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88 that is best known, a Soviet meteorologist, L. S. Gandin, independently de- veloped a framework for inference that is virtually identical (Gandin, 1963~; he chose the term "objective analysis" instead of "geostatistics." Section 5.2 presents the basic ideas behind a geostatistical analysis, in- cluding a brief discussion of splines and conditional simulation. The first part of 5.3 gives several applications of geostatistics, and the second part discusses recent advances and future directions. 5.2 Theory ant! Methods~of Geostatistics Geostatistics is mostly concerned with spatial prediction, but there are other important areas, such as mode! selection, effect of aggregation, and spatial sampling design, that offer fruitful open problems; see 5.3.2. The emphasis in this section will be on a spatial-prediction method known as kriging. Matheron (1963) coined the term in honor of D. G. Krige, a South African mining engineer (see Cressie, 1990, for an account of the origins of kriging). 5.2.1 The Variogram First, a measure of the (second-order) spatial dependence exhibited by the spatial data is needed. A model-based parameter (which is a function) known as the variogram is defined here; its estimate provides such a measure. Statisticians are used to dealing with the autocovariance function. It is demonstrated here that the class of processes with a variogram contains the class of processes with an autocovariance function, and that kriging can be carried out on a wider class of processes than the one traditionally user! in statistics. Let {Z(s): s ~ D C R4) be a real-valued stochastic process defined on a domain D of the a?-dimensional space R4, and suppose that differences of variables lagged in-apart vary in a way that depends only on h. Specifically, suppose var(Z(s + h) - Z(s)) = 2'y(h) for all s, s+ h ~ D; (5.1) typically the spatial index s is two- or three-dimensional (i.e., d = 2 or 3). The quantity 2~), which is a function only of the difference between the spatial locations s and s + h, has been called the variogram by Math- eron (1963), although earlier appearances in the scientific literature can be found. It has been called a structure function by YagIom (1957) in

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89 probability and by Gandin (1963) in meteorology, and a mean-square`d dif- ference by Jowett (1952) in time series. Kolmogorov (1941) introduced it in physics to study the local structure of turbulence in a fluid. Nev- ertheless, it has been Matheron's mining terminology that has persisted. The varion:ram must satisfy the conditional negative semi-definiteness con- ~ v ~ ~ ~ _ Ok Ok _ _ n .~_ _ ~ ~ ~ t~ =_~_ ~ An__ _~ _1 1_ OlllOn, 4~-=1 ~j-=1 aiaj-~tStSj) ~ U' IOr any nmle number OI SDallal 10- ~;~= ~ 1~. ~ 1 ", . k. ~ area rang Flora 1~. ~ 1 Cl,tllVl[O tOt ~ B,.-~^J, ~1~ 1~11~111~1~ fat . ~',...7k} satisfying ~'k_1 ai = 0. When 27(h) can be written as 27(||h||)' for h ~ Rd. the variogram is said to be isotropic. Variogram models that depend only on a few parameters ~ can be used as summaries of the spatial dependence and as an important component of optimal linear prediction (kriging). Three basic isotropic models, given here in terms of the semivariogram (half the variogram), are: Linear mode! (valid in R3, ~ > T) )(h; 0) = { cO + by h 76 0 where t} = (Combo)' cO > 0, be > 0; Spherical mode} (valid in Ri, R2, and R3) Ash; B) = O h= 0 <`, co + Cst32 (~h~/aS) - 2 (~h~/aS)3] 0 < Ah < aS ~ co+Cs Ah > us where ~ = (co,cS'aS), co > 0' cs > 0, as > 0; Exponential model (valid in R4, ~ > 1) Ash; l9) { CO + Ce t!expel~~ hit /ae )] h i O where 6' = (cO,ce,ae), co > 0, ce > 0, ae > 0. Another semivariogram mode! is the rational quadratic mode! Ovoid in R4, ~ > I): ~ O h=0 7(h;~) = i co + i+~l~/ar h 7t 0 where ~ = (co~cr,a~), co ~ O. cr > 0, ar > 0

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go A semivariogram model that exhibits negative correlations caused by periodicity of the process is the wave (or hole-effect) mode! (valid in Ri, R2, and R3~: ~ O h=0 1~ co + Cu' ~ a'U siln(~lhil/aw ) h ~ 0 where ~ = (Co~cu~,atl~' co > 0, cu. > 0, au, > 0. A further condition that a variogram mode] must satisfy is (Matheron, 1971) 2~(h)/~h~2 , O as Ah ~ oo. In fact, the power semivariogram model, 7~; ~ { co+bp~h~> h 7{ 0 where 6, = (cO, bp, A), co > 0, bp > 0, 0 < ~ < 2, is a valid semivariogram mode] in R4, ~ > 1. When the process Z is anisotropic (i.e., dependence between Z(s) and Z(s + h) is a function of both the magnitude and the direction of h)' the variogram is no longer purely a function of distance between two spatial locations. Anisotropies are caused by the underlying physical process evolv- ing differentially in space. Sometimes the anisotropy can be corrected by a linear transformation of the lag vector h. That is, Ash) = 2'y~Ah~), h ~ R3, where A is a ~ x ~ matrix and 2~ is a function of only one variable. Replacing (5.~) with the stronger assumption cov(Z(s + h), Z(s)) = C(h) for all s, s + h ~ D (5.2) and specifying the mean function to be constant, i.e., E(Z(s)) = ,u for all s ~ D, (5.3) defines the class of second-order (or wide-sense) stationary processes in D, with covariance function Ct ). Time series analysts often assume (5.2) and work with the quantity p(~)C(~/C(O). Conditions (5.1) and (5.3) define

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91 the class of intrinsically stationary processes, which is now shown to contain the class of second-order stationary processes. Assuming only (5.2), Ache = C(O) - C(h), (5.4) that is, the semivariogram is related very simply to the covariance function. An example of a process for which 2'y(~) exists but C(~) does not is a one- dimensional standard Wiener process TWO: t > 0~. Here, 2^y~h) = the (-oo < h < off), but cov(W(t), W(u)) = mint, u), which is not a function of itup. Thus, the class of intrinsically stationary processes strictly contains the class of second-order stationary processes. Now consider estimation of the variogram from data {Z(si): i = 1, . . ., n3. Suppose these are observations on an intrinsically stationary process (i.e., a process that satisfies (5.1) and (5.3~), taken at the n spatial locations {Si: i = 1, . . ., n). Because of (5.3), var(Z(s + h)Z(s)) = E(Z(s + h)Z(s)~2. Hence, the method-of-moments estimator of the variogram 2~(h) is 2~(h) _ ~ [Z(si) - Z(sj)42/~N(h)~, h ~ R4, (5.5) N(h) where the average in (5.5) is taken over N(h) = {(si,sj): s'sj = h), and ~N(h)~ is the number of distinct elements in N(h). For irregularly spacer! data, N(h) is usually modified to {(si, sj): si - sj ~ T(h)), where T(h) is a tolerance region of Ret surrounding h. Other estimators, more robust than (5.51, are given in Cressie and Hawkins (1980) and Cressie (1991, sec. 2.4~. Parametric models, 2~;~), can be fit to the estimator (5.5) by various means; as a compromise between efficiency and simplicity, Cressie (1985) advocates minimizing a weighted sum of squares ~ , ~ ~ ':- ~ 2~(h~k)) _ 1 ~ ~1\rth~k) k ~ l 27(h~k);6~) with respect to variogram mode] parameters B. The sequence hill, . . ., h(I{) denotes the "lags" at which an estimator (5.5) was obtained, and which sat- isfy range and replication conditions such as those given by Journel and Huijbregts (1978, p. 194, eq. IIT.42~. Zimmerman and Zimmerman (1991) summarize and compare several methods of variogram-parameter estima- tion based on simulated Gaussian data. They find that the weighted-least- squares approach usually performs well, and never does poorly, against such competitors as maximum likelihood estimation (both ordinary and re- stricted) and minimum norm quadratic unbiased estimation.

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92 5.2.2 Kriging For the purposes of this section, assume that the variogram is known; in practice, variogram parameters are estimated from the spatial data. Suppose it is desired to predict Z(So3 at some unsampled spatial location so using a linear function of the data Z _ (Z(s13, . . , Z(s,,33': n Also) = ~ AiZ(si) i=1 It is sensible to look for coefficients uniformly unbiased and which minimize the mean-squared prediction error E(Z(So)Z(So)32. More generally, one could try to minimize E(~tZ(So)'p(z)~) with respect to predictor p(Z3, where ]; is a loss function. For example, the loss function proposed by Zellner (19863, (5.6) (ii: ~ = 1, . . . , n) for which (5.6) is LtZ(so3' p(Z3] = b {expta(Z(so3 - p(Z33]a(Z(so)p(Z)3 - 1), ~ > 0, allows overprediction to incur a different loss than underprediction. Mini- mizing mean-squared prediction error results from using [tZ(So), p(Z)] = 5tZ(So) _ p(Z)42, b > 0, which is the squared-error loss function. In all that is to follow, squared-error loss is used. The uniform unbiasedness condition imposed on (5.63 is simply E(Z(So)3 = ,u = E(Z(So33' for all ~ ~ R. which is equivalent to rt ~ Ai = 1. i=1 (5.7) Now, assuming (5.73, the mean-squared prediction error can be written in two ways. If the process is second-order stationary' n n n n E(Z(so3 - ~ AiZ(Si332 = C(03 - 2 ~ Aic~si - so3 + ~ ~ Ai~jC(sisj), i=1 i=1 i=1 j=1 (5.8) or' if the process is intrinsically stationary (a weaker assumption)' n n n n E(Z(so)~ AiZ(si))2 = 2 ~ Ai7(siso3~ ~ Ai~j7(si - sj) . (5 9) i=1 i=1 i=] j=l

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93 Using differential calculus and the method of Lagrange multipliers, optimal coefficients ~ = (~1,...,An)' can be found that minimize (5.9) subject to (5.7~; they are v . ~ . . ~ = r-1 [A + (1 ~ irrl~.~)l] and the minimized value of (5.9) (kriging variance) is ak~sO) =,~r-i,,_ (i - 1'r-l~12 . . (5.10) (5.11) In (5.10) and (5.11), fly = [bustsold. ..,~(s,,so)]t, 1 = (1~. is the n X n symmetric matrix with (i, j)th element nisiSj). The kriging predictor given by (5.6) and (5.10) is appropriate if the process Z contains no measurement error. If measurement error is present, then a "noiseless version" of Z should be predicted; Cressie (1988) has details on when and how this should be implemented. Thus far, kriging has been derived under the assumption of a constant mean. More realistically, assume ..,1)', and r Z(s) = Its) + t(S), S ~ D, (5.12) where E(Z(s)) = Cost for s ~ D and [~) is a zer~mean, intrinsically station- ary stochastic process with variers + h) - his)) = var(Z(s + h) - Z(s)) = 2~(h), h ~ R3. In (5.12) the "large-scale variation" p(~) and the "smallL- scale variation" b(~) are modeled as deterministic and stochastic processes, respectively, but with no unique way of identifying either of them. What is one person's mean structure could be another person's correlation struc- ture. Often this problem is resolved in a substantive application by relying on scientific or habitual reasons for determining the mean structure. Suppose ,u(s) = x(s)',`3, a linear combination of variables that could in- clude trend-surface terms or other explanatory variables thought to influence the behavior of the large-scare variation. Thus, p Z(S) = ~ Xj(s)>j + {(S), s ~ D, j=0 (5.13) where,l3 _ (,Bo,...,,dp)' are unknown parameters and [(a) satisfies (5.1) and (5.3) with zero mean. Although the mode! has changed, the problem of predicting Z(so) using an unbiased linear predictor (5.6) remains. The uniform unbiasedness condition is now equivalent to the condition >'X = xO, (5.14)

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94 where xo _ (xo(So), . ., xp(so))' and X is an n x (p+ 1) matrix whose (i, j)th element is z;_l(si). Then, provided (5.7) is implied by (5.14), minimizing the mean-squared prediction error subject to (5.14) yields the universal krig- ing predictor ZU(SO) = ~uZ where (5.15) Ad = Or- + x(x~r-ix)-~`xo - x~r-~)~; (5.16) the (universal) krig;~ng variance is basso) = or- - ~x~r-~ - xO)~x~r-ix)-~x~r-~ - xO). (5.17) Another way to write the equations (5.14) and (5.15) is Z(so) = van + v2xO, (5.18) where vat (an n x 1 vector) and v2 (a (p + 1) x 1 vector) solve Ivy + Xv2 = Z X'v~ = 0. (5.19) Equations (5.18) and (5.19) are known as the dual-kriging equations, since the predictor is now expressed as a linear combination of the elements of (~',xO). From (5.19), it is clear that splice smoothing is equivalent in form to universal kriging (see Watson, 1984, where the relationship between the two prediction techniques is reviewed). Kriging has the advantage that in practice the data are first used to estimate the variogram, so adapting to the quality and quantity of spatial dependence in the data. Furthermore, kriging produces a mean-squared prediction error, given by (5.17), that quantifies the degree of uncertainty in the predictor. Cressie (1989b) presents these two faces of spatial prediction along with 12 others, including disjunctive kriging and inverse-distance-squared weighting. 5.2.3 Conditional Simulation of Spatial Data Simulation of spatial data (Z(si): i = I, . . ., N) with given means {~(Si): i = 1, . . ., N) and covariances (C(si, sj): 1 < i ~ j < N) can be carried out in a number of ways, depending on the size of N and the sparseness of Rev, the N x N symmetric matrix whose (i,j)th element is C(si,sj). One way

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95 is to use the Cholesky decomposition EN = IN[N, where [N iS a lower- triangular N x N matrix (e.g., Golub and Van loan, 1983, pp. 86-90~. Then ZN(Z(SI ), . . ., Z(SN)~' can be simulated by ZN = AN + [NEN, (5.20) where EN (,u~s~),...,~(sN))', and EN is an N x 1 vector of simulated independent and identically distributed random variables, each with zero mean and unit variance. Other methods, including polynomial approx~ma- tions, Fourier transforms, and turning bands, are presented and compared in Cressie (1991, sec. 3.6~. Now consider the simulation of values of {Z(s): s ~ D) conditional on observed values Zn Call this conditionally simulated process {W(s): s ~ D), end suppose (V(s): s ~ D) is an unconditionally simulated process with the same first and second moments as {Z(s): s ~ D). For example, (5.20) might be used to simulate VN- (V(SI),...,V(SN)~/, where N > n. Consider conditional simulation at an arbitrary location so+ in D. Now write En en En+1 = C' C(sr~+l, Sn+1 ) and notice that the two terms of the decomposition Z(sn+~) = CnEn Zn + [Z(sn+l)CnEn Zn] (5.21) are uncorrelated. Hence, the conditional simulation W(Sn+l ~ = C' En 1 Zn + [V(sn+l ~C' En 1vn] ~ Sn+1 ~ D, (5.22) has the same first two moments, unconditionally, as the process {Z(s): s ~ D) and W(si) = Z(si), i = 1,...,n. That is, unconditional simulation of sample paths of V yields, through (5.22), conditionally simulated sample paths of W. It is apparent from (5.20) and (5.21) that when the ci's are Gaussian, so too is the process {W(s): s ~ DI. However, this may not reflect the reality of the conditional process when the original process (Z(s): s ~ D) is "far from" Gaussian even though the first two moments match and the two processes agree at the data locations. There is clearly a danger in using conditional simulation uncritically.

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96 5.3 Applications and Research Frontiers A geostatistical analysis of spatial data has a "nonparametric" flavor to it, in that inferences are based on properties of univariate and bivariate clis- tributions of Z(s) and Ztu), which are estimated from the data. In other words, assumptions are few, although often it helps to transform the data so that they are Gaussian-like. In contrast, Markov-random field models, or simultaneous spatial autoregressive models, have a very rigid structure that is not so well adapted to problems of spatial prediction (kriging). Sec- tion 5.3.1 shows how geostatistics has considerable flexibility in applications across diverse scientific clisciplines. 5.3.1 Applications .. . . . . .. . . ... . The strength of geostatistics over more classical statistical approaches is that it recognizes spatial variability at both the "large scale" and the "small scale," or in statistical parlance, it models both spatial trend and spatial correlation. Trend-surface methods include only large-scale variation by assuming independent errors. Watson (19723 eloquently compares the trio approaches and points out that most geological problems have a small-scare variation, typically exhibiting strong positive correlation between data at nearby spatial locations. The books by David ~1977), Journel and Huijbregts (1978), and Clark (1979) are all aimed at applications of geostatistics in the mining industry. The geostatistica] method has also found favor among soil scientists who seek to map soil properties of a field from a small number of soil samples at known locations throughout the field; soil pH in water, soil electrical conductivity,-exchangeable potassium in the soil, and soil-water infiltration are some of the variables that could be sampled and mapped. Water erosion is of great concern to agriculturalists, since rich topsoil can be carried away in runoff water. The greater the soil-water infiltration, the less the runoff, resulting in less soil erosion and less stream pollution by pesticides and fertilizers. Also, greater infiltration implies better soil struc- ture, which is more conducive to crop growth. Cressie and Horton (1987) describe how double-ring infiltrometers were placed at regular locations in a field that had received four tillage treatments' moldboard, paraplow, chisel, and no-till. From these data, the spatial relationships were characterized; Gotway and Cressie (1990) used the resulting stochastic models to estimate efficiently the tillage effects and to build a spatial analysis of variance table,

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97 from which tilIage differences can be tested. Kriging can be applied in geophysical problems that require accurate mapping of the ocean floor. Data are slopes or depths and a variety of as- sumptions are made about the large-scale and small-scale variations defined by (5.12) (e.g., Shaw and Smith, 1987; Smith and Jordan, 1988; Gilbert, 1989; MaTinverno, 1989~. This area of investigation would benefit from geo- statistical analyses that use the data initially to fit an appropriate variogram mode} and then draw kriging maps based on the fitted variogram. Applications of geostatistics abound in other areas, such as rainfall pre- cipitation (e.g., Ord and Rees, 1979), atmospheric science (e.g., Thiebaux and Pedder, 1987), acid deposition (e.g., Bilonick, 1985), and groundwater flow (e.g., Clark et al., 1989~. Examples from groundwater flow and acid deposition will now be used to illustrate the geostatistical method described in 5.2. Flow of Groundwater from a Proposed Nuclear Waste Site In 1986 three high-level nuclear waste sites were proposed in the United States (in Nevada, Texas, and Washington), thus prompting study of the soil and water-bearing properties of their surrounding regions. The chosen site will probably contain more than 6S,000 high-level waste canisters placed about 30 feet apart in holes or trenches surrounded by salt, at a depth of 2,000 feet. However, leaks could occur, or the radioactive heat could cause the tiny quantities of water in the salt to migrate toward the heat until eventually each canister would be surrounded by about 6 gallons of water. The chemical reaction of salt and water would create hydrochloric acid that could slowly corrode the canisters. Eventually, the nuclear wastes could reach the aquifer and sometime later contaminate the drinking water. Therefore, the types of questions one might ask are: If a nuclear waste site were to be designated for, say, Deaf Srn~th County, Texas, what are the risk parameters for radionucTides contaminating the groundwater? Where would they flow? How long would they take to get there? Here the direction- of-flow question will be addresse~l; kriging will be used to draw ~ ~n~.i~1 anon of potentiometric heads throughout the area of interest. ~ ~r~V^~^ -lot Potentiometric heads in the West Texas/New Mexico region are shown in Figure 5.1, and are given by Harper and Furr (1986~. They were measured by driLing a narrow pipe into the aquifer and letting the water find its own level in the pipe. Measurements are given in feet above sea level. An anisotropic variogram model was fit to the data; in each of two

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98 209.4 1 4 1S9.414 109.414 S9.414 9.~14 _ f ~ ~se,1 L: _ .2611 . ~ ~J: 2891 ~ 2SS] ~ 134 - ~ ~ . 1 2~t 3510 - 1 ~ ~ -145.24 -85~24 ~ t476 I'.~' ~ ~ t4O: - _ t364 htOORt e ~. O _ teas ~or~cR I'rt ~. I J. ~ ~ OLDHA" O it'' _ ~ 22 - RANOALL Al OEA, l - ITH t I I ~ ~ I ~ r 255, t 2,5, BalSCO. | BAILY 1 1 ~' _ DALLA~t 5~AN = Oe~lL _E ~"O - . .102< 148.e l3' ~/ IOC' ~ 1030 CARSON e 1~ 1408 aRA~ .~ct" `RU t TRONG _ 1437 .1, 13 ' ~CELER CASTRO SW15~1ER 1 2116 1 1 ~ ~rse 1 `_ _ .~8 1 2548 ~ 1 .___ ~ ~ 17" 35,S 23S. 2t28 2S't6 2S44 coc~a.. 2739 1433 I.~. er" -4S.24 4.~. S4.~. 104.~' 154.~. 1828 1~ 106 IfLOYo 153C I ~ l," tcoc ~ 17' 1n94 ~ .~ - _~' CROS.Y ~ st ~' 1588 | ~ 180 1 1 . , 1 FIGURE 5.~: Locations en c! levels of piezometric-head data in West Texas/New Mex~co. Amarillo is located close to the "1527" well in Pot- ter Country. Reprinted, by permission, from Cressie (1989a). Copyright (~) l9S9 by American Statistical Association. Orthogonal directions, values of ~ = (~, 82, 83) in 27 0, 82 > 0, and 0 < 83 < 2. From the fitted variogram, kriging predictors {Z(so): so ~ D) given by (5.6) and (5.10) and kriging standard errors {~k(So): So ~ D) given by (5.11) were obtained; see Cressie (1989a). Figure 5.2 shows the predicted surface, from which it can be concluded that contaminated groundwater from Deaf Smith County, Texas, would flow directly "downhill" toward

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99 Doo f Snatch Co. ~41 _ 31q2 2843 ~44 _ _ 946 - 616 - '347 _ piezometric heat (fit) off Buzz la48 - ~ . South ~ . 0.~ _ 1 72. 0~ 1 A; 144.~ 81;00 900 -63.00 -135 00 N ~7f FIGURE 5.2: Three dimensional view of krig~ng surface {Z(so): so ~ D), from the northeast corner of D. Reprinted, by permission, from Cressie (1989a). Copyright ~ 1989 by American Statistical Association.

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100 Amarillo, Texas. However, Amarillans need not be concerned; a decision was made in 1987 by the U. S. Congress to locate the nation's high-level nuclear waste dump site in Nevada, probably at Yucca Mountain. Acid Deposition and Network Design It is generally accepted that an important factor in the relatively recent increase of acid deposition is the emission of industrial by-products into the atmosphere; the consequences for aquatic and terrestrial ecosystems are potentially disastrous. Most fish populations in freshwater lakes are very sensitive to changes in pH (ElFAC, 1969~. More fundamentally, such changes could also adversely affect most other aquatic organisms and plants, resulting in a disruption of the food chain. Acid deposition has also been closely connected with forest decline (Pitelka and Raynal, 1989) in both Europe and the United States. In the United States, acid deposition results mainly from the atmospheric alteration of sulfur and nitrogen air pollutants produced by industrial pro- cesses, combustion, and transportation sources. Total acid deposition in- cludes acid compounds in both wet and dry form. Dry deposition is the removal of gaseous pollutants, aerosols, and large particles from the air by direct contact with the earth (NAPAP, 1988~. Since dry deposition is ctif- ficult to monitor, and attempts at any such monitoring are relatively new, we focus on wet deposition here. Wet deposition, or acid precipitation as it is commonly caned, is defined as the hydrogen ion concentration in all forms of water that condenses from the atmosphere and fans to the ground. Measurement of the total annual amount of hydrogen ion is the end result of a very complicated process begin- ning with the release of pollutants into the atmosphere. They might remain there for up to several days and, depending on a variety of meteorological conditions (e.g., cold fronts or wind currents), they may be transported large distances. While in the atmosphere, the pollutants are chemically altered, then redeposited on the ground via rain, snow, or fog. A model for the spatial distribution of total yearly hydrogen ion (H+), measured on the Utility Acid Precipitation Study Program (UAPSP) net- work in 1982 and 1983, was developed by Cressie et al. (1990~. We present their results for the 1982 data, including implications of the fitted model for network design. Figure 5.3 is a map of the eastern half of the United States, showing the 19 UAPSP monitoring sites. Their latitudes, longitudes, and annual acid

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101 FIGURE 5..3: Monitoring sites of the UAPSP network for the years 1982 and 1983. The square denotes an optimally located additional site. Reprinted, by permission, from Cressie et al. (1990~. Copyright ~ 1990 by Elsevier Science Publishers, Physical Sciences and Engineering Division. ~ 2 (~) 5.83 4.29 2.74 ,~ . _ \ _ _ _ E ~ \ _\ //~1O / ~32 FIGURE 5.4: Median-polish surface obtained from the 1982 data. Units on the vertical axes are in ,umoles H+/cm2. Reprinted, by permission, from Cressie et al. (1990~. Copyright ~ 1990 by Elsevier Science Publishers, Physical Sciences and Engineering Division.

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102 depositions (in Emote H+/cm2) for 1982 (and 1983) are presented in Cressie et al. (1990~. By grouping nearby sites, a 4 x 3 table of acid-deposition data was constructed. The table was then median-polished (e.g., Emerson and Hoaglin, 1983), from which a crude picture of the large-scale variation was obtained; see Figure 5.4. In the east-west direction there appears to be a positive linear trend, reflecting higher acid-deposition levels in the east. However, in the north- south direction, the trend is quadratic, with higher levels in the central region and lower levels in the extreme north and extreme south. The surface in Figure 5.4 was subtracted from the original data to ob- tain residuals, Basil: i = 1,...,19~. Using great-arc distances to de- fine distances between sites, an isotropic (robust) variogram estimator was computed, to which a spherical variogram mode! was fit by weighs Ed least squares (5.2.1~. The fitted parameters were co = 0.608(,umoles H+/cm2~2, cs = 2.041~`umoles H+/cm2~2, and Us = 361.210miles. Figure 5.5 gives a graphical representation of the results. Optimal spatial prediction (ordinary kriging) can be implemented on the residual process b(~) through equations (5.6), (5.10), and (5.11~. A predicted surface of acid-deposition levels can then be obtained by adding back this kriging surface (of the residual process) to the median-polish sur- face shown in Figure 5.4. This is called median-polish kriging by Cressie (1986~. The mean-squared errors of prediction (median-polish-kriging vari- ances) (~k(So): So ~ eastern United States) are given by (5.11) and will now be used to choose the optimal location of a new site. Let SUse, . . ., sn) denote the existing network and let sp _ {Sn+] ~ . . . sn+m) denote m > 2 potential new sites from which one will be chosen. Define S+i _ S U {sit, i = n + 1, . . ., n + m to be augmented networks. Then S+j is preferred if it predicts best the remaining m - 1 sites in So (on the average). Specifically, let Basso; S+i) denote the kriging variance for predicting the acid-deposition level at so using the augmented network S+i, where i = n + 1, . . ., n+ m. For illustration, define the objective function 72+m V(sj) = Ad, Nisi; S+j)/(m1), j = n + 1, . . ., n + m . (5.23) i=n+1 Then the site in Sp that achieves min{V(s'): j = n + 1, . . ., n + m) will be declared the optimal site to add. (Other criteria are considered in Cressie et al., 1990.)

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103 7 2 1 o * * 0 200 400 600 h 6300 1 000 1 200 FIGURE 5.5: Empirical variograms (robust) for median-polished residuals. The superimposed dashed line indicates the weighted-least-squares fit. Units on the vertical axes are in (,umoles H+/cm2~2; units on the horizontal axes are in miles. Reprinted, by permission, from Cressie et al. (1990~. Copyright ~ 1990 by Elsevier Science Publishers, Physical Sciences and Engineering Division. Eleven potential sites (Minneapolis, Minnesota; Des Moines, Iowa; Jef- ferson City, Missouri; Madison, Wisconsin; Springfield, Illinois; Altoona, Pennsylvania; Charlottesville, Virginia; Charleston, West Virginia; Balti- more, Maryland; Trenton, New Jersey; and Knoxville, Tennessee) were cho- sen to improve geographic coverage of the existing network (of 19 sites). From among these eleven sites, Baltimore (marked with a square on Fig- ure 5.3) was chosen as the optimal site to add. Its associated average kriging variance, given by (5.23), was 2.56(,umolesH+/cm2~2, compared to Min- neapolis's 2.59 (the second smallest value); Charlottesville had the largest value of 2.77. 5.3.2 Research Frontiers Change of Support The change-of-support problem remains a major challenge to geostatisti- cians. Although data come as Z = (Z(s~),...,Z(sn))', inference may be required for Z(B) ~~ {B Z~u)du. Kriging adapts very easily to ac-

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104 commodate the change from point support so to block support B. For example, in (5.10) and (5.11), fly is modified to ~(B) [~BjlB:(s~ - U) flu, . . ., ~ iB 7(suu) bull, and in (5.11) ak(~) has the extra term ~ iBiB 7(Uv) MU ~v. But in mining applications and emission com- pliance, for example, the quantity of greatest interest is the conditional distribution Pr(Z(B) ~ z ~ Z). Both disjunctive kriging (Matheron, 1976) and indicator kriging (Journel, 1983) attempt to answer this question based on bivariate distributional properties of the (possibly transformed) process. The problem is important enough to pursue beyond these initial approaches. Multivariate Spatial Data Prediction of a value Z(sO) based on data Z and observations on other processes is known as cohriging. The appropriate generalization of the vari- ogram (5.1) is the cross variogram var(Y(u)Z(v)) = 2:yz~u, v), (5.24) where Y(u) and Z(v) are normalized to have the same units. CoLriging equations for predicting Z(so) from Z and Y can be obtainer! in terms of fizz, Gym, and Ye (Clark et al., 1989~. However, there is a dearth of models for (5.24~; the basic requirement for a valid model is that its parameters can be estimated from the partial realization (Z', Y') of the bivariate process. Variogram Mode] Fitting and its Effect on Inferences The variogram (5.1) has the property of conditional negative-definiteness. Based on a nonparametric estimator 2~( ), say, current practice is to fit a parametric mode! 2~; B], which is guaranteed to be conditionally neg;ative- ~ day , ~ . ~ . ~ ~ . ~ . . ~ ~ ~ ~ ~ . ~ , ~ ~1 definite. Is there a way to find a nonparametr~c fit to 2 yL ) from the set ol all conditionally negative-definite functions? If it can be found, its description is not likely to be very parsimonious. Variogram-model choice should probably balance the closeness of its fit to the data, with its predictive power. For temporal data, Rissanen (1984, 1987) takes such an approach; however, his being able to sequence the observations is important, since the accumulated prediction errors form an integral part of his method. Development of a spatial version is an area worth investigating. Now, having chosen a model 2~; B), what effect does the estimation of 49 have on inferences for Z(so)? Zimmerman and Cressie (1991) have some initial results, but considerable further research is needed to resolve this important problem.

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105 Bibliography [1] Bilonick, R. A., The space-time distribution of sulfate deposition in the northeastern United States, Atmos. Environ. 17 (1985), 2513-2524. [2] Clark, I., Practical Geostatistics, Applied Science Publishers, Essex, England, 1979. A] Clark, I., K. L. Basinger, and W. V. Harper, MUCK: A novel ap- proach to co-kriging, pp. 473-493 in Proceedings of the Conference on Geostatistical, Sensitivity, and Uncertainty Methods for Groundwater Flow and Radionuclide Transport Modeling, B. E. Buxton, ea., Battelle Press, Columbus, Ohio, 1989. [4] Cressie, N., Fitting variogram models by weighted least squares, '7. Int. Assoc. Math. Geol. 17 (1985), 563-586. [5] Cressie, N., Kriging nonstationary data, A. Amer. Stat. Assoc. 81 (1986), 625-634. [6] Cressie, N., Spatial prediction and ordinary kriging, Mathematical Ge- ology 20 (1988), 405-421. [7] Cressie, N., Geostatistics, Am. Stat. 43 (1989a), 197-202. [8] Cressie, N., The many faces of spatial prediction, pp. 163-174, in Geo- statistics, Vol. 1, M. Armstrong, ea., Kluwer, Dordrecht, l989b. [9] Cressie, N., The origins of kriging, Mathematical Geology 22 (1990), forthcoming. [10] Cressie, N., Statistics for Spatial Data, John Wiley and Sons, New York, forthcoming (1991~. t11] Cressie, N., and D. M. Hawkins, Robust estimation of the variogram, I, ]. Int. Assoc. Math. Geol. 12 (1980), 115-125. t12] Cressie, N. A. C., and R. Horton, A robust/resistant spatial analysis of soil-water infiltration, Water Resour. Res. 23 (1987), 911-917.

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108 [37] Smith, D. K., and T. H. Jordan, Seamount statistics in the Pacific Ocean, J. Geophys. Res. 93 (1988), 2899-2918. [38] Thiebaux, H. J., and M. A. Pedder, Spatial Objective Analysis with Applications in Atmospheric Science, Academic Press, London, 1987. [39] Watson, G. S., Trend surface analysis and spatial correlation, Geol. Soc. Amer., Special Paper 146 (1972), 39-46. [40] Watson, G. S., Smoothing and interpolation by kriging and with splines, ]. Int. Assoc. Math. Geol. 16 (1984), 601-615. [41] YagIom, A. M., Some classes of random fields in n-dimensionaI space, related to stationary random processes, Theory Probab. Its Appl. 2 (1957), 273-320. [42] ZeDner, A., Bayesian estimation and prediction using asymmetric Toss functions, '7. Amer. Stat. Assoc. 81 (1986), 446-451. t43] Zimmerman, D. L., and N. Cressie, Mean squared prediction error in the spatial linear mode! with estimated covariance parameters, Ann. Inst. Stat. Math. 43 (1991), forthcoming. [44] Zimmerman-, D. L., and M. B. Zimmerman, A Monte CarIo comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors, Technometrics 33 (1991), forthcoming.