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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 long-term 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 best-known 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 site-pair 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 two-dimensional 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 two-dimensional representation, the spatial covari-

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so Vancouver (BE Hydra bided 12 UBC \ ~PIant Science field lab.) \N 1 . Lan8ara `~ 1 (49th ~ Cambie) _ ~ O ~ - $~ ~ 4~ . ~ No ~ ~ ~0 ., ~ ~& Pitt Meadows /~ ~6 (Airport) ~~ ~ /, ,~ ~x, [~ ~ ~ ~ ~ ~ '' ' ' ~ r ~ ~ ~ a- 0~~ ~ =~4Ai~)/// G ~ ~ _~ ~ ~ ~ ~ =/ ~ TS~T~WaS~ be. ~5 ( FerrY TO ~ C-_; ; con - ~~ ~ - >\ Mission <: city 7~ LCjntgleyln: ~ ~~ ., .,~,1~ \~` - ' I) Abbotsford Airport CANADA \ Matte STATI5 Abbotsford city.8 FIGURE 4.1: The 12-station 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. Bi-orthogonal 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 non-negative-definite. 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 1980-83. 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 (t80~) (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 percentage-of-maximum 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 D-plane 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 long-term 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 G-plane configuration of solar radiation monitoring stations (left) into the D-plane 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 model-dependent, 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 Rodriguez-Iturbe, 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 post-change records with 5 years of pre-change records. Using ~ stations, one still needs 2 years of post-change 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 quasi-controls. 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 time-space interaction. Assuming the standard two-way fixed effects ANOVA model, the F-statistic 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 non-centraTity 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 signal-to-noise 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 long-term, high-quality 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 four-week 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 Space-Time 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 Vere-Jones, 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 OCR for page 71
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 Rodriguez-Iturbe et al. (1974), Valdes et al. (1985), Rodriguez-Iturbe and Eagleson (1987), Sivapalan and Wood (1987), and Phelan (1991~. Bibliography [~] Bookstein, F. L., The Measurement of Biological Shape and Shape Change, Lecture Notes in Biomath. 24, Springer, New York, 1978. [2] Caselton, W. F., and J. V. Zidek, Optimal network monitoring design, Stat. Prob. Lett. 2 (1984), 223-227. [3] Cressie, N., C. A. Gotway, and M. O. Grondona, Spatial prediction from networks, Chemometrics Int. Lab. Syst. 7 (1990), 251-271. [4] DaJey, D. J., and D. Vere-Jones, An Introduction to the Theory of Poin Processes, Springer-VerIag, New York, 1988. [5] Green, R. H., Sampling Designs and Statistical Methods for Environ- mental Biologists, John Wiley and Sons, New York, 1979. [6] Guttorp, P., On binary time series obtained from continuous time point process models describing rainfall, Water Resour. Res. 22 (1986), 897- 904. [7] Guttorp, P., Analysis of event based precipitation data with a view towards modeling, Water Resour. Res. 24 (1988), 35-44.

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84 [8] Guttorp, P., and M. L. Thompson, Nonparametric estimation of inten- sities for sampled counting processes, to appear '7. R. Stat. Soc., B. 1990. [9] Guttorp, P., A. J. Petkau, P. D. Sampson, and J. V. Zi(lek, Enva ronmental Monitoring: Models, Network Design, and Data Analysis, Department of Statistics, University of Washington, SIMS Technical Report 107 (1987~. [10] Guttorp, P., K. Newman, and P. D. Sampson, Nonparametric estima- tion of spatial covariance with an application to monitoring network design, to appear in Statistics in Environmental and Earth Sciences, P. Guttorp and A. Walden, eds., Griffin, London, 1991. [11] Hay, J. E., Solar energy system design: The impact of mesoscaTe vari- ations in solar radiation, Atmosphere-Ocean 21 (1983), 138-157. [12] Hay, J. E., An assessment of the mesoscale variability of solar radiation at the earth's surface, Solar Energy 32 ( 1984), 425-434. [13] Hirsch, R. M., and E. J. Gilroy, (1985), Detectability of step trends in rate of atmospheric deposition of sulfate, Water Resour. Bull. 21 (1985), 773-784. [14] Hobbs, P. V., and J. D. Locatelli, Rainbands, precipitation cores and generating cells in a cyclonic storm, IT. Atmos. Sci. 35 (1978), 230-241. [15] Kavvas, M. L., and K. R. Herd, A radar-based stochastic model for short-time-increment ra~nfaH, Water Resour. Res. 21 (1985), 1437- 1455. [16] Le Cam, L. M., A stochastic description of precipitation, pp. 165-186 in Proceedings Fourth Berkeley Symposium on Mathematical Statistics and Probability 3, J. Neyman, ea., University of California Press, Berkeley, 1961. [17] Ma, H. W., H. Joe, and J. V. Zidek, A Bayesian Nonparametric Univariate Smoothing Method, with Applications to Acid Rain Data Analysis, SIMS Technical Report, 104 (1986), University of British Columbia.

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85 [18] National Research Council, Acid Deposition: Long-Term Trends, Com- mittee on Monitoring and Assessment of Trends in Acid Deposition, National Academy Press, Washington, D.C., 1986. [19] Phelan, M. J., Aging functions and their nonparametric estimation in point process models of rainfall, to appear in Statistics in Environ- mental and Earth Sciences, P. Guttorp and A. Walden, eds., Griffin, London, 1991. [20] Poliack, A. K., A. B. Hudischewskyj, T. S. Stoeckenius, and P. Gut torp, Analysis of Variability of UAPSP Precipitation Chemistry Mea surements, Draft Final Report SYSAPP-89/041, Systems Applications Inc., San Rafael, 1989. t21] Rodriguez-Iturbe, I., The design of rainfall networks in time and space, Water Resour. Res. TO (1974), 713-728. [22] Rodriguez-Iturbe, I., and P. S. Eagleson, Mathematical models of rain- storm events in space and time, Water Resour. Res. 23 (1987), 181-190. [23] Sacks, "T., and S. SchiDer, Spatial designs, in S. S. Gupta and J. O. Berger (eds.), Statistical Decision Theory and Related Topics IV, vol. 2 Springer, New York, 1988. t24] Sampson, P. D., and P. Guttorp, Nonparametric Estimation of Non- Stationary Spatial Covariance Structure, Department of Statistics, Uni- versity of Washington, SIMS Technical Report 148 (1990~. t25] Schumacher, P., and J. V. Zidek' Using prior information in design- ing point impact detection networks, talk at TST satellite meeting on Statistics, Earth and Space Sciences (1989), Leuven (to appear). [26] Sivapalan, M., and E. F. Wood, A multidimensional model of non- stationary space-time rainfall at the catchment scale, Water Resour. Res. 23 (1987), 1289-1299. [27] Sivkov, S. I., Computation of Solar Radiation Characteristics, Israel Program for Scientific Translations, Jerusalem, 1971. t28] Switzer, P., Statistical considerations of network design, Eos (1979), 1712-1716.

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86 [29] Switzer, P., and C. Loader, Spatial covariances, talk at ISI satellite meeting on Statistics, Earth and Space Sciences (1989), Leuven (to appear). [30] Valdes, J. B., I. Rodriguez-Iturbe, and V. K. Gupta, Approximations of temporal rainfall from a multidimensional model, Water Resour. Res. 21 (1985), 1259-1270. [31] Vong, R. J., I.. Moseholm, D. S. Covert, P. D. Sampson, J. F O'[oughlin, M. N. Stevenson, R. J. CharIson, W. H. Zoller, and T. V. Larson, Changes in rainwater acidity associated with closure of a copper smelter, J. Geophys. Res., D 93 (1988), 7169-7179. [32] Waymire, E., V. K. Gupta, and T. Rodriguez-Iturbe, A spectral theory of rainfall intensity at the meso-,B scale, Water Resour. Res. 20 (1984), 1453-1465. [33] Weerahandi, S., and J. V. Zidek, Bayesian nonparametric smoothers. Can. J Stat. 16 (1988), 61-74.