GEOS 5311 Lecture Notes: Interpolation andModel Parameterization

Dr. T. Brikowski

Spring 2013

0interpolation.tex,v, Vers. 1.221

Parameterization:Data Needs

I general categories include (see Fig. 1, or Table 1 Mercer andFaust (1980))

I physical framework: physical extent of the hydrostratigraphicunits to be modeled

I hydrologic frameworkI water table and potentiometric surface mapsI hydrographs of groundwater well and surface water body water

levelsI aquifer hydraulic properties: storativity, hydraulic conductivity,

leakance, etc.I source/sink properties: well pumping rates, evapotranspiration,

recharge estimates, etc.

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Modeling Data Needs

Figure 1: Flow modeling data needs. After Anderson and Woessner(Table 3.1, 1992).

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Obtaining Regionalized Data

I a basic problem in numerical modeling is reconciling point(well) measurements with the averaging implicit in discretizedmodels. Two problem areas:

I obtaining a valid areal average from point measurements ofhighly heterogeneous aquifer properties (e.g. hydraulicconductivity)

I estimating aquifer properties at points away from well or otherdata (interpolation/extrapolation)

I Areal average example:I Snake River Plain, where lava tubes and buried sedimentary

channels provide much faster flow & transport pathways thanareal averages would indicate (Poeter and Gaylord, 1990)

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Hanford Nuclear Reservation Problem

Purpose Big Mess!, must evaluate transport of tritium (3H,radioactive) to Columbia River and accessibleenvironment

Cause careless disposal of radionuclides (Fig. 3), leaky“temporary” storage tanks

Hurdles: Dangerous tanks contain mixedradionuclide-hydrocarbon waste andcan be explosive (e.g. USSR Mayakexplosion, 1957)

Heterogeneity contaminated aquifer highlyheterogeneous mix of Columbia Riverdeposits and a bit of basalt flow (Figs.4–5

Complex Paths observed transport complex (Fig. 6)

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Hanford Site Surface Features

Figure 2: Hanford Nuclear Reservation site features, after (Fig. 1,Poeter and Gaylord, 1990). See state location map.

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Hanford Kriged Water Table

Figure 3: Water table elevations (kriged) for Hanford site, 1951 and1981. Water level rise resulted from disposal of liquid wastes. After (Fig.3, Poeter and Gaylord, 1990).

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Hanford Lithofacies

Figure 4: Lithofacies maps for upper 80 ft. of saturated zone at Hanfordsite. (a) gravel-dominated facies, (b) mud-dominated (Poeter andGaylord, 1990).

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Hanford Cross-Sections

Figure 5: Correlated borehole cross-sections, Hanford Site. After (Fig. 7,Poeter and Gaylord, 1990).

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Hanford Tritium Migration

Figure 6: Observed tritium migration vs. time, Hanford Site. After (Fig.4, Poeter and Gaylord, 1990). N.B. EPA MCL 3H is 20,000pCi

l . Seeother maps.

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Regionalization Approaches

I evaluate likely lithologies present in model area, model each asseparate hydrostratigraphic unit

I ensure that areas of differing hydraulic gradient in the field areassociated with differing conductivity zones orsource/sink/boundary effects in model

I perform a formal inverse model, which estimates thedistribution of rock properties based on hydrologicobservations (usually transmissivity estimates based on headgradients) (Menke, 1984; Townley and Wilson, 1985; Yeh,1986)

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Anisotropy in Averaging

I Recall the effect of layered heterogeneity on directional(anisotropic) hydraulic conductivity, as discussed in theHydrogeology lecture notes

I see also Anderson and Woessner (1992, p. 69-70) or (Freezeand Cherry, 1979)

I When approximating a layered series of units, specification ofanisotropy can be used to incorporate the relative limits ofvertical to horizontal flow through the series as a whole.

I Recall that anisotropy “skews” the flow field in the dominantprincipal component direction (Fig. 8).

I Streamlines are no longer perpendicular to isopotentials,giving anisotropic flow model results an “odd” appearance.

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Anisotropy and Layering

Figure 7: Relationship between layered heterogeneity and anisotropy. Inthis case the homogeneous equivalent Kz = d∑n

i=1diKi

and

Kx =∑n

i=1Kidid . From Freeze and Cherry (Fig. 2-9, 1979).

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Anisotropic Flow

Figure 8: Flow field given various hydraulic conductivity ellipsoidgeometries. After Freeze and Cherry (Fig. 5-9, 1979).

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Introduction to Interpolation (“contouring”)

I in modeling a continuous 2- or 3-D field of values is neededfor many parameters, and must be obtained from scatteredpoint (borehole) data. Usually this take the form ofcontouring the point data.

I Interpolation estimates values between known points, i.e.inside of the convex hull (the line surrounding all the points,Fig. 9)

I Extrapolation estimates values outside the convex hull

I often the choice of interpolation/extrapolation method cangreatly influence model results

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Convex Hull

Figure 9: Convex hull and TIN of a scattered dataset

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Interpolation methods in GMS

Four basic types of interpolation are available in GMS

I TIN-Based: Linear interpolation and Clough-Tocher

I Inverse-Distance Weighted (IDW, only method available inmany packages like ArcView)

I Natural Neighbor (area and distance weighted, good forclustered data)

I Kriging (correlation-length weighted, good for geologic media)

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Linear Interpolation

I simplest method, best suited for huge, smooth datasets

I generates temporary TIN (triangular irregular network), fits aplane to each triangle (e.g. Fig. 9), calculates surface valuefor points lying in the triangle using the plane (i.e. a 4-termequation)

I local method, so fast, but no extrapolation is allowed

I produces “rough” surface (first derivative discontinuous or“C0”)

I doesn’t allow unsampled local minima and maxima to beinferred (all interpolated values lie between extremes ofsampled values)

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Clough-Tocher

I AKA “finite-element” method, also based on temporary TIN

I fits a cubic surface to the triangle (i.e. a 10 term equation,Fig. 10

I local method, so fast, but no extrapolation is allowed

I makes smooth surface (first derivative is continuous or“C1-continuity”)

I good at quickly emphasizing local trends in data (mostly usedto generate realistic artificial topography in animations)

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Clough-Tocher Element

Figure 10: Clough-Tocher discretization, showing subdivisions and inputsused to approximate function surface. After EMRL (2003).

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IDW Methods in GMS

I Inverse distance weighted (IDW): nearby points influence theestimate most strongly.

I expressed mathematically as

F (x ,y) ≈N∑i=1

wi (x ,y) · fi (1)

where wi is the weight associated with the ith point (whichhas function value fi )

I a wide variety of methods can be used to determine theweights, and to select the points that are used to make theestimate

I see also GMS Online Help

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IDW Weight Functions

I Shepard’s Method:I weight function varies linearly over range 0:1 and is radially

symmetric about each pointI fails to infer local extrema, produces rough (C0) surfaceI tends to show “bullseyes” around isolated points (because of

radial symmetry)

I Modified Shepard’s Method:I weight functions are nodal or basis functions defined at each

data point (node).I various function types are used:

I gradient plane: sloping planes reflecting local estimatedgradient in data. Allows inference of local extrema. Producesrough surface (first derivative discontinuous acrossconnections between nodes).

I quadratic function: higher-order nodal functions thatapproximate (don’t necessarily pass through) neighboringnodes. Can be slow with large datasets, produces nicelysmoothed surface, requires at least 5 data points.

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IDW Point Subsetting

All, or a subset of points may be used in IDW estimates (see eqn.1). Subsets use the nearest N points (in any direction, or byquadrant, which can help with clustered data), which are found byone of three methods:

I global: use or examine all points in dataset. Can be slow forlarge datasets

I local: points are crudely sorted into distance “bins” using atemporary TIN, only examines points in the same triangle asthe point of interest, then neighboring triangles, etc. Savesmuch time for large datasets.

I “enclosing triangle”: only TIN triangles are generated usingthe point of interest as a vertex (Fig. 11). Makes methodlocal (i.e. fast) and default basis function makes method C1continuous (smooth surface)

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Enclosing Triangle Subset

Figure 11: Enclosing triangle subset method for IDW, and “S”-shapedbasis function for surface approximation. “A” is the point of interest,bold line shows perimeter of enclosing triangles, which are the only areasused to generate the interpolated value at A. After EMRL (2003).

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Natural Neighbor Method

I essentially a combination of Clough-Tocher and IDW

I data points whose “areas of influence” (in a finite-elementsense) are adjacent to the interpolation point are used (usingDelaunay Triangulation, Fig. 12)

I weights based on size of boundary between areas of influenceof “neighboring” data points and interpolation point

I extrapolates beyond convex hull easily

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Delaunay Triangulation

Figure 12: Delaunay triangulation for natural neighbor method.IDW weights are based on Theissen polygon area for each node.After EMRL (2003).

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Introduction to Kriging

I uses spatial correlation structure in data to compute weightsin IDW-style interpolation (Journel, 1989)

I GMS implementation based on GSLIB (Deutsch and Journel,1992)

I two basic types: ordinary, and universal. The latter attemptsto account for regional trends in the data.

I computes a spatially-varying estimate of interpolation error(estimation variance), so error distribution can be mapped

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GMS Interpolation/Extrapolation Summary

Table 1: Major characteristics of interpolation/extrapolation methodsimplemented in GMS. Continuity: ’C0’ implies only ’zeroth’ derivative(i.e. the function itself) is continuous, producing a rough surface.Continuity ’C1’ indicates the function and its first derivative arecontinuous, making a smooth surface at the data points. All methodsexcept Kriging produce a surface that passes through the data points.Local methods are faster for large datasets, but may yield roughersurfaces when regional trends are present.

Method Local Continuity Comment

Linear√

C0 Faceted surface

Shepard’s(IDW)

C0 Smoother, prone to over-shoot

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GMS Interpolation/Extrapolation Summary (cont.)

ModifiedShepard’s(GradientPlane)

C1

ModifiedShepard’s(Quadratic)

C1 Smoother than gradientplane

Clough-Tocher

√C1

NaturalNeighbor

C1 Weights computed usinginverse distance and rela-tive density of points in anydirection

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GMS Interpolation/Extrapolation Summary (cont.)

Kriging C1 Uses spatial correlations toimprove interpolation esti-mate. Inexact interpola-tion, allowing calculation ofestimation error.

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GMS Interpolation/Extrapolation Summary (cont.)

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Jackknifing

Methods are needed to test the accuracy of an interpolationmethod with a given data set.

I input data can be subsetted,I withhold some points (e.g. 1

3 )I compare interpolated to measured values at those points

I withhold individual points successively (Jacknifing)I re-compute interpolation without the pointI compare interpolated to actual value at the pointI accumulate statistics on the error

I Jackknifing is offered as a menu choice in GMS/[2-3]DScatter Module/Interpolation. Use it!!

I it is a special form of the more general approach towarddetermination of variance termed resampling (Tukey andMosteller, 1977)

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Quantifying Error

A variety of standard formulas are used to quantify error, usuallyfor model calibration. The following three are discussed in thetextbook (Sec. 8.4, Anderson and Woessner, 1992):

ME Mean Error: mean difference between measured head (hm)and simulated head (hs):

ME =1

n

n∑i=1

(hm − hs)i (2)

MAE Mean Absolute Error: mean of absolute value of differences(avoids cancellation by large errors of opposite sign)

MAE =1

n

n∑i=1

|hm − hs |i (3)

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Quantifying Error (cont.)

RMS Root Mean Squared Error: standard deviation

RMS =

√√√√1

n

n∑i=1

(hm − hs)2i (4)

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Types of Kriging

I Simple krigingI assumes the mean (expected) value of the interpolant does not

vary with position (i.e. is stationary)I seeks changes from that expected value, i.e. that the expected

value is zero everywhere in the problem domainI therefore generally produces smoother but less accurate

interpolation

I Ordinary krigingI most commonly applied methodI assumes an unknown but spatially invariant mean value

I Universal krigingI allows for spatially varying mean (drift, e.g. a sloping surface)I in GMS this can be fit by a linear or quadratic function

I Indicator krigingI interpolates an indicator function rather than the dataI e.g. for defining lithology, where function equals 1 if in sand,

zero if in clay

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Variograms

I determines degree of correlation vs. distance. Sample varianceplotted on Y-axis, distance on X-axis. This is called theexperimental variogram in GMS (Fig. 13).

I typically these show points close to one another with lowervariance than points farther apart. Beyond some distance(called the range) variance remains constant provided there isno regional trend in the data

I the experimental variogram is usually rough, and is fit by amodel for use in interpolation 14

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The Mathematics of Kriging

I the model variogram is used to compute weights in an IDWscheme to give minimum estimation error at observationpoints.

I note kriging may not interpolate known points exactlyI this can be an advantage, since estimation error can then be

calculated without jacknifingI for datasets with constant variance (no spatial correlation),

kriging should interpolate as well as any IDW scheme (i.e. youcan’t do worse than most non-kriging approaches, and oftenwill do better)

I Ordinary kriging can be expressed mathematically as:

F (x ,y) ≈N∑i=1

wi (x ,y) · fi (5)

N∑i=1

wi = 1 (6)

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The Mathematics of Kriging (cont.)

where wi is the weight associated with the ith point, whichdepends on spatial correlation and other information.

I technically inclusion of (6) is what distinguishes ordinary fromsimple kriging

I This form is quite similar to the IDW formulation (1), andproduces the same result when no spatial correlation is used.

I the weight function wi (x ,y) is determined by constructing avariogram, which expresses the spatial dependence (i.e.variance) of variability between observations fi

I the calculation of that variance is the topic of Geostatisticsclasses, GMS provides a number of variogram forms (e.g.“Gaussian” which has an “S” shape) that can be selected tobest-fit the data

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Model Variogram

Figure 13: Features of a model variogram. These are used to determinethe model variogram (i.e. inputs to GMS model variogram editor). AfterEMRL (2003).

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Model vs Experimental Variogram

Figure 14: Relation between model and experimental variograms. γ isvariance between samples a distance h apart. After EMRL (2003).

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Anisotropy

I variogram range may depend on direction (Fig. 16)

I software like GMS/GSLIB that allow directional searches maybe used to detect anisotropy through trial-and-error

I the result is to define a variogram with maximumcontribution, and another one perpendicular to it.Interpolation is carried out using both variograms to estimatespatial correlation in the selected directions.

I directional searches are specified using an azimuth andbandwidth (i.e. for a semi-rectangular search area, Fig. 17)

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Geologic Origin of Anisotropy

Figure 15: Geological example of anisotropic variograms. After Journeland Huijbregts (Fig. 1.1, 1989).

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Anisotropic Variograms

Figure 16: Model and experimental variograms in anisotropic case.Experimental variogram designated by filled squares compares data in adirection perpendicular to that designated by open squares. After EMRL(2003).

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Directional Variogram Specification

Figure 17: Geometric parameters for specifying directional variograms inGMS (or GSLIB). After EMRL (2003).

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References

Anderson, M.P., Woessner, W.W.: Applied Groundwater Modeling.Academic Press, San Diego (1992)

Deutsch, C.V., Journel, A.G.: GSLIB: Geostatistical Software Library andUser’s Guide. Oxford University Press, New York (1992)

EMRL: GMS Reference Manual. Environmental Modeling ResearchLaboratory, Brigham Young Univesity, Provo, UT, 4.0 edn. (2003),http://www.bossintl.com/online_help/gms/

Freeze, R.A., Cherry, J.A.: Groundwater. Prentice-Hall, Englewood Cliffs,NJ (1979)

Journel, A.G.: Fundamentals of geostatistics in five lessons. Short Coursein Geology 8, 40 (1989)

Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic Press,San Diego (1989), fourth Printing

Menke, W.: Geophysical Data Analysis: Discrete Inverse Theory.Academic Press, Inc., Orlando, FL (1984)

Mercer, J.W., Faust, C.R.: Ground-water modeling: An overview.Ground Water 18, 108–115 (1980)

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References (cont.)

Poeter, E., Gaylord, D.R.: Influence of aquifer heterogeneity oncontaminant transport at the hanford site. Ground Water 28, 900–909(1990)

Townley, L.R., Wilson, J.L.: Computationally efficient algorithms forparameter estimation and uncertainty propogation in numerical modelsof groundwater flow. Water Resour. Res. 21, 1851–60 (1985)

Tukey, J., Mosteller, F.: Data Analysis and Regression, A Second Coursein Statistics. Addison-Wesley, Reading, MA (1977), out of print?

Yeh, W.W.G.: Review of parameter identification procedures ingroundwater hydrology: The inverse problem. Water Resour. Res. 22,95–108 (1986)

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