lab for remote sensing hydrology and spatial modeling dept of bioenvironmental systems engineering...

Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems Engineering

National Taiwan University 1/90

GEOSTATISTICS

VARIOGRAM MODELING

Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering

National Taiwan University




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.




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.,

xxZE ,)( )()( xZhxZ

. ),(2)()()()( 2 xhxZhxZExZhxZVar

is called the semi-variogram, or simply the variogram.

)(h




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:

)()0()( hCCh

Properties of the covariance function and variogram




)()0(

)(221

)()(2)()(21

)()(2)()(21

)()(21)()(2

1)(

22

22

22

2

hCC

hC

xZhxZExZEhxZE

xZhxZxZhxZE

xZhxZExZhxZVarh




Other properties of the covariance function of a stationary random field include

0)0( C

)()( hChC

)0()( ChC 0)(

hhC




Relation between the covariance and variogram of a stationary random field




Similarly, the following properties hold for variograms:

0)0( 0)()( hh

)0()( Chh




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

nxxx ,,, 21 n ,,, 21

0)(1 1

n

i

n

jjiji xxC




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.




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.




Under intrinsic stationarity

),()(2

1)(21

)()(2)()(21

)()(21)()(2

1,

22

2

jiji

jiji

jijiji

xxCxZVarxZVar

xZxZExZExZE

xZxZExZxZVarxx

),()(21)(2

1, jijiji xxxZVarxZVarxxC




Under the condition that , we have

n

ii

1

0

Rnixx i

n

j

n

ijiji

),,2,1( allfor ,0,1 1

0,21

21

,21)2

1),(

1 11 11 1

1 11 1

n

j

n

ijiji

n

i

n

jjji

n

j

n

iiij

n

i

n

jjijiji

n

i

n

jjiji

xxxZVarxZVar

xxxZVarxZVarxxC




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 .

)(h

)(hC

nixZ i ,,2,1,

n

ii

1

0




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.

n

ii

1

0

xZE




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 .

n

i

n

jjiji xxCWVar

1 1

)()(

n

i

n

jjiji xxWVar

1 1

)()(

)( ji xxC )( ji xx




Variogram characteristicsContinuity 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.

)(h

0)0(




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.

0)( 0 Ch




The nugget effect




Variogram characteristicsBehavior 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.

)(h

2h

0)(

lim 2 h

hh




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.

2h




Variogram characteristicsThe 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.




Importance of influence range in spatial estimation.

Estimation of rainfall at an ungauged site.




38 raingauge locations in northern Taiwan




The inverse distance weighted (IDW) method




Kriging method




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,

nixz i ,,2,1),(

nixi ,,2,1, )(),( ji xzxz




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 .

2)()(21

ji xzxz

2)()(21

jiij xzxz




Calculation of the experimental variogram




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




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”.

0 ,)()(1

i

n

iii ahah




One should also notice that, in using the nested-structure variogram model, we are not limited to combining models of the same shape.




Example of a nested-structure variogram

Linear

Exponential

Nested




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.




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).




Variogram Regularization




Illustration of regularization effect by increase of support




Regularization of variogram models

Example• Contours of the 100-year return period daily rainfall depth based on observed data

and high-resolution downscaled rainfalls.

07/28 - 08/01, 2014 48

Based on site observations Based on high-resolution downscaled rainfalls.

Contours exhibit higher degree of spatial continuity.

(A)

(B)

Department of Bioenvironmental Systems Engineering, National Taiwan University




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.




Types of anisotropy Affine anisotropic (or geometric

anisotropic) Zonal anisotropic (or stratified

anisotropic)




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.




Linear model




Spherical model




Exponential model




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.




Directional and distance tolerances for directional variograms




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.




Rose diagram of an anisotropic random field




Zonal anisotropy Anisotropic variation is due to layered

structure. Sill value changes with direction while the influence range remains the same.




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

)(1 h

1axis variationmaximum theof range

axis variationminimum theof rangek




(1) The anisotropic axes are parallel to the coordinate axes

Assume that X-axis is parallel to the minimum variation axis (direction 1).




(2) The anisotropic axes are not parallel to the coordinate axes




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(x0) centered on the point x0.

)()( xmxZE

)()()(21 xxxZxZVar

)()()( 0xmxmxm




Inside such a neighborhood V(x0), 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(x0), i.e., on the

point x0.

xxh x




The construction of a model of quasi-stationarity thus amounts to building a model of the structural function which depends on the argument x0.

);( 0xh




Proportional Effect




Dimensionless variogram (Relative variograms)

Local relative variograms (LRV) General relative variograms (GRV) Pairwise relative variograms (PRV)




Local relative variograms (LRV)




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 .2

im




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.

)(hm)(h

hh

jiji

ij

vvhN

hm),()(2

1)(

2)(

)()(

hm

hhGR




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.

hhji ji

jiPR

ij

vv

vv

hNh

),(2

2

2

)(2

1)(

Reference: An Introduction to Applied Geostatistics (Isaaks and Srivastava)




38 raingauge locations in northern Taiwan




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.




Examples of design storm variograms




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.




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.




Sills of design storm variograms




Ranges of design storm variograms

These relationships are well consistent with the random characteristics of design storms.




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.




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.




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.




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.




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.

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