Blackbox Kriging: Spatial Prediction Without Specifying Variogram Models Ronald Paul BARRY and Jay M. YE. HOEF This article proposes a new approach to kriging. where I f1eltible family of vllriograJDS is used in lieu of one of the traditionally used parametric models.. This nonpanmetric approach minimizes the problems of misspecifying the nnogram model. 'The ftexible variogram family is developed using the idea of a mO\'ing average function composed of many small rectangles for the one-dimensional case and many small boxes (Of" me IWo- dimensKlnl1 cue. Through .'limulation. we show that the usc of nexible piectwise-linear models can result in lo......er mean squared prediction errocs man the use of traditional models. We then use a flexible piecewise-planar variogrIm model as a step in kriging the two-dimensional Wolfamp Aquifer data, v.ilhoul the need to assume that the under- lying process is isotropic. We pro\"C that. in ODe dimension. any continuous variognm wilh a sill can be apprnximaled ubitJarily close by piecewise-linear variograms. We dis- cuss ways in which the piecewise-linear vanogram modeb can be modified 10 improve the fit of the variogmm estimate near me origin. Key Words: Anisotrop)': Geostatistics; Moving averages. I. KRIGING AND THE CHOICE OF VALID YARIOGRAM FAMILIES Ordinary kriging is a method for predicting the value of a random process al a specific location in a region, given a dataset consisting of measurements of lhe random process at a variety of locations in the region. Specifically, leI Z (.'II) Ii = 1, ... ,71 be a set of measurements at locations .'II, •.. , SJl in an m-dimensional region D. These measurements are assumed to be one realization of a random process Z(·) with the following properties: 1. E[Z(·)J- ,. 2. 21'(h) = var (Z(s) - Z(s - h)), for all s, S - h in D, exists and only depends on h (Cressie 1993, p. 40). These two assumptions fonn the intrinsic stationarity hypothesis (Cressie, p. 60), and the vanogram is defined to be the function 2")'(h) = var(Z(s)- ROIl.Ild Polli Barry is ProfU!Or. of Malhemalical Sciences. University of Alaska Fair- bank$, Fairbllnk.'I., AK 99775-6660, ffrpb@wrora.ala!ka.edIl.Jay M. Ver Hoef i$ a Biometrician. Aluka Depart- mrnl of Fi!l1 and Ga.me. Division of Wildlife Conservation. Fairbanla, AK 99101, ffjmv@lIltOn.llasWdu. @199lS A_ricQlt Slotis'ictll"'JJocUuirHl (IJtl/ 1M /ntemmioMl Biotnllric Sociny JotlnW of Arric"/'"taI. Biolofktll. (IJtl/ EiI"jronntnl/(jf S4UiJtics. J. NwnMr 1. 297-J11 m