INTERPOLATION METHODS

Morrison (1971) characterizes the production of the distribution of a property intwo dimensional space as a two staged process. In the first stage, a pointinterpolation method is used to provide estimates of the property at nodes locatedeither at the intersection of regularly spaced, orthogonal grid lines, or centrallybetween the grid lines. Isolines are then produced by linear interpolation betweenthese nodal values. Morrison (1971) concluded that the error introduced by thesecond stage linear interpolation may be assumed to be insignificant.

There are many methods of interpolating randomly spaced point data. Some ofthese method are global while others are local. Global methods utilize all theknown values to evaluate an unknown value, while in local methods only aspecified number of nearest neighbours are used to evaluate an unknown value.The selected methods are representative of the general classes of methodsavailable. The general purpose of interpolation is to obtain an estimate of aproperty at a point i , Pi, based on values of the property at sampled or measuredlocations.

Inverse-distance weighted method (IDW)The inverse-distance weighted procedure is versatile, easy to program andunderstand, and is fairly accurate under a wide range of conditions (Lam, 1983).Using this method, the property at each unknown location for which a solution issought is given by:

(1)P i =Σj=1

GP j/D ij

n

Σj=1

G1/D ij

n

where Pi is the property at location i; Pj is the property at sampled location j; Dij

is the distance from i to j; G is the number of sampled locations; and n is theinverse-distance weighting power. The value of n, in effect, controls the region of influence of each of the sampled locations. As n increases, the region ofinfluence decreases until, in the limit, it becomes the area which is closer to point ithan to any other. When n is set equal to zero, the method is identical to simplyaveraging the sampled values. As n gets larger, the method approximates the

Voronoi tessellation procedure (Watson and Philip, 1985). Usually, the value of n is set arbitrarily. Values of 1.65 and 2 were used for n by Kelway (1974) andNOAA (1972), respectively, for interpolating rainfall, while the ARMOS model (Kaluarachchi et al., 1990) recommends the use of values ranging from 4 to 8 forinterpolating oil pressure heads.

Watson and Philip (1985) listed some of the limitations of the inverse-distanceweighted procedure. The major limitation is that estimates are bounded by theextrema in the sampled values. Additionally, the radial symmetry which thisprocedure imparts to the data obscures the effect of linear features such as ridgesor valleys. For n 1, the derivative of the interpolated surface is discontinuous at≤the sampled locations, while for n >1, the surface is flat at these sampled locations.Watson and Philip (1985) have presented a variation of the inverse-distanceweighted procedure which removes the limitations mentioned above. In theirprocedure the weights are functions of slopes as well as of distance.

Punctual Kriging (PKG)In the punctual kriging method, a weighted mean of the sampled values isevaluated for each node, such that the estimation error is minimized by solvingthe following set of equations simultaneously (Davis, 1986):

(2) P i = Σj=1

G

WjP j

(3)Σj=1

G

Wj = 1

k = 1,2,...G (4)Σj=1

G

Wjγ(Dkj) + λ = γ(Dki)

where Wj is the weight for location j; is a variable added to minimize theλestimation error; and (Dkj) is the semivariogram value between points k and j. Inγsituations where the nature of the semivariogram is unknown, the set of equationslisted above can still be solved by assuming that there is no nugget effect and thatthe semivariogram is given by:

(5) γ(Dkj) = A ∗ f(Dkj) where f is a known function of Dkj and A is a constant. There is only a smallgroup of functions that can be used to define a semivariogram. McBratney andWebster (1986) have listed these functions and explained why other functions areinappropriate. In the simplest case in which f(Dkj) = Dkj, equation 4 may bereplaced by:

k = 1,2,...G (6)Σj=1

GWjDkj + β = Dki

where β is a constant, since it is not necessary to determine the value of or inλ βorder to solve for Pi. This simplest case, which is essentially the assumption of alinear semivariogram, is useful in situations where the number of sampledlocations is not sufficient to define the actual shape of the semivariogram.Research has shown that while the shape of the semivariogram influences theestimation variance [a statistic whose usefulness is questionable in any case(Philip and Watson, 1986; Shrivastava, 1986)] , it has little effect on theinterpolated values ( Tevis et al., 1991). Cooke and Mostaghimi (1992) used alinear semivariogram to estimate areal mean rainfall and obtained values whichwere not significantly different from those obtained by more conventionalmethods.

Akima's polynomial method (AQP)Akima (1978) developed a method of interpolation in two dimensional space,based on fitting local fifth-order polynomials to triangles whose vertices arelocated at the sampled locations. He used a triangulation method proposed byLawson ( 1972) which maximizes the minimum interior angle of the triangles.Once the triangles are formed, coefficients of the local polynomials are chosensuch that the partial derivatives of the surface is continuous across the sides of thetriangles and at the vertices. The local polynomials are of the form:

(7)P i = Σl=0

5

Σk=0

5−l

q lkx lyk

whereq are the 21 coefficients of each local polygon, x and y are coordinatedistances relative to some local origin, and l and k are dummy variables. Akima'smethod can also be used for extrapolation, though the accuracy decreases for suchan application.

Major limitations of this method are that higher order derivatives are notcontinuous across the sides of the triangles, and that if the property changesdrastically over a short distance, there tends to be some oscillations in theestimated values close to the vertices. Preusser (1990 ) developed a modifiedmethod which has continuous second derivatives across the boundaries andMontefusco and Casciola (1989) developed a modified method which eliminatesany tendencies to oscillate.

Minimum surface curvature method (MSC)The minimum surface curvature method was developed by Briggs (1974). Heshowed that if the total curvature of a surface is minimized under the constraintthat the surface honors the values at sampled locations then the surface satisfiesthe equation:

(8)∂4P∂x4

+ 2 ∂4P∂x2∂y2

+ ∂4P∂y4

= 0

at all unsampled locations. On a discrete, gridded surface, this corresponds to solving simultaneously the finite difference equation:

Pi+2,j + Pi,j+2 + Pi-2,j + Pi,j-2 + 2(Pi+1,j+1 + Pi-1,j+1 + Pi+1,j-1 + Pi-1,j-1)

- 8(Pi+1,j + Pi-1,j + Pi,j-1 + Pi,j+1) + 20 Pi,j = 0 (9)

for each grid point which is not sampled or which is not at nor immediatelyadjacent to the boundary of the domain. Similar difference equations werepresented for those grid intersection points not included in Equation 9. Sampledlocations are excluded from the solution as their values are always honored. Thesesimultaneous equations can be solved iteratively.

REFERENCESAkima, H. 1978. A method of bivariate interpolation and smooth surface fittingfor irregularly distributed data points. ACM Trans. on Math. Software 4: 148-159.

Briggs, I.C. 1974. Machine contouring using minimum curvature. Geophysics 39:39-48.

Cooke, R.A. and S. Mostaghimi. 1992. A microcomputer based routine forobtaining mean watershed precipitation from point values. Computers andGeosciences 18: 823-837.

Davis, J. C. 1986. Statistics and data analysis in geology (2nd ed.). New York:John Wiley & Sons.

Kaluarachchi, J.J., J.C. Parker and R.J. Lenhard. 1990. A numerical model forareal migration of water and light hydrocarbon in unconfined aquifers. Adv. WaterResour. 13: 29-40

Kelway, P. S. 1974. A scheme for assessing the reliability of interpolated rainfallestimates. Jour. Hydrology 21: 247-267.

Lam, N. S. 1983. Spatial interpolation methods review. The AmericanCartographer 10: 129-149.

Lawson, C.L. 1972. Generation of a triangular grid with application to contourplotting. Tech. Memo. 299, Sect. 914, Jet Propulsion Lab., Caltech, Pasadena,California.

McBratney, A.B., and R. Webster 1986. Choosing functions for semi-variogramsof soil properties and fitting them to estimates. Jour. Soil Science 37: 617-639.

Montefusco, L.B. and G. Casciola. 1989. C1 surface interpolation. ACMTransactions on Mathematical Software 15: 365-374.

Morrison, J. 1971. Method-produced errors in isarithmic mapping. TM CA-5,American Congress on Surveying and Mapping, Washington DC.

N.O.A.A. 1972. National Weather Service River Forecast System forecastprocedures. TM NWS HYDRO-14, US Department of Commerce, Washington DC.

Philip, G.M. and D.F. Watson. 1986. Matheronian geostatistics - Quo vadis?.Math. Geol. 18: 93-117.

Preusser, A. 1990. C1- and C2- interpolation on triangles with quintic and nonicbivariate polynomials. ACM Transactions on Mathematical Software 16:253-257.

Srivastava, R.M. 1986. Philip and Watson - Quo vadunt?. Math. Geol. 18:141-149.

Tevis, J.W., A.D. Whittaker and D.J. McCauley. 1991. Efficient use of data in thekriging of soil pH. ASAE paper 91-7074. St. Joseph, MI: ASAE.

Watson, D.F. and G.M. Philip. 1985. A refinement of inverse distance weightedinterpolation. Geo-Processing 2: 315- 327.

Arithmetic Mean : (A+B+C)/3

Inverse Distance:

n = 1:

n=2n = 8

A

B

C

8

3

12(4)

(11)

(7)

X

P i =Σj=1

GP j/D ij

n

Σj=1

G1/D ij

n

x =48n + 11

3n + 712n

18n + 1

3n + 112n

x =48 + 11

3 + 712

18 + 1

3 + 112

x =482 + 11

32 + 7122

182 + 1

32 + 1122

n = 0.5

x =488 + 11

38 + 7128

188 + 1

38 + 1128

x =4

80.5 + 1130.5 + 7

120.5

180.5 + 1

30.5 + 1120.5