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25 Nov 2009 (Updated )

fits different theoretical variograms to an experimental variogram

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variogramfit performs a least squares fit of various theoretical variograms to an experimental, isotropic variogram. The user can choose between various bounded (e.g. spherical) and unbounded (e.g. exponential) models. A nugget variance can be modelled as well, but higher nested models are not supported.

The function works best with the function fminsearchbnd available on the FEX. You should download it from the File Exchange (File ID: #8277). If you don't have fminsearchbnd, variogramfit uses fminsearch. The problem with fminsearch is, that it might return negative variances or ranges.

Supported bounded models:
'blinear' -- bounded linear
'circular' -- circular model
'spherical' -- spherical model, =default
'pentaspherical' -- pentaspherical model

Supported unbounded functions :
'exponential' -- exponential model
'gaussian' -- gaussian variogram
'whittle' -- Whittle's elementary correlation (involves a modified Bessel function of the second kind).
'stable' -- stable models sensu Wackernagel 1995). Same as gaussian, but with different exponents. Supply the exponent alpha (<2) as an additional pn,pv-pair 'stablealpha',alpha (default = 1.5).
'matern' -- Matern model. Requires an additional pn,pv pair. 'nu',nu (shape parameter > 0, default = 1). Note that for particular values of nu the matern model reduces to other authorized variogram models. (nu = 0.5 : exponential model, nu = 1 : Whittles model, nu -> inf : Gaussian model)

Wackernagel, H. (1995): Multivariate Geostatistics, Springer.
Webster, R., Oliver, M. (2001): Geostatistics for Environmental Scientists. Wiley & Sons.
Minsasny, B., McBratney, A. B. (2005): The Matérn function as general model for soil variograms. Geoderma, 3-4, 192-207.


Fminsearchbnd, Fminsearchcon, Experimental (Semi ) Variogram, and Parseargs: Simplifies Input Processing For Functions With Multiple Options inspired this file.

This file inspired Ordinary Kriging and Kriging And Inverse Distance Interpolation Using Gstat.

MATLAB release MATLAB 7.8 (R2009a)
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Comments and Ratings (37)
18 Jan 2017 Diego Rivera

somebody can help me with a example using funtion by exponential model, or gaussian maybe?.

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13 Dec 2016 kons geor

Just a realisation of some sort. To model a nugget effect use as initial range and sill, zero (a0=c0=0). This produces the respective model. However I have not tested if it is actually accurate.

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14 Jun 2016 Silpakorn D

09 Feb 2016 Ahmed Zidan


I have a data in a regular grid (dx=dy) and x and y have different length.

I am facing a problem to create a covariance matrix, because of the different length and regular grid.

I tried to meshgrid, but the matrix will be very large that matlab can't accept it.

I will be grateful for any suggesting.

Thank you all.

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25 Apr 2015 zigang guo

I tried to change the headline to this:
function [a,c,n,S] = variogramfit(h,gammaexp,a0,c0,tau2,eps,numobs,varargin)
And add several lines to the % variogram function definitions Section as:
case 'multifractal'
type = 'bounded';
func = @(b,h)b(2)*eps.^(tau2-1)*(1-.5*(((h./eps)+1).^(tau2+1)-2*(h./eps).^(tau2+1)+(h./eps-1).^(tau2+1)));

As you can see i am trying to use the same routine as you did for the other models. However, these two extra inputs are the issue atm.
Look forward for you reply!
Have a nice day!

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02 Feb 2015 Karrock

Hi Wolfgang!
I think i might have the same issue as
25 Sep 2012 C. Gutierrez
Even when running the "run example"-code in the function header. Did you identify the problem back then?

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22 Dec 2014 Wolfgang Schwanghart

@Mani Ahmadian: Hi Mani, thanks for your comment. I don't have a good method for dealing with outliers using variogramfit except using an optimization scheme other than "least squares". E.g., least absolute deviations may be less sensitive to outliers. In case you are supplying variogramfit with the binned, experimental variogram, I'd better care for outliers during binning. For example, you might want to adapt the function variogram ( ) to compute the trimmed mean or median when aggregating binned variogram values.

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22 Dec 2014 Mani Ahmadian

Dear Wolfgang

Thanks for your great code. It's very complete, but what's your idea about outliers remove and then perform best fit?

Have a nice day

Comment only
23 Oct 2014 steve

steve (view profile)

function [a,c,n,S] = variogramfit(h,gammaexp,a0,c0,numobs,varargin)

h=[2.238; 2; 3; 5.657; 2.238; 1.414;3.606; 3.606; 4.472; 4.123];
gammaexp=[12.5; 12.5; 0; 112.5; 0; 12.5; 50; 12.5; 50; 112.5];

% check input arguments

if ~exist('a0','var') || isempty(a0)
a0 = max(h)*2/3;
if ~exist('c0','var') || isempty(c0)
c0 = max(gammaexp);
if ~exist('numobs','var') || isempty(a0)
numobs = [];
% check input parameters
params.model = 'spherical';
params.nugget = [];
params.plotit = true;
params.stablealpha = 1.5;
params.solver ='fminsearch';
params.weightfun = 'cressie85'; = 1;

% check if fminsearchbnd is in the search path
switch lower(params.solver)
case 'fminsearchbnd'
if ~exist('fminsearchbnd.m','file')==2
params.solver = 'fminsearch';
'fminsearchbnd was not found. fminsearch is used instead')

% check if h and gammaexp are vectors and have the same size
if ~isvector(h) || ~isvector(gammaexp)
'h and gammaexp must be vectors');
% check size of supplied vectors
if numel(h) ~= numel(gammaexp)
'h and gammaexp must have same size');
% remove nans;
nans = isnan(h) | isnan(gammaexp);
if any(nans);
h(nans) = [];
gammaexp(nans) = [];
if ~isempty(numobs)
numobs(nans) = [];
% check weight inputs
if isempty(numobs);
params.weightfun = 'none';
b0 = [a0 c0 params.nugget];

switch lower(params.model)
case 'spherical'
type = 'bounded';
func = @(b,h)b(2)*((3*h./(2*b(1)))-1/2*(h./b(1)).^3);

case 'exponential'
type = 'unbounded';
func = @(b,h)b(2)*(1-exp(-h./b(1)));
error('unknown model')

% nugget variance
if isempty(params.nugget)
nugget = false;
funnugget = @(b) 0;
nugget = true;
funnugget = @(b) b(3);

% create weights (see Webster and Oliver)
switch lower(params.weightfun)
case 'cressie85'
weights = @(b,h) (numobs./variofun(b,h).^2)./sum(numobs./variofun(b,h).^2);
case 'mcbratney86'
weights = @(b,h) (numobs.*gammaexp./variofun(b,h).^3)/sum(numobs.*gammaexp./variofun(b,h).^3);
weights = @(b,h) 1;

% create objective function: weighted least square
objectfun = @(b)sum(((variofun(b,h)-gammaexp).^2).*weights(b,h));
% call solver
switch lower(params.solver)
case 'fminsearch'
% call fminsearch
[b,fval,exitflag,output] = fminsearch(objectfun,b0);
case 'fminsearchbnd'
% call fminsearchbnd
[b,fval,exitflag,output] = fminsearchbnd(objectfun,b0,lb,ub);
error('Variogramfit:Solver','unknown or unsupported solver')

% create vector with initial values
a = b(1); %range
c = b(2); %sill
b0 = [a0 c0 params.nugget];

if nugget;
n = b(3);%nugget
n = [];
% Create structure array with results
if nargout == 4;
S.model = lower(params.model); % model
S.func = func;
S.type = type;
switch S.model
case 'matern'; =;
case 'stable';
S.stablealpha = params.stablealpha;

S.range = a;
S.sill = c;
S.nugget = n;
S.h = h; % distance
S.gamma = gammaexp; % experimental values
S.gammahat = variofun(b,h); % estimated values
S.residuals = gammaexp-S.gammahat; % residuals
COVyhaty = cov(S.gammahat,gammaexp);
S.Rs = (COVyhaty(2).^2) ./...
(var(S.gammahat).*var(gammaexp)); % Rsquare
S.weights = weights(b,h); %weights
S.weightfun = params.weightfun;
S.exitflag = exitflag; % exitflag (see doc fminsearch)
S.algorithm = output.algorithm;
S.funcCount = output.funcCount;
S.iterations= output.iterations;
S.message = output.message;
% if you want to plot the results...
if params.plotit
switch lower(type)
case 'bounded'
hold on
fplot(@(h) funnugget(b) + func(b,h),[0 b(1)])
fplot(@(h) funnugget(b) + b(2),[b(1) max(h)])

case 'unbounded'
hold on
fplot(@(h) funnugget(b) + func(b,h),[0 max(h)])
axis([0 max(h) 0 max(gammaexp)])
xlabel('lag distance h')
hold off

% fitting functions for fminsearch/bnd
function gammahat = variofun(b,h)

switch type
% bounded model
case 'bounded'
I = h<=b(1);
gammahat = zeros(size(I));
gammahat(I) = funnugget(b) + func(b,h(I));
gammahat(~I) = funnugget(b) + b(2);
% unbounded model
case 'unbounded'
gammahat = funnugget(b) + func(b,h);
if flagzerodistances
gammahat(izero) = funnugget(b);


Comment only
23 Oct 2014 steve

steve (view profile)

look at that code and please advice me accordingly. i am trying to come up with a variogram for modeling but i just cannot get through. thanks

Comment only
23 Oct 2014 Wolfgang Schwanghart

@steve: thanks for your comment. variofun is a subfunction that you'll find below the main function.

Comment only
23 Oct 2014 steve

steve (view profile)

nice work there. what am wondering is what does variofun mean? since am trying to run that code am being told that it is an undefined variable. please help

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26 Feb 2014 Wolfgang Schwanghart

@Roque Santos: Hi, see here:

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23 Feb 2014 Roque Santos

Hello friend,
I used your code in a job (Masters), wanted to know how do I quote your code.
I await

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06 Jan 2014 Wolfgang Schwanghart

@Roque Santos: thanks for your comment. Please check the documentation for varargin here:

Comment only
06 Jan 2014 Roque Santos

Hello, ... great code! Congratulations on your initiative!
Could not identify the input variable "varargin".
Could you help me?

Comment only
29 Aug 2013 Jeff

Jeff (view profile)

Very helpful, thank you. I made a small change so that the nugget can be specified directly and not optimized:
line 299: funnugget = @(b) params.nugget;
line 336: [b,fval,exitflag,output] = fminsearchbnd(objectfun,b0(1:2),lb(1:2),ub(1:2),options);
line 346: n = params.nugget;

It's not pretty, but it works.

09 Aug 2013 Wolfgang Schwanghart

@Ludwig: Yes. The output is the function parameter, which means that b(1) (or a or S.range) is 1/3 of the range. Sorry for the confusion. Perhaps I should at least provide two different values (S.range and b(1)) in the structure array output.

Comment only
09 Aug 2013 Ludwig

Ludwig (view profile)

@Wolfgang: My issue was around the output. For unbounded models, won't S.range (or a) be 1/3 of the range since it is just assigned b(1)?

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08 Aug 2013 Wolfgang Schwanghart

@Ludwig: I know that the way range is used for unbounded models is confusing. When supplying the initial values you should enter the range where the model reaches about 95% of the sill variance. I decided to do so, since it can be easier visually determined from the experimental variogram and can better compared to bounded models. The parameter b(1) in the exponential variogram model
gamma = b(2)*(1-exp(-h./b(1)));
should be approx. 1/3 of this range.

Comment only
08 Aug 2013 Ludwig

Ludwig (view profile)

I am working with fitting exponential models to my data and was a bit confused about the outputted range. Should the range be 3x the fitted parameter that is currently being called the range? When I ask the function to include a plot I can see that the given range is not 95% of the sill.

It is possible I am having a misunderstanding.

Comment only
01 Aug 2013 Aditi

Aditi (view profile)

24 Apr 2013 meo

meo (view profile)

27 Dec 2012 Wolfgang Schwanghart

@ C Gutierrez: I think I haven't seen this error so far. Can you send me more information on how you have called variogramfit via the "contact author" interface?

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25 Sep 2012 C. Gutierrez

When I try to run this function on either the example data or my own data, I get the following set of errors:
Error using variogramfit>@(b)sum(((variofun(b,h)-gammaexp).^2).*weights(b,h))
Too many input arguments.

Error in fminsearchbnd>@(x,varargin)fun(xtransform(x),varargin{:}) (line 233)
intrafun = @(x, varargin) fun(xtransform(x), varargin{:});

Error in fminsearch (line 191)
fv(:,1) = funfcn(x,varargin{:});

Error in fminsearchbnd (line 264)
[xu,fval,exitflag,output] = fminsearch(intrafun,x0u,options,varargin);

Error in variogramfit (line 336)
[b,fval,exitflag,output] = fminsearchbnd(objectfun,b0,lb,ub,options);
Have you seen this before?

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13 Sep 2012 Wolfgang Schwanghart

@Vandita: I had requests for the inclusion of a hole-effect model earlier, but I currently don't have the time to implement it. However, it should be fairly easy to modify the function to include the hole-effect.

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04 Sep 2012 Vandita

I wish to fit a hole effect model to my data using this function. Is there anything existing or if I have to mmodify, what should I take care in addition to adding an equation for that in this code?

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01 May 2012 Hugoptimus

excelent code, it works succesfully and it is clear to review

18 Jan 2012 Wolfgang Schwanghart

Hi Aditi,

yes, this is possible, but you have to make some changes to the function.

Note that the general variogram parameters are in vector b.

% b(1) range
% b(2) sill
% b(3) nugget

So let's say you fit your horizontal data such that
b(1) = 1;
b(2) = 5;
b(3) = []; % no nugget

Then, before fitting your next model, edit to the following lines (302-313)

% generate upper and lower bounds when fminsearchbnd is used
switch lower(params.solver)
case {'fminsearchbnd'};
% lower bounds
% lb = zeros(size(b0));
lb = [0 5];
% upper bounds
if nugget;
ub = [inf max(gammaexp) max(gammaexp)]; %
% ub = [inf max(gammaexp)];
ub = [inf 5];

This should work. Make the obvious changes if you employ a nugget variance.

Is this a common problem and should it be implemented in variogramfit? If yes, let me know.

Best regards,

Comment only
18 Jan 2012 Aditi

Aditi (view profile)

Can I use this to simultaneously fit two variograms using some of the same parameter values? E.g., I want to fit both the horizontal and vertical data such that they have the same sample variance parameter.

Comment only
18 Jan 2012 Wolfgang Schwanghart

Hi Adam, right!

d.val = gammaexp
d.dist = h

I know it's a little pedestrian...

Regards, Wolfgang

Comment only
18 Jan 2012 Adam

Adam (view profile)

question not function,excuse me :)

Comment only
18 Jan 2012 Adam

Adam (view profile)

Please one stupid function,if I am doing right.
For this function I use output from function variogram d(val)=gammaexp and d(dist)=h?
Thank you a lot for those two functions!

10 Sep 2011 Wolfgang Schwanghart

@ James,

I think that this should be feasible without too much effort. I would, however, not want to implement it in the existing variogramfit function, but write a wrapper, that evaluates the function for various models and selects the one that has the highest coefficient of determination.

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01 Sep 2011 James Ramm

As you know I've used this in conjuction with kriging by GSTAT for large datasets.

I'm now using this to create 3D grids from scattered XYZC data ( - layered geophysical/geological data). This requires many interpolations in a loop. To this end, I wonder if it is possible to automate the model selection process so that the 'best' model may be chosen for each new sample variogram?

29 Sep 2010 Patrick A.

This submission is absolutely perfect in my opinion. Clear and clean code, well commented, nice code, efficient,... thanks a lot for this !

13 Jul 2010 Christian

Great program Wolfgang, simplicity of use, great code commenting, everything I could have wanted...(and I didnt have to do it myself!)

08 Feb 2010 1.1

The matern model was added as theoretical variogram.

23 Feb 2010 1.2

Seems I forgot to include the m file in the previous update. Here it is.

07 Oct 2010 1.4

removed a bug concerning weight functions. So far the weight function (either Cressie85 or McBratney86) was not invoked when called.

14 Oct 2010 1.5

forth output argument now contains the anonymous function of the variogram model. This is needed for kriging (to come).

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