Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 1/90 GEOSTATISTICS VARIOGRAM MODELING Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 2/90 In the example of Brownian motion, we have seen that it is possible that a random process or random field has an infinite capacity of dispersion, i.e., it has neither an a priori variance nor a covariance. For such random fields the covariance function is not practically applicable, and the variogram based on the assumption of intrinsic stationarity is preferred.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 3/90 Intrinsic stationarity A random function Z(x) is said to be intrinsically stationary if the expectation exists and does not depend on the support point x, i.e., for all vectors h the increment has a finite variance which does not depend on x, i.e., is called the semi-variogram, or simply the variogram.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 4/90 Let Z(x) be a stationary random field with expectation , variance  2, covariance C(h) and variogram  (h). The covariance function and variogram are related by the following equation: Properties of the covariance function and variogram

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 5/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 6/90 Other properties of the covariance function of a stationary random field include

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 7/90 Relation between the covariance and variogram of a stationary random field

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 8/90 Similarly, the following properties hold for variograms:

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 9/90 Semi-positive definite condition for covariance functions The covariance function of a stationary random field must be positive definite in the sense that for any set of and all real, the following inequality holds

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 10/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 11/90 The semi-positive definite condition of covariance functions indicates that not any function can be considered as the covariance function of a stationary random field. It is imperative that it be positive semi-definite.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 12/90 Conditional negative definite for variograms Although the variogram and covariance function of stationary random fields are interchangeable, there are situations for which covariance functions are support- point dependent while variograms are not, as in the case of 1-D Brownian motion. The variogram is defined under the assumption of intrinsic stationarity which does not require stationarity for variance.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 13/90 Under intrinsic stationarity

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 14/90 Under the condition that, we have

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 15/90 Authorized linear combinations (ALC) When only the intrinsic stationarity is assumed, the variogram, which is independent of the support-points, exists but the covariance function may not exist (as illustrated by the Brownian motion). The only linear combinations of, that have a finite variance are those satisfying the weight condition.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 16/90 If only the authorized linear combinations (ALC), i.e. linear combinations satisfying are considered, there is no need to calculate the a priori variance C(0), nor to know the expectation or the covariance C(h). Knowing the variogram is enough for kriging estimation if the intrinsic stationarity holds.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 17/90 Note that initially we have If only the authorized linear combinations are considered for W, the variance of W becomes This indicates that for ALC, we can pass from a formula written in terms of covariance to the corresponding formula written in terms of variogram by replacing with.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 18/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 19/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 20/90 Variogram characteristics Continuity at the origin The continuity in space of a random field Z(x) is reflected by the rate of growth of for small values of h. Theoretically, however, an experimental variogram (variogram estimated from observed data) may have a value of significantly different from 0 near the origin.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 21/90 Such discontinuity of the variogram near the origin is called a “nugget effect” and is possibly due to measurement errors of Z(x), or micro-scale variability of the random field under investigation. In particular, if for all h, it is called pure nugget effect.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 22/90 The nugget effect

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 23/90 Variogram characteristics Behavior at infinity Using the property that is a conditional positive definite function, it can be shown that the variogram necessarily increases more slowly at the infinity than does, i.e.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 24/90 An experimental variogram which increases at least as rapidly as for large distances h is incompatible with the intrinsic stationarity hypothesis. Such an increase in the variogram most often indicates the presence of a trend or drift, i.e., a non-stationary expectation.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 25/90 Variogram characteristics The influence range The influence range is the minimum distance between two independent random variables. It is also the zone of influence of a random variable.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 26/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 27/90 Importance of influence range in spatial estimation. Estimation of rainfall at an ungauged site.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 28/90 38 raingauge locations in northern Taiwan

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 29/90 The inverse distance weighted (IDW) method

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 30/90 Kriging method

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 31/90 Modeling the sample variogram Given a set of observations, the variogram can be estimated through the following procedures: define a certain number of distance classes between measurement points, calculate the number of pairs ( ) present in each distance class,

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 32/90 calculate the average distance in each distance class, calculate the average value of the experimental (or sample) variogram is then constructed using average distance and average value of.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 33/90 Calculation of the experimental variogram

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 34/90 Basic variogram models Sample variograms usually do not satisfy the conditional negative definite condition, therefore, theoretical variogram models are needed. Some of the commonly used admissible variogram models include Power model Spherical model Exponential model Gaussian model others

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 35/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 36/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 37/90 An important property of admissible variogram models is that any linear combination of admissible variogram models with positive coefficients is also an admissible model, i.e. is also an admissible variogram model and random field with such variogram model is said to have a “nested structure”.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 38/90 One should also notice that, in using the nested-structure variogram model, we are not limited to combining models of the same shape.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 39/90 Example of a nested-structure variogram Linear Exponential Nested

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 40/90 On issues of variogram fitting Parameters (sill and range) of a semi- variogram model are often estimated by the ordinary least squares fitting method. The method of ordinary least squares assumes that different data pairs involved in semi- variogram fitting are uncorrelated. Except for realizations with very short range, such assumption is a clear violation of the realizations under investigation which are associated with certain spatial correlation structure.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 41/90 Thus, it inevitably introduces higher degree of uncertainty in the process of parameter estimation. Other semi-variogram fitting methods such as the weighted least squares (Cressie 1985; Gotway 1991; Pardo-Iguzquiza 1999) and the generalized least squares (Pardo-Iguzquiza and Dowd 2001) have also been proposed, and the uncertainty of semi- variogram parameter estimation using these methods has also been addressed (Pardo-Iguzquiza and Dowd 2001).

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 42/90 Variogram Regularization

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 43/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 44/90 Illustration of regularization effect by increase of support

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 45/90 Regularization of variogram models

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 46/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 47/90

Example Contours of the 100-year return period daily rainfall depth based on observed data and high-resolution downscaled rainfalls. 07/ /01, Based on site observationsBased on high-resolution downscaled rainfalls. Contours exhibit higher degree of spatial continuity. (A)(B) Department of Bioenvironmental Systems Engineering, National Taiwan University

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 49/90 Anisotropic variogram modeling Spatial variation of a random field may not be the same in different directions. For example annual total rainfall may change more rapidly in the E-W direction than in the N-S direction. When the spatial variation structure (variogram) of a random field varies with directions, it is said to be anisotropic.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 50/90 Types of anisotropy Affine anisotropic (or geometric anisotropic) Zonal anisotropic (or stratified anisotropic)

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 51/90 Affine anisotropy (Geometric anisotropy) The variation to be expected between two points h distance apart in one direction (direction 2) is equivalent to the variation expected between two points k times h distance apart in another direction (direction 1). For geometric anisotropy, the sill remains the same in all directions while the influence range varies with directions.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 52/90 Linear model

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 53/90 Spherical model

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 54/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 55/90 Exponential model

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 56/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 57/90 Directional variograms For random fields with geometric anisotropy, spatial variation structures or the variograms vary with directions and are termed directional variograms. Directional variograms can be estimated using data pairs fall in the direction under consideration. It is also a common practice to impose directional and distance (or lag) tolerances in determining directional variograms.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 58/90 Directional and distance tolerances for directional variograms

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 59/90 Finding anisotropic axes by constructing the rose diagram Anisotropic axes are referred to as the axes of minimum and maximum variation axes. They can be determined by constructing an ellipse of directional influence ranges, commonly known as the rose diagram.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 60/90 Rose diagram of an anisotropic random field

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 61/90 Zonal anisotropy Anisotropic variation is due to layered structure. Sill value changes with direction while the influence range remains the same.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 62/90 Finding the variogram for any direction under geometric anisotropy Suppose that the variogram along the minimum-variation axis (direction 1) is known, say, and we are interested in knowing the variation between two points which are h distance apart and may not fall along direction 1. Assume the anisotropic ratio

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 63/90 (1) The anisotropic axes are parallel to the coordinate axes Assume that X-axis is parallel to the minimum variation axis (direction 1).

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 64/90 (2) The anisotropic axes are not parallel to the coordinate axes

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 65/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 66/90 Quasi-stationarity Without the hypothesis of stationarity, let Z(x) be a random field and Under the hypothesis of quasi-stationarity, we have when two points x and are inside the neighborhood V(x 0 ) centered on the point x 0.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 67/90 Inside such a neighborhood V(x 0 ), the structural function (the covariance or variogram function) depends only on the vector of separating distance and not on the two locations x and ; however, this structural function depends on the particular neighborhood V(x 0 ), i.e., on the point x 0.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 68/90 The construction of a model of quasi- stationarity thus amounts to building a model of the structural function which depends on the argument x 0.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 69/90 Proportional Effect

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 70/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 71/90 Dimensionless variogram (Relative variograms) Local relative variograms (LRV) General relative variograms (GRV) Pairwise relative variograms (PRV)

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 72/90 Local relative variograms (LRV)

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 73/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 74/90 The commonly observed linear relationship between the local mean and the local standard deviation leads to the common assumption that the local variogram is proportional to the square of the local mean. If the relationship between the local mean and the local standard deviation is something other than linear, one should consider scaling the local variograms by some function other than.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 75/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 76/90 General relative variograms (GRV ) is the mean of all data values that are used to calculate and can be expressed as Note that no separate regions are required for calculation of general relative variograms.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 77/90 Pairwise relative variograms (PRV) The adjustment is done separately for each pair of sample values, using the average of the two values as the local mean. Reference: An Introduction to Applied Geostatistics (Isaaks and Srivastava)

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 78/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 79/90 38 raingauge locations in northern Taiwan

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 80/90 Design storm depths of the 38 rain-gauge sites were considered as measurements of the 48 design storm random fields, and were used to interpolate design storm depths at ungauged locations. Design storm depths at the 38 rain-gauge sites were used to calculate the experimental semi- variograms with respect to specific design storms.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 81/90 Examples of design storm variograms

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 82/90 Spatial variation characteristics of design storms Parameters of the semi-variogram represent certain spatial variation characteristics of the random field. We examine values of the variogram parameters and their implications for the spatial variation characteristics of design storms.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 83/90 Remarks about the random characteristics of design storms The design storms, by definition, characterize the characteristics of extreme events. The degree of extremity increases with the increase of return period and decrease of design storm duration. The degree of spatial independence increases (or the spatial correlation decreases), as the extremity of the storm event increases.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 84/90 Sills of design storm variograms

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 85/90 Sills of design storm variograms

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 86/90 Ranges of design storm variograms These relationships are well consistent with the random characteristics of design storms.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 87/90 Climatological variogram The sill ω represents variance of the random field under investigation, i.e. the total rainfall depth of the design storm in this study. Storm events of various seasons and intensities have different spatial variation structures; therefore, it would be unrealistic to adopt a unique variogram for all storm events irrespective of the season, meteorological conditions and rainfall intensity.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 88/90 Bastin et al. (1984) adopted a climatological variogram model of the form where m is an index for storm events, h is the Euclidian distance and  (m) is the scaling factor. With this structure, all the time non-stationarity is concentrated in the scale factor  (m), whereas the component is time invariant and is called the scaled climatological variogram.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 89/90 Lebel et al. (1987) pointed out that in a region of relatively regular weather patterns, the scaling factor  (m) mainly reflects the seasonal variation of the spatial structure of the rainfall field. In Taiwan annual maximum rainfall depths are mostly produced by two dominant storm types: mesoscale thunderstorms and synoptic scale tropical cyclones, during the period between May and November. Therefore, if the climate pattern persists,  (m) accounts for the difference in spatial structure of storm types.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 90/90 In regions where the climatic variability is stronger, the factor  (m) mainly accounts for the scale effect as a result of the variation in time of the mean rainfall intensity. The scale factor is equivalent to the sill ω, or the variance of the rainfall field; therefore, design storms with higher total rainfall depths will have higher sill values, as depicted in previous figures.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 91/90 For design storms with duration tr smaller than 6 h, the increase in the recurrence interval or decrease in the storm duration corresponds to decreases in the influence range and spatial correlation of design storm depth. For example, rainfall depths of the 1 h, 200-year design storm are highly independent in space, and the semi-variogram shows a near pure nugget effect.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 92/90 More on the proportional effect

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 93/90

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 94/90 Each calculated semi-variogram is divided by the square of the arithmetic mean of the samples which were used in its calculation.

Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 95/90 Since the levels now appear to have comparable structures we can combine them to form one overall experimental semivariogram for the whole section, and to fit a model for use in the later estimation procedures.