variogram modeling

8/3/2019 Variogram Modeling

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Variogram modeling


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Introduction:

Characteristics of semivariogram:


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Sill: The semi variance value at which the

variogram levels off. Range: the maximum distance beyond which

there is no correlation between pair of

samples.


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Nugget: The nugget represents variability at

distances smaller than the typical samplespacing, including measurement error.


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Need of variogram model:

For the sake of kriging we need to replace

the empirical semivariogram with anacceptable semivariogram model.

The reason is that semivariogram models

used in the kriging process need to obey

certain numerical properties in order for thekriging equations to be solvable.


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It should be noted that the variogram models

introduced here are those that we consider tobe the basic models.

They are simple, isotropic models,

independent of direction.

The basic variogram models can be

conveniently divided into two types; those

that reach a plateau and those that do not.


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Variogram models of the first type are often referred

to as transition models. The plateau they reach iscalled the silland the distance at which they reach

this plateau is called the range.

Variogram models of the second type do not reach a

plateau, but continue increasing as the magnitude of

h increases.


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Using hto represent lag distance, a to

represent (practical) range, and ctorepresent sill, the most frequently used

models are:

Spherical Model: Perhaps the most

commonly used variogram model is thespherical model.


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It has a linear behavior at small separation

distances near the origin but flattens out atlarger distances, and reaches the sill at a.

In this type we have a large range. i-e the sill

is reached at high range.


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The Exponential Model: Another commonly

used transition model is the exponentialmodel.

This model reaches its sill asymptotically,

with the practical range a.


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Like the spherical model, the exponential

model is linear at very short distances nearthe origin, however it rises more steeply andthen flattens out more gradually.

In fitting this model to a sample variogram itshelpful to remember that the tangent at theorigin reaches the sill at about one fifth of therange.


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The Gaussian Model: The Gaussian model

is a transition model that is often used tomodel extremely continuous phenomena.

Like the exponential model, the Gaussianmodel reaches its sill asymptotically.


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The distinguishing feature of the Gaussian

model is its parabolic behavior near theorigin.

The Linear Model: The linear model is not a

transition model since it does not reach a sill,

but increases linearly with h.

All the four models are given in the following

figure.


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Fitting model to variogram:

The actual process of fitting a model to an

empirical semivariogram is much more of anart than a science, with different authorities

suggesting different methods and protocols.


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Since empirical semivariograms are often

quite noisy, quite a bit of subjective judgmentgoes into selecting a good model.


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Example:

Here is the semivariogram for porosity data,

with three fitted models. In each case the sillvalue was fixed at the overall variance of

0.78 and the range was estimated using

weighted nonlinear regression (weighting by

number of data pairs for each lag):


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The fitted ranges for the three models are

4141 m for the spherical, 5823 m for theexponential, and 2884 m for the Gaussian.

The Gaussian model gives the best fit, but

the spherical is a close second.


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variogram modeling

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