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4
Spatial Statistics in
Environmental Science
Peter Guttorp
University of Washington
4.l Introduction
During the last 15 years much attention has been focused on environmen
tal problems, such as tree and lake death from acidic precipitation, global
warming due to increased carbon dioxide concentration, and a possible re
duction of the ozone layer in the stratosphere. For example, the problem
of longterm trends in atmospheric deposition was the subject of a recent
report of the National Research Council (1986~. Many statistical problems
are emerging from research in the environmental sciences. This chapter ad
ciresses the estimation of spatial covariance, with an application to a solar
radiation network. Also discussed briefly are some aspects of monitoring
network design and the usefulness of point process models in developing
global climate models.
4.2 Estimating Spatial Covariance
The fundamental problem of environmetrics is that the observable processes
of interest are highly variable. Noise typically overwhelms the signal. For
example, when studying wet deposition of sulfate or nitrate at a location, the
variability of rainfall constitutes a large fraction of the observed variability
(PolIack et al., 1989~. Statistically precise methods for signal extraction are
vital for policymakers.
71
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72
In order to assess the severity of an environmental insult, the researcher
typically has access to monitoring data from a relatively sparse network
of stations, while assessment of the mean level (averaged both temporally
and spatially) is needed over unobserved locations. Thus it is necessary to
use spatial interpolation methods. The most common such method, namely
king, is discussed in Chapter 5 of this report. A Bayesian nonparametric
method for interpolation, caned regularization, has been developed by Zidek
and coworkers (Weerahandi and Zidek, 1988; Ma et al., 1986) with environ
mental applications in mind. Common to these methods is the necessity to
determine the spatial covariance.
The development of nonparametric procedures for interpolating observed
spatial covariances of a random function sampled at a finite number of Toca
tions has lagged wed behind the development of interpolation methods for
the expected value of the underlying function. The kriging and regulariza
tion methods mentioned above depend explicitly on the spatial covariance or
variogram functions. Most approaches to modeling spatial covariance struc
ture have been parametric and have assumed isotropy and/or stationarity.
The bestknown models are parametric forms for the variogram originating
in Matheron's theory of regionaTized variables. The common assumption of
a spatially stationary variogram in kriging analyses was called the "intrin
sic dispersion law" by Matheron. Switzer and Loader (1989) propose a less
parametrically oriented method to fit empirical dispersion or covariances.
Since the empirical sitepair covariances may themselves be subject to sam
pling variability, some degree of parametric mo(leling is required, which at
the same time respects the apparent heterogeneity in the covariance field.
Basically, a parametric covariance mode] is forced on the available empirical
covariances, and modified covariance estimates are obtained by shrinking
toward the parametric covariances.
A nonparametric approach to global estimation of the spatial covariance
structure of a random function Zig, t) observed repeate(lly at a finite num
ber of sampling stations xi, a = 1, 2, . . ., N. in the plane has been developed
by Sampson and Guttorp (1990). The true covariance structure is assumed
to be neither isotropic nor stationary, but rather a smooth function of the
geographic coordinates of station pairs (xi, xj). Using a variant of multidi
mensional scaling (MDS), a twodimensional representation for the sampling
stations is computed for which the spatial `dispersions Var(Y(xi)—Y(xj))
are approximated by a monotone function of interpoint distances. That is,
in terms of this second twodimensional representation, the spatial covari
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so Vancouver
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\ Matte STATI5
Abbotsford
city.8
FIGURE 4.1: The 12station solar radiation monitoring network in Lower
Mainland, British Columbia, Canada. Reprinted, by permission, from Hay
(1984~. Copyright (I) 1984 by Pergamon Press.
ance structure as represented by the spatial dispersions is stationary and
isotropic. (These variances are usually fitted by parametric models for the
variogram.) Thinplate splines are applied to compute a smooth mapping
of the geographic representation of the sampling stations onto the MDS
representation. Biorthogonal grids, introduced by Bookstein (1978) in the
field of morphometrics, can be used to depict the mapping. This mapping
yields a nonparametric method for estimating Var(Y(xa)  Y(xb)) for any
two unsampled locations pa and xb in the geographic plane, and a graphi
cal representation of the global spatial covariance structure. The resulting
nonparametric models for spatial covariance are constrained to be positive
definite or, in the terminology of geostatistics, the variogram models are
conditionally nonnegativedefinite. This is obtained by fitting a mixture of
covariance functions of Gaussian type in the MDS step of the algorithm.
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74
To
0 200 400 600 800 1000 1200 1400
Day
0 200 400 600 800 1000 1200 1400
Day
. _
FIGURE 4.2: Daily solar radiation totals for Vancouver International Air
port (site 4~: (a) raw; (b) transformed.
4.2.1 Example: Spatial Variation in Solar Radiation
We present here a preliminary analysis of data collected from a solar radi
ation monitoring: network in southwestern British Columbia, Canada (Hay,
~ feasibility of solar power genera
tion in British Columbia. This example manifests a somewhat extreme but
easily understood form of nonstationarity in the spatial covariance structure
of the solar radiation field. Figure 4.1, taken from Hav (19831. disDla.vs the
locations of the 12 monitoring stations.
19841' with a view toward determining the
wet v~J, I Vies
The data consist of daily solar radiation totals (MJ m2day~) for the
years 198083. Figure 4.2 plots the data for the monitoring station at Van
couver International Airport. Note the relatively sharp upper bound on the
maximum solar radiation as a function of season. Sivkov (1971, Chap. 7)
explains how and why the maximum solar radiation (observed on cloudless
days) varies approximately as a sine function with minimum at the vernal
equinox. A reasonable stochastic model for observations at one location is
thus
Zi,~ = bi,' act + ,l) sin ~ 365 (t—80~) (1 + ci,'),
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75
where observations are taken daily (t = 1,2,...,365), Pi, is a random vari
able taking values on the interval (0,1] to express atmospheric attenuation
effects, and ei,~ represents a mean zero measurement error effect. Cloudi
ness is the principal factor determining 8~. As the first step in our analysis,
we estimate the parameters ~ and A, which define the maximum expected
solar radiation as a function of day of year. We then scale all the data as
a percentage of the estimated seasonally adjusted maximum possible solar
radiation. Thus we attempt to focus on analyzing the spatial structure of
8~. These data have a concentration of values near the maximum of looked,
and so we compute covariances among monitoring stations using a logit
transformation of the percentageofmaximum data. These transformations
removed the major aspect of seasonality associated with the orientation of
the earth with respect to the sun. However, the spatial covariance structure
retains seasonal structure because of variation in the atmospheric processes.
We therefore analyze the spatial structure of the data separately by season.
Here we present only the results for the combined spring and summer quar
ters (vernal equinox, March 22, through autumnal equinox, September 22~.
Tnterstation correlations are very high for these data, and the dispersions
are closely related to geographic distances among the stations. Figure 4.3
shows the distribution of monitoring stations in the Dplane as determined
by MDS applied to the matrix of dispersions. The most obvious deviation
between the two planar representations is in the relative location of station
1, Grouse Mountain. The Grouse Mountain station is at an elevation of 1128
meters while Al other stations lie below 130 meters. This orographic feature
explains the relatively high dispersions (Iow covariances) between station 1
and aD the others as reflected in the scaling in Figure 4.3.
4.3 Network Design
The purpose of a monitoring network is to detect potential changes in key
environmental parameters. The designer of a longterm monitoring network
cannot fully foresee all of the benefits that may be derived from the network
by its future users. Environmental engineers, resource developers, biologists,
human health agents, and so on, wiD need the data for a variety of purposes,
some of which wiD not even have been identified. In addition to the hypoth
esis testing mentioned above, there is a need for inference about changes in
areal averages, and about the areal maximum of such changes. The network
may be regarded simply as an information gathering device.
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76
00 
~ .
N
of
Cal
CO
~ 
......................... l
.............................
.~ ~
............................
N 
O 
...... ~3
~ .
. ~
. ~ ~ ~ I
. O . ~.
....... ............................ .......
.......................
.......................
1 .......................
.........................
.........................
.........................
. ~
............ ...................................
2 4
N .
.~
1 0 1 2 3 4
FIGURE 4.3: Transformation of the Gplane configuration of solar radiation
monitoring stations (left) into the Dplane configuration (rights.
There are objectives where the choice of design may not be critical.
Switzer (1979) argues that for estimating are al averages, the search for a
design that minimizes mean squared estimation error is unnecessary, since
the criterion is relatively insensitive to design changes among sensible de
signs. The optimal design is very modeldependent, and the mathematics
are invariably difficult. He argues that designs intended for this purpose
might better be chosen on a priori grounds, avoiding clustering and with
regard to topography and subregions of greater variability. Unfortunately,
the situation is not always so simple. In impact detection, for example, the
choice of the design is critical.
Kriging has its attendant theory of design, based on minimizing mean
squared estimation error (Cressie et al., 1990~. For impact design, this
criterion may not be the most natural. Rather, one wants to maximize the
power of the test.
In general, the appropriate design criterion is as uncertain as the objec
tive itself (see RodriguezIturbe, 1974, for a discussion). CaseTton and Zidek
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77
(1984) argue that a reasonable design criterion will be based on an index
of the information transmitted. A particular set of monitoring stations is
good if it provides a lot of information (in the sense of Shannon) about
unmonitored sites (see §4.3.2~.
4.3.1 Impact Design
Suppose we need to assess the effect of a potential impact taking place at a
known time. Typical examples are changes in environmental requirements,
closure or startup of potential pollution sources, and environmental disas
ters. The nub hypothesis is that of constant mean before and after the
change. Suppose that it is feasible to make observations at any point on
the grid of potential monitoring sites before and after the known time of
potential change. According to an emerging body of evidence, it is very
difficult to detect even fairly large changes in ambient levels with high prob
ability. For example, Hirsch and Gilroy (1985) use a certain nonparametric
testing procedure, a sulfate deposition mode! fitted to data from New York
state, and simulated sulfate deposition experiments with step changes of
various magnitudes, including 20~o. They show that with one monitoring
station, 90% power requires 15 years of postchange records with 5 years of
prechange records. Using ~ stations, one still needs 2 years of postchange
records, and adding more stations does not yield appreciable reductions.
Much of the difficulty is the result of the large component of meteorological
variability in deposition. In the work of Vong et al. (1988), a design based
on meteorological criteria was used to reduce this variability, which yielded
unambiguous evidence of the local deposition effect of a copper smelter.
Regard a design D as a set of labels designating the sampling sites. The
region of interest is overlain with an imaginary grid of potential sites from
which D is to be chosen. An impact is regarded as a random field Z. covering
the whole region. At site i, Zi is the size of the change owing to development
and other uncontrolled factors. Only Zi with i in D will in fact be measured
(with error) once D is specified.
Suppose that I: replicate measurements of Zi are taken at each site in D.
Their variability is assumed constant over i, and indicates the precision of the
process of measurement. Changes wiD be measured against this variance. A
strategy (suggested, e.g., by Green, 1979) can be used to reduce the impact
of temporal effects. Sites outside areas of likely impact are admitted as
possible quasicontrols. These do increase the power of tests, even though
they, strictly speaking, are not controls. The null hypothesis (again following
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78
Green, 1979) is that of no timespace interaction. Assuming the standard
twoway fixed effects ANOVA model, the Fstatistic has power depending
on the noncentrality parameter which can be estimated by
~ 2a2 ~
where ZD IS the average of the observations. In some special cases it is
possible to maximize Ed. Suppose that the area of potential impact
can be divided into a collection of homogeneous zones (this has to be clone
using expert knowledge). Then the problem of maximizing the expected
noncentraTity parameter is reduced to that of finding the optimal sampling
fractions, which is a quadratic integer programming problem (Schumacher
and Zidek, 1989~. Simulated annealing is being explored as an alternative
approach to the optimization (Sacks and Schiller, 1988~.
4.3.2 Information Transmission Network Design
The future benefits that may be derived from a network cannot ah be speci
fied in advance. Even when a network is designed with a particular objective
in mind, it is quite common that the answer to very different questions must
be elicited from the data once the network is operational. CaseTton and
Zidek (1984) suggest circumventing these difficulties by an approach that
may be suboptimal in specific cases but has overall merits for these types of
networks.
We let Z denote a random field of measurable quantities indexed by
potential site labels i. We decompose Z into the gauged sites G = (Zi, i ~ D)
and the ungauged sites U. The choice of D will be made to maximize the
amount of information in G about U. Here the information measure is
taken to be ItU, G) =E(Iog;(f(U~G]/f(U)~), Shannon's index of information
~ , _ ~ ~ . ,, ~ , . . .
. · · ~ ~/ TT I ~\ · ~ 1 1 · ~ · 1 1 · ~ ~ AT · ~ ~
transmission, where J(UIQ) IS one COnaIt10na1 aEnS11Y OI u given Q' and
f(U) the a priori density of U.
A simple special case is when the random field is multivariate normal,
when [(U,G) =  2 Tog ~!  Rat, where ~ is the identity matrix, and R the
diagonal matrix whose elements are the squared canonical correlation co
efficients between U and G. These can be obtained from estimates of the
spatial correlations, for example, using the method of Sampson and Guttorp
mentioned in the previous section.
For particular patterns of the covariance matrix of Z. derived from mo(l
els of acidic deposition (such as that used by Vong et al., 1988), it is possi
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79
~
~ ~
Aim
~ 7 _:
1 V~
,rl
_~
~ }~=: ~
\ ~

FIGURE 4.4: The MAP3S monitoring network.
1 Lewes, DE
2 Illinois, IL
3 Whiteface, NY
4 Ithaca, NY
5 Brookhaven, NY
6 Oxford, OH
7 Penn State, PA
8 Virginia, VA
9 Oak Ridge, TN
ble to develop workable approximations to the canonical correlations and to
solve the design problem in terms of signaltonoise ratios at gauged and un
gauged sites, respectively. The analysis suggests the importance of replicate
measurements at gauged sites (Guttorp et al., 1987, section 3.3~.
Example: Finding the Least Informative Station in a Network
The Multistate Atmospheric Power Product Pollution Study/Precipitation
Chemistry Network (MAP3S/PCN) of nine monitoring stations (Figure 4.4)
in the northeastern United States was initiated in 1976 with the objective of
creating a longterm, highquality data base for the development of regional
transport and deposition models. There is substantial seasonal variability in
the data, and we concentrate here on log deposition of H+, using fourweek
totals for January through April. Guttorp et al. (1991) has further details.
In order to decide which station carries the least information in the network,
we need to compute the information in the network leaving out each station
in turn. Thus the station left out is considered ungauged, and Al the other
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80
TABLE 4.1: = r~~
ltiole Correlation Coefficients
I(U, G) standard error
Station(U)
Lewes, Del.
Illinois, ]11.
Ithaca, N.Y.
Whiteface, N.Y.
Brookhaven, N.Y.
Oxford, Ohio
Penn State, Pa.
Virginia, Va.
Oak Ridge, Tenn.
.26
.66
.49
.40
.42
.58
.57
.31
.29
.08
.10
.09
.09
.09
.10
.10
.08
.08
stations are gauged.
the other stations in the network.
For each station left out, we compute I(U,G) from
The analysis of canonical correlations
(which for one ungauged site simplifies to the multiple correlation coefficient)
indicates that the three stations in Illinois, Ohio, and Pennsylvania each
have significantly higher multiple correlations with the remainder of the
network than have any other stations. The results are listed in Table 4.1,
where it is seen that Illinois, 111., is the least informative station in the
network, in the sense of being best predicted by the other stations. In
other words, the gauged stations have the highest information about the
(presumed) ungauged station at Illinois.
It is worth noting that the stations at Oxford, Ohio, and Penn State,
Pennsylvania, are not significantly different from the Illinois station. On
the other hand, the geographically extreme stations in Delaware, Virginia,
and Tennessee are all poorly predicted, and are therefore highly informative
stations.
4.4 Modeling Precipitation Using SpaceTime
Point Processes
An environmental problem of enormous potential impact is the global warm
ing due to increased CO2 concentration in the atmosphere. Much effort has
been extended to develop realistic models of global climate in order to be
able to assess the potential impact of changes in atmospheric gasses on dif
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81
ferent aspects of weather patterns. In order to do this, hydrologists have
found it useful to employ stochastic models of precipitation, which is an im
portant factor in climate change, and also itself affected by climate change.
Such models have also found important applications in assessing the risk of
flash floods and in design of dams.
A realistic stochastic mode! of rainfall must take into account the physi
cal structure and organization of storms, such as the description of cyclonic
storms in Hobbs and LocateDi (1978~. In essence, the storm system contains
mesoscale rainbands, which contain smaller mesoscaTe regions, or precipita
tion cores, which are characterized by higher rainfall rates. These cores
originate in generating cells aloft (in warm frontal bands) or within layers of
potentially unstable air (in cold frontal bands). This description was used
by Waymire et al. (1984) and by Kavvas and Herd (1985) to construct ap
propriate stochastic models, following the work of Le Cam (1961~. In what
follows, we essentially follow the Waymire et at. description.
The essence of the Waymire et al. (1984) model is the following stochastic
representation of the rainfall intensity ~ at time t and location z:
((t,Z) = /R2 /(o ~ gift T; Z Y) X(1T, EYE,
where 9~ is a dispersion function, representing the rainfall intensity from a
given cell born at (T. y) depending on the random variable 71, and X(~r, any)
counts the rain cells alive in an infinitesimal neighborhood of (T. y). Thus X
is a point process that has the structure of a cluster process (see DaTey and
VereJones, 198S, and the discussion in chapter 7 of this report). From this
representation, it is easy to write down formulae for the mean and covariance
of the random field A. In order to get useful results, one needs to make a
few more assumptions. If it is reasonable to assume that the dispersion of
a rain cell is independent of the occurrence of rain cells, then the expected
value can be written
E6(t,z) = //E[gn~t T; Z Y)]Px (T,y) y,
where puke is the kth order product moment density for the point process X,
measuring the joint probability density of k events. It may be reasonable to
assume that spatial and temporal features are separable, in the sense that
pit y) = pt )(T)P2 (Y)
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82
and
gn(uiv) = ~gl(u)g2(v).
With these assumptions, it is easy to see that
E((t, z) = E(~)[g~ * pt (at) [92 * Pt )~(Z),
where Aft * f2] is the convolution of fi with f2.
Similar computations yield that
Em, Z1)~(~2, Z2) = E (77) ~ // 91(~1 —r1 )91 (~2 —T2)p1 (T1, T2) Ale Ale
x t//92(z~  Y~)92(Z2  Y2)P(2~(Y~,y2) dye
83
the number of events in each time interval are recorded. Generally, because
of the intractable nature of the likelihood function, estimation is usually
based on the method of moments. Further discussion of problems involved
in spatial and temporal averaging of precipitation data and the attendant
problems of parameter estimation can be found, e.g., in RodriguezIturbe
et al. (1974), Valdes et al. (1985), RodriguezIturbe and Eagleson (1987),
Sivapalan and Wood (1987), and Phelan (1991~.
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