UNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY RESTON, VA. 22092 In Reply Refer To: October 7, 1980 EGS-Mail Stop 411 GROUND WATER BRANCH TECHNICAL MEMORANDUM NO. 81.01 Subject: A computer program for interpolation of geohydrologic data using the method of "kriging" The purpose of this memo is to describe briefly the interpolation procedure known as kriging, some of its applications and limitations in hydrology, and the availability of a documented computer program for kriging. Although kriging techniques have potential applicability to a variety of fields, we particularly ask that this memo be brought to the attention of those persons involved in ground-water studies. Frequently there is a need to estimate the areal characteristics of geohydrologic variables such as aquifer thickness, transmissivity, storage coefficient, etc., continuously over a region given the value of the variable at discrete points. The characteristics of the kriging method are that: (1) correlation information inferred from the data themselves are used in the analysis; (2) the interpolation estimates are "statistically" unbiased with minimum variance; and (3) such variance can be computed and provides an assessment of the estimation error. The ability to give an assessment of the interpolating error is the reason the method has attracted much attention. Geohydrologic applications: Kriging is applicable to regionalized variables which have some spatial correlation structure (see next section). Applications in hydrology to date include estimating the areal distribution of: (1) precipitation, by single event or annual mean; (2) transmissivities; and (3) potentiometric head. Examples of other uses include estimating areal thickness of confining beds, ET distribution in a basin, or spatial distribution of water-quality parameters. Delhomme (1978) gives several applications of kriging to hydrology and describes the mathematical development. He also discusses the applicability of kriging to the analysis of data networks. Karlinger and Skrivan (1980) applied kriging to estimation of the areal distribution of mean annual precipitation. Development of the kriging system: A kriging estimate (see original) of a "regionalized variable at an arbitrary location is determined as a weighted sum of the observed data. (see original for equation) where the n observations z(subscript i) and their locations (x(subscript i), y(subscript i)) are given by the hydrologist, and the gamma(subscript i) terms are unknown weights to be determined. The term "regionalized" is used to convey that the spatial variation of the variable possesses a certain structure. In other words, that the variation is not completely random. The kriging procedure determines the n weights, gamma(subscript i), such that these conditions hold: (1) the estimate is unbiased--in the sense that the average error will be zero for a large number of such estimates, i.e., (see original), where E[] is the expectation; and (2) the variance of the estimation, i.e. (see original) is minimal. These conditions lead to a system of linear equations involving the weighting terms gamma(subscript i). An inherent part of the resulting equations is a function called the semi-variogram which is a measure of the average difference of data values for points a given distance apart. The semi-variogram obtained directly from the basic data is considered empirical. Partly because of the varying degrees of uncertainty of the basic data, the empirical semi-variogram often will not be a smooth function. An important part of the kriging analysis are the inferences made from the empirical semi-variogram concerning: (1) the distance beyond which there is little or no correlation among the data; and (2) the regularity or continuity of the data. If the semi-variogram indicates "essentially" no spatial correlation, then kriging offers no advantage over other data fitting procedures. However, if a correlation structure is indicated, several theoretically acceptable smooth semi-variogram models are tried before selecting a best fit to the empirical variogram. With a fitted semi-variogram, the linear system of equations is solved for the unknown gamma(subscript i) values. As a byproduct of the solution, the variance of the kriging estimate, or kriging error, is also calculated. Interpolated values for the variable can be computed at a grid of arbitrary locations for contouring of estimates and kriging errors. The errors can be used for defining confidence intervals of the estimates. For each point where an interpolated value is needed a linear system of equations must be solved for a corresponding set of gamma(subscript i) values. Validation procedure: A validation procedure is used to test the choice of the fitted semi-variogram on the resulting interpolations. This involves: (1) The data for point (x(subscript 1), y(subscript 1)) is ignored. Based on the data at the other (n-1) points a kriged estimate z*(subscript 1) is made at (x(subscript 1), y (subscript 1)) and compared to z(subscript 1). The same process is followed for each of the data points being ignored one at a time. In this way a residual (z*(subscript i)-z(subscript i)) is obtained for each of the n data points. (2) The average residual for the n observations is calculated and compared to zero for unbiasedness. (3) The squared residuals are compared to the kriging errors, which should approximately equal each other for theoretical consistency. If these comparisons are favorable, then the fitted semi-variogram is acceptable. If not, then the semi-variogram model is adjusted. Some practical considerations: To define the semi-variogram, probably at least 20-30 data points are necessary. Because the inferences from the semi-variogram are a significant part of the analysis, enough points are needed to properly select a final form of semi-variogram model. On the other hand, because the size of the resulting system of linear equations for validation is dependent on the number of observations, it may not be practical to use more than 100-200 points in the validation procedure. Cases with many more data might use some subsets of the observational data for validation. If there is an independent reason to expect a general trend (i.e., drift) in the variable, that trend can be made part of the analysis. Because a system of equations must be solved for a set of gamma(subscript i) values for each points where an interpolation is needed, it is practical to seek ways to reduce the number of data points used to compute the kriged estimate. In some cases it is acceptable to consider only those points within a certain neighborhood. Limitations: Gambolati and Volpi (1979) state that the kriged estimates are not necessarily more reliable than those gotten from other methods of estimating. The primary appeal to the kriging technique is its ability to provide some assessment of the estimation error. The criteria used to obtain kriging estimates are that the estimation be unbiased and the variance of the estimation be minimized. Satisfaction of those criteria does not assure that the resulting estimates reproduce the spatial fluctuations of the true variable. The spatial variability of the kriged estimate tends to be a smooth version of the real variability. The density of data points probably influence how good a representation of the true variability is possible with kriging. The use of the kriged estimates will influence the extent to which this "smoothing" characteristic may limit their usefulness. Hydrologists have only very recently begun to develop experience with the kriging method. It has some powerful features--but as all methods, does have limitations. Recognizing those limitations and being guided by basic hydrologic principles, kriging should be a useful addition to the set of tools used to interpolate between data points. Computer Program K603: The FORTRAN program K603 is documented in the enclosed Computer Contribution (Skrivan and Karlinger, 1980) and is available on SYS1.LOADLIB on both RE1 and RE3. The documentation includes details on data preparation and time estimates. In addition, the report has annotated bibliography on kriging development and applications. Additional copies of the report are available on request from the Ground Water Branch. References: Delhomme, J. P., Kriging in the hydrosciences: Advances in Water Resources, v. 1, no. 5, p. 251-266. Gambolati, G., and Volpi, G., 1979, A conceptual deterministic analysis of the kriging technique in hydrology: Water Resources Research, v. 15, no. 3, p. 625-629. Karlinger, M. R., and Skrivan, J. A., 1980, Kriging analysis of mean annual precipitation, Powder River Basin, Montana and Wyoming: U.S. Geological Survey, Water-Resources Investigations (Approved May, 1980). Skrivan, J. A., and Karlinger, M. R., 1980, Semi-variogram estimation and universal kriging program; U. S. Geological Survey Computer Contribution, 98 p. (s) Charles A. Appel for Eugene P. Patten Acting Chief, Ground Water Branch Enclosure WRD Distribution: A (Memo only) B (NR, 12 copies w/enclosures) (SR, 6 copies w/enclosures) (CR, 21 copies w/enclosures) (WR, 12 copies w/enclosures) S (Memo only) Copy w/enclosure to District Offices Copy (Memo only) to Subdistrict OfficesA computer program for interpolation of geohydrologic data using the method of "kriging"