TEACHING AID

Ricardo A. Olea

A six-step practical approach to semivariogram modeling

Published online: 9 May 2006� Springer-Verlag 2006

Abstract Geostatistical prediction and simulation arebeing increasingly used in the earth sciences and engi-neering to address the imperfect knowledge of attributesthat fluctuate over large areas or volumes—pollutantconcentration, electromagnetic fields, porosity, thicknessof a geological formation. Central to the application ofsuch techniques is the need to know the spatial conti-nuity, knowledge that is commonly condensed in theform of covariance or semivariogram models. Theirpreparation is subdivided here into the following steps:(1) Data editing, (2) Exploratory data analysis, (3)Semivariogram estimation, (4) Directional investigation,(5) Simple modeling, (6) Nested modeling. I illustratethese stages practically with a real data set from a geo-physical survey from Elk County, Kansas, USA. Theapplicability of the approach is not limited by thephysical nature of the attribute of interest.

Keywords Geostatistics Æ Continuity Æ Uncertainty ÆSemivariogram Æ Model fitting

1 Introduction

Shortage of information is common in the modeling andevaluation of natural resources. Geostatistics has beendeveloped in recent decades to assist in the process andto quantify the uncertainty associated with such imper-fect knowledge (e.g. Goovaerts 1997; Chiles and Delfiner1999; Olea 1999).

A fundamental difference between geostatistics andclassical statistics is the assumption by geostatistics of theexistence of spatial autocorrelation, which matches thecommonly held notion that in the vicinity of small valuesthere are other small values, while large values tend to beclose to other large values. The assessment of such

autocorrelation is a prerequisite in most applications ofgeostatistics, which is generally done by modeling thesemivariogram based on the information provided by asampling of the attribute of interest. Equivalent resultscan be obtained employing covariances instead, whichcan be readily derived from semivariograms.

Despite numerous publications on the subject, mod-eling a semivariogram remains to the uninitiated themost difficult and intriguing aspect in the application ofgeostatistics. What follows is a hands-on approach in-tended to teach by example, by breaking the task ofmodeling into six sequential steps. Proper execution ofthe task requires computer software for calculations andgraphical display, for which I provide references. Theuse of a two-dimensional gravimetrical survey here ismerely pedagogical. The basic steps are the same,regardless of the physical nature of the attribute and thedimensionality of the sampling. The focus is on handlingtypical peculiarities of spatial continuity as revealed by aproper sampling, rather than in complications derivedfrom insufficient data, blunders in their preparation, or acombination of the two.

2 Step 1: Data editing

To remain as focused as possible, I will start byassuming that there is a sampling already available. Thiswill avoid going into sampling design, a major stepworthy of a separate paper.

Typically, a geostatistical sampling comprises onerecord per measurement. Each record gives the observedvalue and its spatiotemporal location, which may be inone, two, or three spatial dimensions. The most frequentcase—and the one to which I will devote my atten-tion—is the atemporal, two-dimensional case, in whichlocation comprises a couple of geographic coordinates.In cases such as locations given as legal description oflocation or latitude and longitude, it is necessary toconvert them to Cartesian coordinates using programssuch as the one prepared by Collins (1999).

R. A. OleaInstitut fur Ostseeforschung Warnemuende,18119 Rostock, GermanyE-mail: ricardo@oleageostats.com

Stoch Environ Res Risk Assess (2006) 20: 307–318DOI 10.1007/s00477-005-0026-1

Where every observation is some type of average overa line, area, or volume—porosity, chemical concentra-tion, ore grade, crop yield—all measurements must havethe same type of underlying line, area, or volume interms of shape, size and orientation. Observations hav-ing significant variations in the underlying line, area, orvolume should form different sub-samplings that mustbe treated separately.

The first task is to review every record to eliminateany possible reading, recording, or processing errors,either in the locations or in the measurement of theattribute. Human errors are more common than mostpeople imagine. Final results will be particularly sensi-tive to anomalous fluctuations; one needs to scrutinizethem as much as possible. The sampling should retainonly those observations denoting real, natural fluctua-tions, within instrumental precision, of course. Failureto eliminate sampling blunders can sometimes be de-tected later in the modeling. Then it is necessary to re-peat the work after taking corrective action. Undetectedblunders usually degrade the representativity of the re-sults.

A posting of values or preliminary mapping, such asthe one in Fig. 1, is helpful to gain familiarity with thedata and detect dubious sampling sites. The sampling inFig. 1 is a medium-size survey with 668 measurementsand will be used extensively to illustrate the methodol-ogy. The sampling is part of a larger geophysical studycomprising more than 27,000 stations that cover all ofthe state of Kansas (Lam 1987; Xia et al. 1992). Thesurvey was done to detect Bouguer gravity anomalies inthe gravitational force relative to the standard geoid,which in this case mainly indicate depth to bedrock.Values of Bouguer anomaly are gravity averages overthe same vertically elongated volumes, with cross-sec-tions that are regarded as points relative to the size ofthe sampling area. The coordinates in this case are thoseof the instrument on the surface of the earth.

The reader is encouraged to download and work withthe sampling available at URL http://www.kgs.ku.edu/Mathgeo/Books/Elk/index.html. As rendered in Fig. 1,the survey is free of errors, ready for reliable use, withina combined measurement and processing precision of0.1 mgal.

3 Step 2: Exploratory data analysis

Before calculating the semivariogram, the user needs toexamine both the spatial distribution of sampling sitesand the cumulative distribution of the measurements toassess any need to modify the original data.

First, for the proper modeling of a semivariogram,the sampling should not have preferential areas, as isthe case when some measurements concentrate inclusters with a much greater sampling density that therest of the sampling area. If that is the situation, thesampling needs preprocessing to eliminate the influ-ence of clusters. One way to do this is by assigning

weights (Isaaks and Srivastava 1989, pp. 241–247),which can be done by the program declus (Deutschand Journel 1998, pp. 213–214). The Elk countygravimetric survey was designed to take one mea-surement at every intersection of the almost perfectlyregular network of roads every mile in the east–westand north–south directions. Hence the demonstrationsampling is as free of clustering as any samplingcan be.

The second decision relates to the requirement orconvenience of transforming the data to increase itsunivariate normality, namely the ability of the mea-surements to approximate a normal distribution. Themost common practice is to convert the data to normalscores. The transformed data will have a normal distri-bution with a mean of zero and a variance of one,transformation that is also known as a Gaussian ana-morphose. Often the transformation is optional andmakes sense solely in the case of clear deviation fromnormality. Some applications, however, such assequential Gaussian simulation, require normal scores,and so the transformation is mandatory. For more de-tails, see Verly (1986). If the reader needs to make anormal score transformation, then see for exampleDeutsch and Journel (1998, pp. 223–226).

Figure 2 shows the cumulative univariate distributionfor the Elk data on a normal probability scale. In thisinstance, the maximum deviation from normality occursfor �56 mgal and is 5.7% points. As a practical rule,there are no clear advantages to working with normalscores unless the deviation from normality is above 10%points. For cases such as the Bouguer values from Elkcounty, one should avoid a normal score transforma-tion. Most statistical libraries have programs tocalculate maximum deviation between cumulative dis-tributions, such as the one in Press et al. (1992, pp. 617–619). Given the nonlinearity of the probability scale, it issafer to run this simple calculation than trying to readthe maximum discrepancy from a graphical comparisonof the distributions like the one in Fig. 2.

4 Step 3—Semivariogram estimation

At this stage it seems opportune to define what thesemivariogram is. Given two sites h units apart and thedifference for a variable of interest at those sites, thesemivariogram, c(h), is half the variance of this differ-ence. The semivariogram has the property of measuringthe degree of dissimilarity between pairs of measure-ments in terms of how far apart they are and the ori-entation of the line between those two sampling sites.

Statistics and geostatistics are sciences of the un-known. Therefore, it follows that the true semivario-gram is never known, and as in statistics generally, allthat it is customarily known is an estimate of the semi-variogram. Although there are several semivariogramestimators, the predominant practice is to use the fol-lowing unbiased estimator:

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Fig. 1 Location of Elk Countyin the state of Kansas, UnitedStates (star), posting ofobservation stations (+), andcontour map of Bougueranomaly in Elk County. Linesclose to the margins are the ElkCounty boundary lines.Distances along the axes are inkilometers and contour intervalis 1 mgal

Fig. 2 Cumulative distributionfor the Bouguer gravityanomaly, Elk County, Kansas,denoted by the dots, and anormal distribution with thesame mean and variance, givenby the straight line. The samplemean is �61.3 mgal, thestandard deviation 4.8 mgal,and the maximum discrepancywith a normal distribution withthese same parameters is 5.7percentage points

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c hð Þ ¼ 1

2 n hð ÞXn hð Þ

i¼1z xi þ hð Þ � z xið Þ½ �2 ; ð1Þ

where z (xi) is a measurement taken at location xi andn(h) is the number of pairs h units apart in the directionof the vector. In the geostatistical jargon, c hð Þ is knownas experimental semivariogram and h is the lag. Animportant assumption for the validity of this estimator isthe absence of any systematic variations, that is to say,there should be no trend. Note that the estimator isconveniently independent from the individual sites xi.

The lag is written in bold to denote that the argumentsimultaneously has a magnitude defined by a distanceand an orientation, which in two dimensions is any ofthe points in the compass. Location is also in bold todenote that it is a vector of coordinates, as many as thedimensions in the sampling space. In two dimensions,they are commonly denoted as easting and northing.

Given an orientation, the estimator in (1) is strictlyapplicable to a sampling at regular intervals, d. c 0ð Þ isalways zero, as the difference of any measurement withitself is zero. c dð Þ is half the square mean of all differ-ences of a measurement with its immediate neighbor,c 2dð Þ is half the square mean of all differences of ameasurement with the neighbor that results from skip-ping one measurement, and so on. For a numericalexample, see, e.g., Olea (1999, p. 74).

In irregular sampling schemes, for the purpose ofestablishing pairs, z(xi+h ) is regarded as a centroid of adistance class. Figure 3 illustrates the situation of a two-dimensional irregular sampling, in which any measure-ment inside the shaded area is considered in the calcu-lation of c hð Þ; although it is not exactly a distance h fromxi. The lag spacing is typically taken close to the value of

the average sampling distance; the lag tolerance, th, is setto half this spacing; the lateral tolerance, tb, 0.5 to 2times the lag spacing; and the angular tolerance, d, 0.25–0.5 times the increment in the azimuth. The increment inthe azimuth is customarily between 22.5� and 45�.Clearly there is a trade-off between the resolution ofsmall distance classes with few observations per class,and the reliability and smoothing of large distanceclasses. Options become more numerous as the samplingsize increases. In practice, the pairing of measurementsand calculation of half-square averages is better donewith the assistance of computer programs, such asVARIOWIN (Pannatier 1996) or gamv (Deutsch andJournel 1998, pp. 53–55).

Figure 4 is an example of a typical experimentalsemivariogram, which in this case shows the results ofprocessing the Elk data with gamv using an incrementallag of 1.6 km (1 mi), an angular tolerance of 10� and alateral tolerance of 1 km. The degree of dissimilarityprovided by the semivariogram often shows a boundedincrease. The distance at which the semivariogramreaches the limiting value is called the range and thebound is denoted as the sill. In the case of experimentalsemivariograms, the range and the sill are hard to defineaccurately because of irregularity in the fluctuations.

Let me conclude this section with some remarksabout the significance of the point estimates comprisingthe experimental semivariogram. The autocorrelationassumption at the core of geostatistics is both a blessingand a problem. It is an advantage in that allows forbetter characterizations than would be possible withoutspatial continuity. Autocorrelation, however, introducesenough theoretical complications that it has beenimpossible to develop any kind of test of significance inthe style of classical statistics. So, for example, the an-swer to the question as to the minimum number of pairsin c hð Þ required to provide a reliable estimation, can be

Fig. 3 Definition of the distance class in the estimation of asemivariogram. The shaded area is defined by an angular toleranced, lag tolerance th, and a lateral tolerance tb, relative to the pointhere appearing in the upper left corner. Any measurement insidethe shaded area can be used in the calculation of c hð Þ despite notbeing exactly h units apart from xi

Fig. 4 Experimental semivariogram for demonstration samplingalong N63E showing a range of about 21.3 km and a sill of about1.07 mgal2

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addressed solely by the following couple of recommen-dations:

1. The minimum number of pairs in semivariogramestimation must be 30, according to Journel andHuijbregts (1978, p. 194), and 50 if one is going tofollow the advice of Chiles and Delfiner (1999, p. 38).Please see Webster and Oliver (1992) for a discussion.

2. If it is necessary to estimate the semivariogram for alarge lag close to the diameter of the sampling area,then the pairs of observations that are that far apartare only those located at opposite extremes of thesampling area, thus excluding the central points fromthe analysis. Hence, the justification for a secondpractical rule that advises to limit the lag of theexperimental semivariogram should be limited to halfthe extreme distance in the sampling domain for thedirection of interest (Journel and Huijbregts 1978,p. 194).

Indirectly, these two rules collectively make it difficultto properly estimate a semivariogram with less than 50measurements at different locations.

5 Step 4: Directional investigation

In more than one dimension, the semivariogram gener-ally has directional properties. Hence the user shouldnot stop at estimating a semivariogram for a singledirection, such as the one in Fig. 4. In general, the datapermitting, the more directions are investigated, thebetter. In two dimensions, the bare minimum is theinvestigation of three azimuths (Goovaerts 1997, p. 98).In three dimensions, there is the additional need to run asensitivity analysis in the declination.

One might observe at least three basic types ofbehavior in a directional semivariogram survey. In thesimplest situation, there is no significant differenceamong experimental semivariograms for the differentdirections tested. In this circumstance, one speaks ofisotropy and it is acceptable to average all experimentalsemivariograms regardless of orientation. The result isusually a smooth semivariogram—an omnidirectionalsemivariogram—which is smoother than individualdirectional semivariograms.

If the sampling is trend free and the average size ofthe anomalies is smaller than the maximum length of thesampling area, then the typical situation is that oneobtains semivariograms with different rates of increasefor short lags that level off at a common sill thatapproximates the sampling variance (Barnes 1991), suchas in Fig. 5. This is the second basic type of behavior, asemivariogram with what is called a geometric anisot-ropy, which is a true function of distance and direction.

Figure 6 shows the third type of basic behaviorthrough eight semivariograms for the Elk data set atincrements of 22.5�, starting from the east–west direc-tion. Now, instead of observing a sill for every semi-variogram, one can see an exponential increase without

bound at different incremental rates. Many of the esti-mated values surpass the value of the variance, which inthis instance is 23 mgal2 . What we have in such situa-tions are a collection of artifacts, not genuine experi-mental semivariograms. Even a casual inspection ofFig. 1 reveals that there is a systematic increase in theBouguer gravity from northwest to southeast, whichviolates the assumption of no-trend for the proper use ofthe estimator in Eq. 1. Hence, the curves in Fig. 6 areneither semivariogram estimates, nor the kind of semi-variograms that will be required in a geostatisticalcharacterization of an attribute with a trend. In thepresence of trend, the semivariogram required formodeling is that computed on the residuals obtainedafter removing the trend. Yet to remove the trend, it isnecessary to have the semivariogram of the residual. Thesimplest, yet most effective way out of this conundrum,is to find a trend-free direction—namely a direction thaton average has a constant mean—and use the semi-variogram in that direction as the semivariogram for theresidual. The justification for this is based on the factthat the experimental semivariogram depends on dif-ferences z(xi+h ) � z(xi) in which the addition orsubtraction of a constant to each term does not changethe increment. The trend-free direction is perpendicularto the direction of maximum dip and coincides with thedirection of minimal increase in the pseudo-semivario-grams of a directional survey. In the case of the ElkCounty data, this direction is about N67E.

A second directional analysis around the approxi-mately trend-free direction—such as the one inFig. 7—helps to narrow the solution, which in this caseis approximately N63E. Other more sophisticated ap-proaches not considered here include iterative modelingof the trend and the semivariogram (Chiles and Delfiner1999, pp. 115–128) and removal of the trend by filteringthrough the calculation of increments (Chiles and Del-finer 1999, chap. 4).

Var(Z)

lag

sem

ivar

iogr

am

Direction 1

Direction 2

Direction 3

Direction 4

Fig. 5 Schematic example of geometrically anisotropic semivario-gram

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6 Step 5: simple modeling

If all that the user wants to obtain are some conclusionsfrom the inspection of the experimental semivariogram,step 4 is the end of the process. Yet, more often than not,

the semivariogram is required for kriging estimation orfor some form of stochastic simulation involving kri-ging. In these situations, modeling of the semivariogrambecomes mandatory.

Kriging is the solution to the quadratic minimizationproblem of finding weights that minimize the estimation

Fig. 6 Directionalsemivariogram investigation forthe Elk County gravity data

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error in a mean square sense. Any quadratic minimiza-tion problem has a unique, positive solution, providedthat the coefficient matrix is not singular, which in thecase of the kriging minimization problem introduces therequirement that the semivariogram be negative definite.By a positive solution, it is meant that the objectivefunction—the estimation error—be positive, which is

essential to avoid imaginary standard errors. A semi-variogram model is any negative definite analyticalexpression of a shape likely to capture and emulate thestyle of variation of some experimental semivariogram.By replacing the experimental semivariogram by a neg-ative definite model, the user avoids singular krigingmatrices no matter what the combination of arguments.

Fig. 7 Directionalsemivariogram investigationaround the trend-free direction

313

This explanation is given to dispel any notion that theuse of semivariogram models is an unnecessary compli-cation introduced solely to make life more miserable.The use of negative definite models is utterly more effi-cient than the alternative to test the non-singularity ofevery kriging coefficient matrix for each particular set ofvalues derived from direct interpolation of the table ofexperimental semivariogram values, even though thiswould be a valid approach.

Although there is an infinite number of negativedefinite functions, the basic shape of the semivariogramrising from zero to reach a limiting value restricts to afew the negative definite functions that are of interest.Those most commonly employed are defined in Table 1

and displayed in Fig. 8. Parameters C and a conve-niently relate directly to the sill and the range. A specialcase of the negative definite model is the pure nuggetmodel, N, which can be considered a limiting case ofsome of the other models when the range is infinitesi-mally small.

N ¼ C0 1� H 0ð Þð Þ;

where H(0) is the Heaviside function, which is 1 at lag 0and 0 otherwise. In this particular case, the constant C0

is not called the sill but the nugget effect, a term derivedfrom the modeling of semivariograms of gold deposits.The sum of a simple model and a pure nugget effectmodel is also negative definite. The higher the nuggeteffect relative to the nugget minus sill, the poor thespatial continuity. To the limit, just a pure nugget effectsemivariogram indicates complete absence of spatialcontinuity, making the use of geostatistics meaninglessas it produces the same results as classical statistics. Suchlack of continuity can be real, the result of too largesampling space, or the consequence of numerous blun-ders in the data. For example, an attribute with a truerange in its semivariogram of 100 m will have a purenugget effect semivariogram when the attribute is sam-pled at intervals of 1 km. In my experience, this iscommonly the case of geochemical data.

Modeling of a semivariogram is the process ofreplacing the collection of estimated values by the closestnegative definite model, namely, the selection of themost adequate model type and the determination of itsparameters. This can be done by:

(a) trial and error (Goovaerts 1997, 97–104);(b) maximum likelihood (Kitanidis 1997, chap. 4); and(c) weighted least squares (Jian et al. 1996).

Considering that this paper presents a practical ap-proach, I refer the reader to the references for theoreticaldiscussions. For years, I have had the most satisfactory

Table 1 Most commonly usedsimple semivariogram models 0 < a, 0 < C

Power model : P hð Þ ¼ a hb; 0\b\2

Exponential model : Ex hð Þ ¼ C 1 � e�3ha

� �

Gaussian model : G hð Þ ¼ C 1� e�3hað Þ

2� �

Spherical model : Sp hð Þ ¼ C 32ha� 1

2ha

� �3� �; 06 hj j\ aj j

C; aj j6 hj j

(

Pentaspherical : Pe hð Þ ¼ C 158ha� 5

4ha

� �3 þ 38

ha

� �5� �; 06 hj j\ aj j

C; aj j6 hj j

(

Cubic model : Cu hð Þ ¼ C 7 ha

� �2 � 354

ha

� �3 þ 72

ha

� �5 � 34

ha

� �7� �; 06 hj j\ aj j

C; aj j6 hj j

(

Sine hole effect : S hð Þ ¼ C 1� sin phað Þ

pha

� �

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0

lag

sem

ivario

gra

m

Power

Exponential

Gaussian

Spherical

Pentaspherical

Cubic

Sine hole effect

Fig. 8 Most common negative definite semivariogram modelswhen the two parameters are equal to 1

314

Fig. 9 Best fits for experimental semivariogram along N63E employing each one of the models in Table 1 plus a pure nugget model whennecessary

315

results employing weighted least squares, particularlywhen the experimental semivariogram follows a typicalbehavior devoid of anomalous fluctuations, such as thecase of the semivariogram in Fig. 4.

Regardless of the method used to find the modelparameters, it is particularly hard for the novice to pickthe best type of model by simple inspection. Hence, thesafest approach is to fit all models and then select themodel with the best goodness of fit, which is a trivial andinstantaneous undertaking when the modeling is notdone by trial and error. Figure 9 shows the results forthe Elk data using a program described in Jian et al.(1996) that is available from the Internet at http://www.iamg.org/CGEditor/index.htm. Table 2 containsthe optimal parameters and the sum for the squares ofthe weighted differences, Rm.

In a weighted least squares sense, the best model isthe Gaussian one, closely followed by the cubic model.

In the absence of a trend, if there is anisotropy, onehas to model as many directions as dimensions in thesampling space. For two dimensions, one has to modelone semivariogram for the direction of maximum rangeand another one for the direction of the minimum range.Programs making use of anisotropic models generallywill expect that:

(a) the two directions are perpendicular;(b) the type of model is the same; and(c) both models have the same sill; which in most cases

forces some approximations. Those programs auto-matically model the range for intermediate directionsas the radius of an ellipse in which the axes are theminimum and maximum ranges.

For cases with trend and modeled only in thetrend-free direction, such as the demonstration dataalong N63E, if the user wants to investigate the pos-sibility of anisotropy, the investigation may be doneindirectly making use of crossvalidation. Crossvalida-tion is a verification process in which each observationis removed with replacement to produce an estimate atthe same site of the removal. Each estimate is thenused to calculate a difference with the correspondingcensored measurement, thus generating a set of errorsthat one can use to investigate their sensitivity to theselection of the estimation method and some of itsparameters (Olea 1999, Chap. 7). If one employs an

estimator involving a semivariogram model—such asthe most adequate form of kriging, which in the caseof the Elk data may be universal kriging because ofthe presence of a trend—one can use crossvalidationto study the sensitivity of the errors to changes in thesemivariogram. Conclusions derived from the analysisof the errors, however, must be taken cautiously be-cause the errors are not independent.

Table 3 shows the sensitivity of crossvalidation errorsto two sets of anisotropic models, one with a maximumrange 10% larger than the range in the best simplemodel and a minimum range 10% less than the range inthe best simple model (16.38, 13.40) and another set witha discrepancy of 20% (17.86, 11.91). Although there is asystematic improvement that suggests a minimum meansquare error for a largest range oriented approximatelyin a north-south direction, for this example theimprovement is not large enough to be considered sig-nificant or to justify the complications of an anisotropicmodel.

7 Step 6: nested modeling

A sum of negative define semivariograms is also negativedefinite. The sum of a simple model plus a pure nuggeteffect model is just a special case. This property ofnegative definite semivariograms opens infinite possi-bilities of semivariogram mixing, which in geostatisticaljargon is called semivariogram nesting. In practice, thegoodness of fit rapidly reaches a saturation point,explaining why one rarely sees nested models involvingmore than a pure nugget effect model plus two simplemodels, not necessarily of the same type.

Table 2 Results of fitting simple models by weighted least squares

Model Rm

(mgal4)C0

(mgal2)a or C(mgal2)

b or a(km)

Gaussian 0.081 0.038 0.990 14.885Cubic 0.085 0.033 0.991 20.525Spherical 0.202 0.000 1.083 25.449Pentaspherical 0.217 0.000 1.100 31.958Exponential 0.349 0.000 1.663 72.665Sine hole effect 0.410 0.038 0.909 15.904Power 0.565 0.000 0.069 0.857

Table 3 Sensitivity of Elk County Bouguer gravity to anisotropicsemivariogram models

Maximumrange (km)

Minimumrange (km)

Orientation oflargest range

Mean squareerror (mgal2)

14.89 14.89 0.23116.38 13.40 N63E 0.233

N83E 0.235N77W 0.235N57W 0.234N37W 0.232N17W 0.230N3E 0.229N23E 0.229N43E 0.231

17.86 11.91 N63E 0.236N83E 0.240N77W 0.241N57W 0.238N37W 0.234N17W 0.230N3E 0.227N23E 0.228N43E 0.231

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To keep an eye on the parsimony of nested modeling,one can use the Akaike information criterion (AIC)from time series analysis. The AIC is a measure ofgoodness of fit involving not only the weighted errors,but the number of points used for the fitting, n, and thenumber of parameters, p, as well (Tong 1983, p. 135):

AIC ¼ n lnRm

n

� �þ 2p

The smaller the Akaike information criterion, the betteris the fit. Given an experimental semivariogram, when nand p remain constant, such as in Table 3, the rankingby Rm or AIC is the same.

By looking at the best simple model in Fig. 10a, onecan see that between a lag of 8 and 20 km there are 8experimental points below the curve, two of them clearlybelow, which justify trying a more complex model to aim

Fig. 10 Best semivariogrammodels. (a) Simple Gaussianmodel. (b) Best double nestedGaussian model.

Table 4 Best simple and double nested models

Model Rm (mgal4) AIC C0 (mgal2) C (mgal2) a (km)

Simple Gaussian 0.081 �104.2 0.038 0.990 14.885Nested Gaussian 0.025 �123.4 0.010 0.6900.380 22.3118.042

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for a better fit. Table 4 and Fig. 10b provide the answer:a double nested Gaussian model.

The AIC for the nested model is indeed smaller thanthat for the simple one. Hence, the improvement isworth increasing the number of parameters from 3 to 5.Considering that the crossvalidation does not show anysignificant evidence of anisotropy for this nested modeleither, the isotropic model:

cðhÞ ¼ N 0:01ð Þ þG h; 0:69; 22:3ð Þ þG h; 0:38; 8:0ð Þ

is the best model for the Bouguer gravity anomaly datafrom Elk County.

If there is a sill, then the covariance is easily obtainedby subtracting the semivariogram from the total sill,which in this case would be:

CovðhÞ ¼ 1:08�N 0:01ð Þ �G h; 0:69; 22:3ð Þ�G h; 0:38; 8:0ð Þ

The main advantage of this indirect way to model thecovariance is that it does not require knowledge of themean.

8 Concluding remarks

I hope the novice reader, for which this paper is in-tended, is now less intimidated and more confident ofbeing able to model a semivariogram or a covariance.

Sophistications and variants abound. The six stepsdescribed here are by no means the absolute way to go.The ultimate test of understanding for the reader will beto feel confident enough to try her or his own version ofthese basic steps.

Acknowledgements I am grateful to John H. Doveton and twoanonymous reviewers for critical reading of the manuscript thatresulted in suggestions that improved the presentation.

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