Class Note for C&PE 940 at KU
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Date Created: 02/06/15
KRIGING CampPE 940 19 October 2005 Geoff Bohling Assistant Scientist Kansas Geological Survey geoffkgskuedu 8642093 Overheads and other resources available at http peoplekuedugbohlingcpe940 What is Kriging Optimal interpolation based on regression against observed 2 values of surrounding data points weighted according to spatial covariance values Pronunciation Hard g as in Danie Krige or soft g a la Georges Matheron take your pick What is interpolation Estimation of a variable at an unmeasured location from observed values at surrounding locations For example estimating porosity at u 2000 m 4700 m based on porosity values at nearest six data points in our Zone A data 5700 e 0 4 1253 5200 7 5 1441 13 1287 15 1459 E O E 4700 e E e o z 4200 7 2 1215 3700 e o39 01 1384 3200 1 1 1 1 1 1 1 1 0 500 1000 1500 2000 2500 3000 3500 4000 Easting m It would seem reasonable to estimate 1 by a weighted average 2 lad with weights 2 given by some decreasing function of the distance d from u to data point 0 All interpolation algorithms inverse distance squared splines radial basis functions triangulation etc estimate the value at a given location as a weighted sum of data values at surrounding locations Almost all assign weights according to functions that give a decreasing weight with increasing separation distance Kriging assigns weights according to a moderately datadriven weighting function rather than an arbitrary function but it is still just an interpolation algorithm and will give very similar results to others in many cases lsaaks and Srivastava 1989 In particular If the data locations are fairly dense and uniformly distributed throughout the study area you will get fairly good estimates regardless of interpolation algorithm If the data locations fall in a few clusters with large gaps in between you will get unreliable estimates regardless of interpolation algorithm Almost all interpolation algorithms will underestimate the highs and overestimate the lows this is inherent to averaging and if an interpolation algorithm didn t average we wouldn t consider it reasonable Some advantages of kriging Helps to compensate for the effects of data clustering assigning individual points within a cluster less weight than isolated data points or treating clusters more like single points Gives estimate of estimation error kriging variance along with estimate of the variable Z itself but error map is basically a scaled version of a map of distance to nearest data point so not that unique Availability of estimation error provides basis for stochastic simulation of possible realizations of Z 11 Kriging approach and terminology Goovaerts 1997 All kriging estimators are but variants of the basic linear regression estimator Z u de ned as 2u mu ma moan with uua location vectors for estimation point and one of the neighboring data points indexed by 05 nu number of data points in local neighborhood used for estimation of Z u mumua expected values means of Z 11 and Z no la 11 kriging weight assigned to datum 2ua for estimation location 11 same datum will receive different weight for different estimation location Z 11 is treated as a random field with a trend component mu and a residual component Ru Z u Kriging estimates residual at u as weighted sum of residuals at surrounding data points Kriging weights la are derived from covariance function or semivariogram which should characterize residual component Distinction between trend and residual somewhat arbitrary varies with scale Development here will follow that of Pierre Goovaerts 1997 Geoslalislz39cs for Natural Resources Evaluation Oxford University Press We Will continue working with our example porosity data including looking in detail at results in the sixdatapoint region shown earlier Porosity a o o isoum 9 O 7 39 o o o a e o o o e 00 c 3 o 1250 e 0 0 l5 0 E O 0 o w a g39 9 0 9 ea 0 15 g 7500 0 60 w o o O o 0 4 p SIXPoint 39 0 l Example 0 o 375w 0 o e 3 9 Z o I I o o o o 0 O 12 o 5000 IOODO 15000 20000 Easting m Basics of Kriging Again the basic form of the kriging estimator is nu Z 11 mu Ella Zua mua The goal is to determine weights la that minimize the variance of the estimator a u VarZu Zu under the unbiasedness constraint E Z u Z u 0 The random field RF Zu is decomposed into residual and trend components Z 11 Ru mu with the residual component treated as an RF with a stationary mean of 0 and a stationary covariance a function of lag h but not of position 11 E ROD 0 CovRuRu h ERu Ru h CR The residual covariance function is generally derived from the input semivariogram model CR h 2 CR 0 7h Sill yh Thus the semivariogram we feed to a kriging program should represent the residual component of the variable The three main kriging variants simple ordinary and kriging with a trend differ in their treatments of the trend component mu Simple Kriging For simple kriging we assume that the trend component is a constant and known mean mu m so that zKugtm zKuzuagt mi This estimate is automatically unbiased since EZua m 0 so that ElZK uJ m The estimation error ZK u Zu is a linear combination of random variables representing residuals at the data points ua and the estimation point u ZKu Zu 23K u m Zu m ampKuRua RuRKuRugt Using rules for the variance of a linear combination of random variables the error variance is then given by 0 u Var RK Var RSK u 2 Cov RK uRSK quotl0 quot 0 quot 0 Z Z AKu KuCR ua u CR 0 2 Z liKuCR ua u 1 31 061 To minimize the error variance we take the derivative of the above expression with respect to each of the kriging weights and set each derivative to zero This leads to the following system of equations mu E112KuCRua u CRua u 0lnu Because of the constant mean the covariance function for Zu is the same as that for the residual component C h C R h so that we can write the simple kriging system directly in terms of C h n ls KuCua u Cua u alnu 31 This can be written in matriX form as K XSK u k where K SK is the matriX of covariances between data points with elements KL 2 C u u j k is the vector of covariances between the data points and the estimation point with elements given by k1 C u u and ASK u is the vector of simple kriging weights for the surrounding data points If the covariance model is licit meaning the underlying semivariogram model is licit and no two data points are colocated then the data covariance matriX is positive definite and we can solve for the kriging weights using ASK K lk Once we have the kriging weights we can compute both the kriging estimate and the kriging variance which is given by Gina clto mm clto 39zquotfzKucua u after substituting the kriging weights into the error variance expression above So what oloes all ths rnath do397 It nds a set ofwelghts for estarnatrng the vanable value at the loeataon from values at a set ofnelghbonng olata pornts The werght on eaeh olata pornt generally oleereases wrth lncreaslng dlstance to that pornt m aeeorolanee wrth the decreaslng datartoresumauon eovananees speclfled m the nghtrhand veetor k However the set of werghts ls also olesrgneolto aeeount forredundancy arnong the data polnts representeol m the data polntrtordata pornt covarlances m the mamx K Multrplyrng kby K on the lelt wlll downwerght pornts falllngln elusters relatave to rsolateol pornts at the sarne dlstance We wlll apply slmple knglng to ourporosrty olata uslng the spheneal semlvanogram thatwe tbefore wth zero nugget a slll ofU 78 and arange of4141m Semivanance 2mm Anon auuu sum mum Lag meters Since we are using a spherical semivariogram the covariance function is given by ChC0 yh 0781 15h4141 05h41413 for separation distances h up to 4141 m and 0 beyond that range The plot below shows the elements of the righthand vector k 038 056032 049 046 037 T obtained from plugging the datatoestimationpoint distances into this covariance function 5700 7 9 k4 C1044m 049 5200 k5 C1170m 046 O E k3 C1746m 032 E 4700 7 k6 C1513m 037 E o O z 4200 7 k2 C806m 056 3700 7 0 k1 C14s7m 030 3200 w w w w w w w w 0 500 1000 1500 2000 2500 3000 3500 4000 Easting m The matrix of distances between the pairs of data points rounded to the nearest meter is given by Point 1 Point 2 Point 3 Point 4 Point 5 Point 6 Pointl 0 1897 3130 2441 1400 1265 Point2 1897 0 1281 1456 1970 2280 Point3 3130 1281 0 1523 2800 3206 Point4 2441 1456 1523 0 1523 1970 Point 5 1400 1970 2800 1523 0 447 Point 6 1265 2280 3206 1970 447 0 10 This translates into a data covariance matrix of 078 028 028 078 006 043 K 017 039 040 027 043 020 rounded to two decimal places Note in particular the relatively high correlation between points 5 and 6 separated by 447 m The resulting vector of kriging weights is 11 x12 1 1 1 2 6 L 4 Lquot Notice that data point 6 is assigned a very small weight relative to data point 1 even though they are both about the same distance from the estimation point and have about the same datapointto estimationpoint covariance k1 038 kg 037 This is because data point 6 is effectively screened by the nearby data point 5 Data points 5 and 6 are fairly strongly correlated with each other and 5 has a stronger correlation with the estimation point so data point 6 is effectively ignored Note that the covariances and thus K lk 006 043 078 037 011 006 017 039 037 078 037 027 040 027 011 037 078 065 01475 04564 00205 02709 02534 00266 043 020 006 027 065 078 the kriging weights are determined entirely by the data configuration and the covariance model not the actual data values 11 The porosities at points 5 and 6 could in fact be very different and this would have no in uence on the kriging weights The mean porosity value for the 85 wells is 1470 and the porosity values at the siX example wells are 1384 1215 1287 1268 1441 and 1459 The estimated residual from the mean at u is given by the dot product of the kriging weights and the vector of residuals at the data points Ru 39R 086 255 183 015 046 002 027 025 003 201 187 028 010 Adding the mean back into this estimated residual gives an estimated porosity of Zu Ru m 187 1470 1283 Similarly plugging the kriging weights and the vector k into the eXpression for the estimation variance gives a variance of 0238 squared Given these two pieces of information we can represent the porosity at u 2000 m 4700 m as a normal distribution with a mean of 1283 and a standard deviation of 049 Note that like the kriging weights the variance estimate depends entirely on the data configuration and the covariance function not on the data values themselves The estimated kriging variance would be the same regardless of whether the actual porosity values in the neighborhood were very similar or highly variable The in uence of the data values through the fitting of the semivariogram model is quite indirect 12 Here are the simple kriging estimates and standard deviation on a 100X80 g1id with lOOmeter spacing using the spheiical semivaiiogram model and estimating each g1id value from the 16 nearest neighbor data points well locations Estimated Porosity Using Simple Kriging 15000 17 16 E 10000 V 15 07 5 o 14 Z 5000 13 12 5000 10000 15000 Easting m Simple Kriging Standard Deviation 15000 0 8 E 10000 0 6 m V E g 0 4 D Z 5000 0 2 0 0 5000 10000 15000 Easting m 13 Some characteristics to note Smoothness Kriged surface will basically be as smooth as possible given the constraints of the data in many cases probably smoother than the true surface Bullseyes Because kriging averages between data points local extremes will usually be at well locations bullseyes are inevitable This is true of almost all interpolation algorithms Extreme form of this is artifact discontinuities at well locations when semivariogram model includes significant nugget Error map re ects data locations not data values Map of kriging standard deviation depends entirely on data configuration and covariance function essentially a map of distance to nearest well location scaled by covariance function 14 Ordinary Kriging For ordinary kriging rather than assuming that the mean is constant over the entire domain we assume that it is constant in the local neighborhood of each estimation point that is that mua mu for each nearby data value Z um that we are using to estimate Z In this case the kriging estimator can be written forwmu woku mmn uuzwa humlmw and we filter the unknown local mean by requiring that the kriging weights sum to 1 leading to an ordinary kriging estimator of quotIO nu ZOKU 213KUZUa with 213 1 061 06l In order to minimize the error variance subject to the unitsum constraint on the weights we actually set up the system minimize the error variance plus an additional term involving a Lagrange parameter uOK u nu L mwmump ztm 06l so that minimization with respect to the Lagrange parameter forces the constraint to be obeyed 1 6L W 2 311 1 a u 15 In this case the system of equations for the kriging weights turns out to be nu 13ng u CRua u uOKuCRua u 06 lnu nu 3ng u l where C R h is once again the covariance function for the residual component of the variable In simple kriging we could equate C R h and C h the covariance function for the variable itself due to the assumption of a constant mean That equality does not hold here but in practice the substitution is often made anyway on the assumption that the semivariogram from which C h is derived effectively filters the in uence of largescale trends in the mean In fact the unitsum constraint on the weights allows the ordinary kriging system to be stated directly in terms of the semivariogram in place of the C R h values above In a sense ordinary kriging is the interpolation approach that follows naturally from a semivariogram analysis since both tools tend to filter trends in the mean Once the kriging weights and Lagrange parameter are obtained the ordinary kriging error variance is given by can C0 39 amp3Kucua u u0K u 16 In matrix terms the ordinary kriging system is an augmented version of the simple kriging system For our siXpoint example it would be 078 028 006 017 040 043 100 11 038 028 078 043 039 027 020 100 22 056 006 043 078 037 011 006 100 23 032 017 039 037 078 037 027 100 2L4 049 040 027 011 037 078 065 100 25 046 043 020 006 027 065 078 100 26 037 100 100 100 100 100 100 000 11 100 to which the solution is 21 01274 12 04515 13 00463 14 K 1k 02595 15 02528 16 00448 11 00288 The ordinary kriging estimate at u 2000 m 4700 m turns out to be 1293 with a standard deviation of 0490 only slightly different from the simple kriging values of 1283 and 0488 17 Again using 16 nearest neighbors for each estimation point the ordinary kriging porosity estimate and standard deViation look very much like those from simple kriging Estimated Porosity Using Ordinary Kriging 15000 17 16 E 10000 V 15 01 5 0 14 Z 5000 13 12 5000 10000 15000 Easting m Ordinary Kriging Standard Deviation 1 0 15000 0 8 E 10000 0 6 01 E g 0 4 D Z 5000 0 2 0 0 5000 10000 15000 Easting m 18 Kriging with a Trend Kriging with a trend the method formerly known as universal kriging is much like ordinary kriging except that instead of fitting just a local mean in the neighborhood of the estimation point we fit a linear or higherorder trend in the xy coordinates of the data points A local linear aka firstorder trend model would be given by 17111 mxy Clo a1xa2y Including such a model in the kriging system involves the same kind of extension as we used for ordinary kriging with the addition of two more Lagrange parameters and two extra columns and rows in the K matrix whose nonzero elements are the x and y coordinates of the data points Higherorder trends quadratic cubic could be handled in the same way but in practice it is rare to use anything higher than a firstorder trend Ordinary kriging is kriging with a zerothorder trend model If the variable of interest does exhibit a significant trend a typical approach would be to attempt to estimate a de trended semivariogram using one of the methods described in the semivariogram lecture and then feed this into kriging with a first order trend However Goovaerts 1997 warns against this approach and instead recommends performing simple kriging of the residuals from a global trend with a constant mean of 0 and then adding the kriged residuals back into the global trend 19 A few of the many topics I have skipped or glossed over Cokriging Kriging using information from one or more correlated secondary variables or multivariate kriging in general Requires development of models for crosscovariance covariance between two different variables as a function of lag Indicator Kriging Kriging of indicator variables which represent membership in a set of categories Used with naturally categorical variables like facies or continuous variables that have been thresholded into categories eg quartiles deciles Especially useful for preserving connectedness of high and low perineability regions Direct application of kriging to perm will almost always wash out extreme values Artifact discontinuities Kriging using a semivariogram model with a signi cant nugget will create discontinuities with the interpolated surface leaping up or down to grab any data point that happens to correspond with a grid node estimation point Solutions factorial kriging filtering out the nugget component or some other kind of smoothing as opposed to exact interpolation such as smoothing splines Or if you really want to do exact interpolation use a semivariogram model without a nugget Search algorithm The algorithm for selecting neighboring data points can have at least as much in uence on the estimate as the interpolation algorithm itself I have used a simple nearest neighbor search A couple of alternatives include quadrant and octant searches which look for so many data points within a certain distance in each quadrant or octant surrounding the data pomt 20
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