Dr. SaMeH S. Ahmed

Civil and Environmental Engineering Department


يعد الاحصاء الجيولوجي واستخدام طرق الكريجينج من احب التخصصات لي وتتركز معظم ابحاثي في هذا المجال

Kriging is function of the geographic location) at an unobserved location from observations of its value at nearby locations.

The theory behind interpolation and extrapolation by kriging was developed by the French mathematician Georges Matheron based on ta group of geostatistical techniques to interpolate the value of a random field (e.g., the elevation, z, of the landscape as ahe Master's thesis of Daniel Gerhardus Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex in South Africa. The English verb is to krige and the most common noun iskriging; both are often pronounced with a hard "g", following

the pronunciation of the name "Krige".

Simple krging

Simple kriging is mathematically the simplest, but the least general. It assumes the expectation of the random field to be known, and relies on a covariance function. However, in most applications neither the expectation nor the covariance are known beforehand.

Simple kriging assumptions

The practical assumptions for the application of simple kriging are:

Simple kriging equation

The kriging weights of simple kriging have no unbiasedness condition and are given by the simple kriging equation system:

egin{pmatrix}w_1  vdots  w_n end{pmatrix}=
egin{pmatrix}c(x_1,x_1) & cdots & c(x_1,x_n) 
vdots & ddots & vdots  
c(x_n,x_1) & cdots & c(x_n,x_n) 
egin{pmatrix}c(x_1,x_0)  vdots  c(x_n,x_0) end{pmatrix}

This is analogous to a linear regression of Z(x_0) on the other z_1 , ldots, z_n.

Simple kriging interpolation

The interpolation by simple kriging is given by:

hat{Z}(x_0)=egin{pmatrix}z_1  vdots  z_n  end{pmatrix}'
egin{pmatrix}c(x_1,x_1) & cdots & c(x_1,x_n)  
vdots & ddots & vdots   
c(x_n,x_1) & cdots & c(x_n,x_n)   
egin{pmatrix}c(x_1,x_0)  vdots  c(x_n,x_0)end{pmatrix}

Simple kriging error

The kriging error is given by:

underbrace{egin{pmatrix}c(x_1,x_0)  vdots  c(x_n,x_0)end{pmatrix}'
c(x_1,x_1) & cdots & c(x_1,x_n)  
vdots & ddots & vdots  
c(x_n,x_1) & cdots & c(x_n,x_n) 
egin{pmatrix}c(x_1,x_0)  vdots  c(x_n,x_0) end{pmatrix}}_{mathrm{Var}(hat{Z}(x_0))}

which leads to the generalised least squares version of the Gauss-Markov theorem (Chiles & Delfiner 1999, p. 159):


Ordinary kriging

Ordinary kriging is the most commonly used type of kriging. It assumes a constant but unknown mean.

Typical ordinary kriging assumptions

The typical assumptions for the practical application of ordinary kriging are:

The mathematical condition for applicability of ordinary kriging are:

  • The mean E[Z(x)]=mu is unknown but constant
  • The variogram gamma(x,y)=E[(Z(x)-Z(y))^2] of Z(x) is known.

Ordinary kriging equation

The kriging weights of ordinary kriging fulfill the unbiasedness condition

sum_{i=1}^n lambda_i = 1

and are given by the ordinary kriging equation system:

egin{pmatrix}lambda_1  vdots  lambda_n  mu end{pmatrix}=
egin{pmatrix}gamma(x_1,x_1) & cdots & gamma(x_1,x_n) &1 
vdots & ddots & vdots  & vdots 
gamma(x_n,x_1) & cdots & gamma(x_n,x_n) & 1 
1 &cdots& 1 & 0 
egin{pmatrix}gamma(x_1,x^*)  vdots  gamma(x_n,x^*)  1end{pmatrix}

the additional parameter mu is a Lagrange multiplier used in the minimization of the kriging error sigma_k^2(x) to honor the unbiasedness condition.

Ordinary kriging interpolation

The interpolation by ordinary kriging is given by:

hat{Z}(x^*)=egin{pmatrix}lambda_1  vdots  lambda_n  end{pmatrix}'
egin{pmatrix}Z(x_1)  vdots  Z(x_n) end{pmatrix}

Ordinary kriging error

The kriging error is given by:

egin{pmatrix}lambda_1  vdots  lambda_n  mu end{pmatrix}'
egin{pmatrix}gamma(x_1,x^*)  vdots  gamma(x_n,x^*)  1end{pmatrix}


(Cressie 1993, Chiles&Delfiner 1999, Wackernagel 1995)

  • The kriging estimation is unbiased: E[hat{Z}(x_i)]=E[Z(x_i)]
  • The kriging estimation honors the actually observed value: hat{Z}(x_i)=Z(x_i) (assuming no measurement error is incurred)
  • The kriging estimation hat{Z}(x) is the best linear unbiased estimator of Z(x) if the assumptions hold. However (e.g. Cressie 1993):
    • As with any method: If the assumptions do not hold, kriging might be bad.
    • There might be better nonlinear and/or biased methods.
    • No properties are guaranteed, when the wrong variogram is used. However typically still a 'good' interpolation is achieved.
    • Best is not necessarily good: e.g. In case of no spatial dependence the kriging interpolation is only as good as the arithmetic mean.
  • Kriging provides sigma_k^2 as a measure of precision. However this measure relies on the correctness of the variogram.

Related terms and techniques


A series of related terms were also named after Krige, including kriged estimate, kriged estimator, kriging variance, kriging covariance, zero kriging variance, unity kriging covariance, kriging matrix, kriging method, kriging model, kriging plan, kriging process, kriging system, block kriging, co-kriging, disjunctive kriging, linear kriging, ordinary kriging, point kriging, random kriging, regular grid kriging, simple kriging and universal kriging.

Related methods

Kriging is mathematically closely related to regression analysis. Both theories derive a best linear unbiased estimator, based on assumptions on covariances, make use of Gauss-Markov theorem to prove independence of the estimate and error, and make use of very similar formulae. They are nevertheless useful in different frameworks: kriging is made forinterpolation of a single realisation of a random field, while regression models are based on multiple observations of a multivariate dataset.

In the statistical community the same technique is also known as Gaussian process regression, Kolmogorov Wiener prediction, or best linear unbiased prediction.

The kriging interpolation may also be seen as a spline in a reproducing kernel Hilbert space, with reproducing kernel given by the covariance function.[12] The difference with the classical kriging approach is provided by the interpretation: while the spline is motivated by a minimum norm interpolation based on a Hilbert space structure, kriging is motivated by an expected squared prediction error based on a stochastic model.

Kriging with polynomial trend surfaces is mathematically identical to generalized least squares polynomial curve fitting.

Kriging can also be understood as a form of Bayesian inference.[13] Kriging starts with a prior distribution over functions. This prior takes the form of a Gaussian process: Nsamples from a function will be normally distributed, where the covariance between any two samples is the covariance function (or kernel) of the Gaussian process evaluated at the spatial location of two points. A set of values is then observed, each value associated with a spatial location. Now, a new value can be predicted at any new spatial location, by combining the Gaussian prior with a Gaussian likelihood function for each of the observed values. The resulting posterior distribution is also Gaussian, with a mean and covariance that can be simply computed from the observed values, their variance, and the kernel matrix derived from the prior.

الملفات المرفقة

The Best Book

Prayer Times

Click here

Office hours

Sunday:      10-1

Monday:      12-1

Wednesday: 10-11

You are welcome to contact me by at any time

You  may also contact me via WhatsApp group

Ext. 2524

Time Table 2018-19S

Civil Eng. Students

DON'T miss!

Love your department and be proud of your field of study

As we approaching the second midterm exams..  Get ready and remember our offices are open to answer you . do not hesitate to .contact me if you have difficulties

For CE 370 students : Your second midterm exam will be on Wednesday 27 March 2019. In the surveying lab

Good Luck


My Dear Students

Please visit the  announcement box for the relevant course and make sure that you can download and upload easily, otherwise

.contact me  to sort it out

There will be an orientation session in the midell of this term to help you select your track in Civil and Environmental Engineering

Special Event

Special Issue

Electrical Power Resources: Coal versus Renewable Energy

Published Online on January 2016

Mining Engineering

Mining engineering is an engineering discipline that involves practice, theory, science, technology, and the application of extracting and processing minerals from a naturally occurring environment. Mining engineering also includes processing minerals for additional value.

Environmental Engineering

Environmental engineers are the technical professionals who identify and design solutions for    environmental problems. Environmental engineers provide safe drinking water, treat and properly dispose of wastes, maintain air quality, control water pollution, and remediate sites contaminated due to spills or improper disposal of hazardous substances. They monitor the quality of the air, water, and land. And, they develop new and improved means to protect the environment.


Actions speak louder than words


College of Eng. MU



Universities I've worked in

Assuit University (Home University), Egypt

Imperial College, London, UK

Faculty of Engineering, Al-Mergeb University, Libya

King Saud University, KSA

,Majmaah University

Majmaah University has a very nice web page that provide  all information regarding deanships, colleges, activities,  campuses and many others. Search your request in

Also visit

My Academic Pages


and give me




Ahly vs Zamalek, 30 March 2019

Africa Nation Cup, June 2019

Course 2018/19-2

  1. Surveying 1 CE 370
  2. Surveying II  CE 371
  3. Senior Design II  (round 6) CE 499

The Rules of Life

Rule #1


Member of the Editorial Board: " Journal of Water Resources and Ocean Sciences"  2013 till now


Participating in The Third International Conference on Water, Energy and Environment,(ICWEE) 2019 - American University of Sharjah, UAE 26-28 March 2019 with a Paper and Poster

New article

Coming soon

2nd Midterm Exams on 3/4/2019

Student Conference

Participation  in the 6Th Student Conference

With a paper and oral presentation

From the Senior Design Project CE499 -35

See inside, the paper, and presentation

CE 370 Course

Surveying I


Power point


Lecture notes


:Second Midterm Exam


Model Answer
Inside, please follow

Working  marks is available -Click on Surveying I

CE 371 Course

Surveying II



First midterm exam



See Inside

Working marks out of 60 is  available- Click on Surveying 2

CE 474 Course



Available 0-1-2-3-4 Power point


Quizzes 2 and 3 with model Answer

Available Chapter 1,2,3,4,5 Lecture notes

Report 1  Cameras Report


60 marks Exams

See Inside

Student Performance Records

CE 499 Course

Senior Design 2

Second Best paper from Senior Design Projects in 2015

Paper title:

Evaluation of Groundwater Quality Parameters using Multivariate Statistics- a case Study of Majmaah, KSA


Abdullah A. Alzeer

Husam K. Almubark

Maijd M. Almotairi

Engineering Practice

A must for Engineer

Useful Links


See what can you get from Google!! more than translation and locations






?What is your opinion

Email me in:  [email protected]

Saudi Vision 2030


Learn more about it

Riyadh Metro

A job opportunity 

Safe Fly

Surveying 2-19-3


Conversion-Hegira to AD



Saudi Digital Library


Visit me in RG


Contact me

Mobile: 00966598311652

[email protected]

Civil and Environmental Engineering Department
College of Engineering, Majmaah University
Majmaah, P.O. 66, 11952, KSA

Thank you

خالص شكري وتقديري

إحصائية الموقع

عدد الصفحات: 2066

البحوث والمحاضرات: 2046

الزيارات: 113077