enviroinfo 2011: application of kriging algorithms for...

Application of Kriging Algorithms for Solving some Water Nets Management Tasks

Jan Studzinski1

Abstract

The kriging methods and computer algorithms have been developed by Krige (1951), Sichel (1952) and Matheron (1962) in the middle of the 20th century as a tool to estimate the size and localization of mineral resources like gold deposits and oil pools and to estimate their contents changeability. In the paper these algorithms have been successfully used to solve some problems connected with the management of the communal water networks. In this case the result of the kriging approximation are contour line maps showing the value distributions of the parameters essential for the water net management, like the water flows and pressures, or the estimation of such the variables like the altitude coordinates of the network nodes and the network sensibility with regard to changes of the net work conditions.

Introduction

The kriging algorithms have been developed by Krige (1951), Sichel (1952) and Matheron (1962) as some methods of spatial statistics and in the beginning they were intended for applications in the mining for identification of raw material resources. Their essential advantages are simplicity, efficiency and reliability and also a big robustness on the measurement disturbances. The kriging algorithms are used to make the 2D and 3D approximation of the variables investigated and their main mathematical apparatus is a method of static optimization applied to model a discreet curve won from the measurements data. In the System Research Institute (IBS PAN) some successful studies have been made while using these algorithms to approximate some environmental data like the rain and snow falls, the soil composition and air temperature (Bogdan, Studzinski, 2006A,B,C,D, 2007). Parallel with these calculations an information system to complex management of communal water networks have been developed in IBS PAN. It consists of three standard programs, i.e. of a GIS (Geographical Information System) system, SCADA (System of Control And Diagnostics Analysis) system and a CIS (Customers Information System) system and of several mathematical models of the water net objects and processes. The main idea of this information system is to solve the tasks connected with the water net operation with the help of these different mathematical models (Studzinski, 2006, 2007). While developing the system and independently the kriging algorithms the idea arose to use this kind of approximation to help the solution of some problems of the water net management. The first approach was to calculate with the kriging algorithms the distribution of the water net flows and pressures (Bogdan, Studzinski, 2007). After the successful results have been obtained the application of the kriging approximation in the waterworks branch has been extended and some new findings in this field are reported in this paper.

Kriging approximation

The kriging approximation consists in estimation of unknown values of a variable in some selected points of an area (called calculation points) on the base of the known values of this variable in other points of this area (called measurement points. The formula to estimate the unknown value of variable z in calculation point xo is

1 Systems Research Institute, Polish Academy of Sciences. Newelska 6, 01-447 Warsaw, Poland e-mail: [email protected]

EnviroInfo 2011: Innovations in Sharing Environmental Observations and InformationCopyright 2011 Shaker Verlag Aachen, ISBN: 978-3-8440-0451-9

)()(1

i

N

iio xZλxZ ∑=

= (1)

where z(xi) are the variable values measured in N measurement points and λi are some weights coefficients which are to calculate. The estimation error can be written down as the relation

∑ −=−==

N

ioiiooo xZxZxZxZxR

1)()()()()( λ (2)

with )( oxZ the real value of the variable. The assumption is made that the middle value of the estimation error is equal to zero, from which the relation

0)()()]([)]([])()([)]([111

=−∑=−∑=∑ −====

ZEZExZExZExZxZExREN

iioi

N

ii

N

ioiio λλλ (3)

and subsequently the condition

11

=∑=

N

iiλ (4)

result. Another assumption made is that the variance of the estimation error is minimal from which the following formula

)](var[)]()(cov[2)](var[)](var[ 2ooooRo xZxZxZxZxR +−== σ (5)

arises. For the individual elements of (5) we can make the following transformations

jij

N

i

N

jijij

N

i

N

ji

N

iiio CxZxZxZxZ λλλλλ ∑ ∑=∑ ∑=

∑=

= == == 1 11 11)]()(cov[)(var)]([var (6)

io

N

iioi

N

ii

N

ioiioo CZZZZxZxZ ∑=∑=

∑==== 111

2)cov(2)(cov2)]()(cov[2 λλλ (7)

and

2)](var[ σ=oxZ (8)

for the variance of )( oxZ is not a function of iλ and it can be considered as constant. From (6, 7, 8) we get the relation

io

N

iijij

N

i

N

jiR CC ∑−∑ ∑+=

== = 11 1

22 2 λλλσσ (9)

that is to minimize against iλ and the derivations received have to be zeroed. In order to take condition (4) into account relation (9) can be converted into equation

)1(22111 1

22 −∑+∑−∑ ∑+==== =

N

iiio

N

iijij

N

i

N

jiR CC λµλλλσσ (10)

with µ the Lagrange parameter, from which the following equations system

101 1

11011

20222

)(CCCC

N

j

N

jjjjj

R =+∑ ∑⇒=+−=∂

∂= =

µλµλλ

σ

01 1

0

20222

)(i

N

j

N

jijjiijj

i

R CCCC =+∑ ∑⇒=+−=∂

∂= =

µλµλλ

σ

Copyright 2011 Shaker Verlag Aachen, ISBN: 978-3-8440-0451-9

………………………………………………………………………………………… (11)

01 1

0

20222

)(N

N

j

N

jNjjNNjj

N

R CCCC =+∑ ∑⇒=+−=∂

∂= =

µλµλλσ

11

=∑=

N

iiλ

can be derived. This system has the following matrix forms

011

1

1

1

111

L

L

MMOM

L

NNN

N

CC

CC

µλ

λ

N

M

1

=

10

10

NC

C

M (12)

or

cC =λ (13)

and from them the quested weights coefficients iλ are to calculate

cC 1−=λ (14)

The calculation of iλ from (14) is however not possible for the unknown value of variable Zo in calculation point xo is contained in vector c and we shall find this value using iλ . In order to solve the problem the following semivariogram function

])[(2

1 2jiij ZZE −=γ (15)

is formulated that can be transformed as follows

ijjiijijiij CmZZEmZEZZEZEZE −=−⋅−−=⋅−+= 222222 ])([])[()()(2

1)(

2

1 σγ (16)

with m = E(Zi) = E(zj). Equations (11) can be written down in the form

NiCC i

N

jijj ,,10

1K==+∑

=µλ

11

=∑=

N

iiλ (17)

and from (16) and (17) the following relations

20

1

2ij )(- σγµσγλ +−=+∑ +

=i

N

jj (18)

20

1 1

2ij )()( σγµλσγλ +−=+∑ ∑+−

= =i

N

j

N

jjj (19)

01

ij i

N

jj γµγλ =∑ −

= (20)

result. Equation (20) can be written down in the new matrix form


011

1

1

1

111

L

L

MMOM

L

NNN

N

γγ

γγ

µλ

λ

N

M

1

=

10

10

Nγ

γM

(21)

from which the quested weights coefficients iλ can be already calculated with the formula similar to (14).

The computer algorithm of the kriging approximation consists of 4 following steps:

Step 1: Calculation of the experimental and discreet semivariogram function on the base of the measurement data.

2

11 )(

21

)( ihn

ih

hzz

nhγ −∑=

=+

where zi, zi+h are the values of the variable investigated in the measurements points which are remoted one from the other up to the distance h and nh is the number of pairs of the points remoted each other up to h.

Step 2: Modeling the discreet semivariogram function with different analytical functions using the methods of static optimization (Fig. 1).

Step 3: Calculation of the weight coefficients using equation (21).

Step 4: Calculation of the investigated variable value in the calculation point using formula (1).

Figure 1. Examples of an experimental discreet semivariogramm and its mathematical model.

Using the above computer algorithm computer program KRIPOW has been developed in IBS PAN in which 8 analytic functions for semivariogram modeling have been implemented. These functions are:

exponential functions ))exp(1()( BhCh −−=γ and ))exp(1()( 22hBCh −−=γ ; square function

)221(22)( hBhCBh +=γ ; trigonometric function )/)sin(1()( BhBhCh −=γ ; spherical function

))5,05,1()( 22 hBCBhh −=γ ; power functions 4/1)()( BhCh =γ and 2/1)()( BhCh =γ ;

and linear function CBhγ(h) = . The identification of the coefficient values in these modeling

functions occurs with the Marquardt algorithm of static gradient optimization. In Figure 2 one can see what form can take the experimental descreet semivariogram constructed from a given set of measurement points in Step 1 of the algorithm. In Figure 3 the mathematical models are shown that have been obtained in Step 2 of the algorithm for the descreet semivariogram presented in Figure 2, with the use of the analytical functions mentioned above and of the Marquard optimization method. By modeling the experimental semivariogram 8 different models are calculated and only one of them


is chosen for farther calculation steps of the algorithm and the choice criterion is the model’s residuum value.

Figure 2. Exemplary screens of program KRIPOW with a cloud of measurement points (left) and the descreet semivariogram function calculated for them.

Figure 3. Examplary screens of program KRIPOW with 8 analytical models made for a descreet semivariogram function.

Water net management tasks solved with the kriging approximation

With program KRIPOW the usefulness of kriging algorithms for the approximation of several variables which are important for the water net management has been checked. The end results of the investigation are the contour line maps showing the value distributions of some variables or the values estimation of other variables and these results are of importance for the water net modeling, operation and planning. In the following three applications of the kriging approximation supporting the general comprehended water net management are described.


• Drawing the contour line maps of the water net flow and pressure distributions

By the operation of a water net it is of importance for the network operator to have the up-to-date information about the water flows in the pipes and water pressures in the nodes of the network. With the use of a hydraulic model fitted good to the network he can calculate these values exactly but the large number of them ordered mostly in the form of very big tables makes it quite impossible to recognize quickly in which part of the water net the values of these variables are incorrect. This incorrectness means that the pressures are too high or too low or the flows are too slow or too fast. Very low water pressures in the end nodes of the water net mean practically the lack of water by the end users of the network and too high pressures can cause failures in the water net. Too slow water velocities are mostly responsible for creation of algae in the network pipes what results in worsening the water quality. The most visible signs of it are bad smell and taste of the drinking water. In such the cases some improvement procedures have to be quick executed by the water net operator to avoid possible network failures or claims of the water consumers. Because of that there is useful to give the operator an information tool for a quick assessment of the quality of the water net work. If using this tool he could state fast and reliably the incorrectness of the network work then he could analyse the reasons of it more detailed studying the tables with the hydraulic calculation results. The idea was to use the kriging approximation to design the distributions of water pressures and of water flows in the water net in form of contour line maps on which the right or bad functioning network parts would be marked with different colors. Designing of such the maps needs a close cooperation between the kriging algorithm and the network model that defines the set of the measurement points used for constructing the discreet semivariogram. Beside the KRIPOW program also a hydraulic model of water nets called MOSUW has been developed in IBS PAN and these both programs cooperate each other via special data files. With MOSUW and KRIPOW one can simulate the work of the investigated water net and draw the contour line maps on which the results of this work are qualitatively illustrated.

Figure 4: Export of a water net hydraulic graph to the water net model (left) and the results of

hydraulic calculation made with the model.

The MOSUW and KRIPOW programs are key elements of an integrated information system that has been developed in IBS PAN to support the complex management of communal water networks. Other basic elements of this system are GIS and SCADA. One of important tasks of GIS software is to create the hydraulic graphs of the water net under consideration and to export them to MOSUW. An important task of SCADA software is to gain the data for making the calibration of the water net model, i.e. for fitting the model to the network. In Figure 4 the screens of GIS and MOSUW programs


are shown and also the way of exporting the water net graph from GIS to the hydraulic model of the network is explained.

Figure 5: The water pressure distribution (left) and the water flow distribution in the water net

calculated with the kriging approximation.

In Figure 5 the results of KRIPOW program with designed water pressure and flow distributions are presented. Looking on the maps the water net operator can recognize very fast the work quality of the network. On our exemplary maps we can state that the water net works wrong: the water pressure values are too high (these parts are marked with brown and red colors on the map) and the water flow values are too slow (these parts are marked with green and blue colors on the map). In the real cases of the water net management the operator has to undertake in such situations some improvement procedures to decrease the pressure and to accelerate the water flows.

• Calculation of the water net nodes altitudes

The water net graphs which are represented on the classical geodetic maps have a lot of disadvantages that make impossible to use them directly for hydraulic calculation of the relevant networks. These disadvantages are discontinuities at the water net design, lack of water net nodes which are not marked on geodetic maps, and as the consequence the lack of the node altitudes. If such the map will be transported in a digital form to a GIS system then the obtained numerical map of the network will be in state to visualize the water net but not to model it mathematically. To get the water net graph adapted for hydraulic calculation the disadvantages mentioned must be eliminated, i.e. some algorithms have to be developed to remove the network discontinuities, to create a new layer in GIS system that will contain the nodes as the new defined water net objects, and to estimate the node altitudes. For the estimation of the net node altitudes the kriging approximation has been used and the appropriate program has been included into the structure of the information system developed in IBS PAN. To make this approximation the municipal geodetic points are used as the measurements points. These geodetic points are conventionally defined in a big number for each city and for example in Rzeszow, which is a Polish city of middle size with ca. 170.000 inhabitants, there are 120.000 geodetic points defined against ca. 20.000 nodes of the municipal water net. In Figure 6 the geodetic measurement points used for estimating the node altitudes and the resulted contour line map designed with the kriging algorithm are shown. To verify the node values calculated with the kriging approximation another algorithm has been used that creates a grid of triangles defined on the geodetic points (see Figure 7).


Knowing the equations of all the triangles fixed on the geodetic points one can approximate the altitude coordinate of each water net node that is usually located inside of the relevant triangle.

Figure 6: The cloud of measurement points for kriging approximation of the node altitudes (left) and the approximation results.

Figure 7: The grid of triangles covering the water net area (left) and the result differences for

both algorithms of the altitude calculation.

The comparison of the calculation results won with the kriging approximation and with the triangles grid algorithm show the negligible differences between them. In Figure 7 on the right side there are two curves designed for the node altitudes calculated for a water net with the kriging algorithm (blue line) and with the triangles grid (green line) and the red line below means the curve of differences. One can see that both algorithms give almost the same altitude values. If we take results of the triangles grid algorithm as the reference values then the errors of calculation made with the kriging algorithm are less then 1% of the referenced altitude values. It means that the kriging approximation proves useful by the estimation of the node altitudes of communal water nets.


• Planning of SCADA systems for water nets

SCADA systems which are in the last years commonly installed in the waterworks are used mainly as a data source for the water net operator providing him with the current information about the work quality of the network. Knowing the momentary values of flows and pressure in the key points of the water net the operator can estimate the state of the water net operation and undertake some emergency activities when the situation is not satisfied. These key points of the water net providing the operator with the data making him possible to run his operational work are usually the water take out stations on the input of the network and the retention reservoirs and the pump stations inside the network. The commonly not numerous measurement points included into a SCADA system are unfortunately insufficient for realizing other tasks connected with the water net management and not only with its up to day operation. The other tasks that could be made with an otherwise designed SCADA system are for example the automatic calibration of the water net hydraulic model as well as detection and indicating of places of hidden water leaks in the network. These both tasks mentioned are of big importance by the complex management of communal water nets. The purposeful enlargement of SCADA system means its widening about the measurement points which are located in so called sensitive network places. In a sensitive measurement point the flow or pressure changes can be registered occurring in a long distance from it as opposed to so called dead points in which only local changes of values of the network variables are recorded. As a result of such the extension of a SCADA system the relatively small number of the measurement points can supply the operator with possibly large amount of useful information. The algorithm of determining the sensitive measurement points for SCADA systems consists in simulation of water leaks in the successive network nodes and in calculation of the network sensitivity in the remaining nodes. The simulation runs are made with a hydraulic model of the network after its successful calibration. For the SCADA system is presently in the planning stage and it can not be used to calibrate automatically the water net model, then this initial calibration step is usually realized using the data gaining from a measurement experiment realized on the water net. There is to stress that the accuracy of the water net model, i.e. the accuracy of its initial calibration decides about the correctness of defining the well situated monitoring points. The network sensitivity is calculated from the following relations (Straubel, Holznagel, 1998) regarding the water flow and pressure values which are computed with the hydraulic model MOSUW:

∑

∑

=

≠

≠

mkkm

mkkmmm

pm L

L/p∆pS

)( (22)

∑

∑

=

≠

≠

mkkm

mkkmmm

qmL

L/q∆qS

)(

(23)

where: k – the node with the water leak simulated, m – the measurement point considered, p – water pressure, q –water flow, m∆p and m∆q – the differences in measurements for normal and emergency states of the water net, L – the distance between the points k and m.

The correct measurement points are these ones with the highest sensitivity values. The sensitivity calculations can be made regarding the flow and pressure values separately or joining them together. After the calculation is made, the contour line maps for the obtained sensitivities can be designed using program KRIPOW with its kriging approximation. In Figures 8 and 9 the approximation results concerning the sensitivity of a water net under


consideration are shown. The sensitivity values can be visualized on the maps using the linear or logarithmic scales.

Figure 8: The contour line maps of the water net sensitivity regarding the pressure (left) and flow changes; presentation in the logarithmic scale.

Figure 9: The contour line maps of the water net sensitivity regarding the joined pressure and

flow changes; presentation in the logarithmic (left) and linear scale.

Conclusions

In the paper the possibilities of using the kriging algorithms to support the management of communal water nets are presented. The cases described concern the preparation of contour line maps designing the water flow and pressure distributions, the calculation of altitudes for the water net nodes and the allocation of sensitive measurement points for SCADA systems. To realize these goals own computer programs have been developed, i.e. program KRIPOW to make the kriging approximation and program MOSUW for hydraulic calculations of water networks. These programs cooperate each other via special data files. The both programs are standard elements of an information system developed in the System Research Institute as a tool improving the complex management of water nets.

By testing the KRIPOW program some practical observations have been made concerning the quality of kriging approximation. While modeling the experimental semivariogram in Step 2 of the kriging algorithm 8 different analytical models are calculated and only one of them is chosen for farther


calculation steps. As the choice criterion the model’s residuum value is used. This way of proceeding is simple but unfortunately not always reliable for there are many practical cases when the semivariogram models with lowest residuum value induce bigger errors by the following kriging approximation of the variable investigated than these once with higher residuum. It means that a worse approximation on the step of semivariogram modeling can result with a better approximation on the step of calculating the unknown variable values. The assessment of quality of different approximation runs can be made by calculating the variable values in some selected measurement points concerned as the calculation points in the kriging algorithm and by comparing them with the measurement data which are known for these points. The practical experience shows that while modeling the discreet semivariogram an accurate approximation of only its beginning points decides about the success of the following kriging approximation. An exact modeling of farther semivariogram points is not important for the accuracy of the whole kriging. It means that if there is decided about the choice of the semivariogram model used subsequently for farther calculations one shall concentrate on observations how the model fits to the beginning semivariogram points and he or she can be less attentive by the estimating the accuracy of modeling the subsequent semivariogram points.

On the other side the KRIPOW program developed is handy and simple in operation and fast in calculations and its algorithms are robust against inaccuracies in the measurement data. There is to stress that KRIPOW program shall be used to carry out the kriging approximation only in cases of big communal water nets with a big number of nodes and pipes. These objects are then concerned as the measurement points in the kriging algorithm and if their number is small then the results of approximation can be unreliable.

In the paper three applications of the kriging approximation for the water net management have been shown. The spectrum of such applications can be extended and now we are working on an algorithm in which the kriging algorithm and KRIPOW program will be applied to detect and indicate the places of hidden water leaks in the water net. The realization of this goal requires the cooperation of GIS and SCADA systems and of MOSUW and KRIPOW programs and all these items are already available and included into the structure of the information system developed in IBS PAN.

References

Bogdan L., Studzinski J. (2006A) Anwendung der Kriging-Approximation zur Anfertigung von Regenfallkarten für Polen. In: Wenkel K.O., Wagner P., Morgenstern M., Luzi K., Eisermann P. (Hrsg.) GI-Edition, Lectures Notes in Informatics, Vol. P-78, Koellen Druck Verlag, Bonn.

Bogdan L., Studzinski J. (2006B) Entwicklung von Algorithmen zur Krigingsapproximation zur Modellierung von Umweltdaten. In: Wittmann J., Mueller M. (Hrsg.) Simulation in den Umwelt- und Geowissenschaften. Shaker Verlag, Reihe Umweltinformatik, Aachen.

Bogdan L., Studzinski J. (2006C) Mathematische Modellierung und Kriging-Algorithmen zur Approximation monatlicher und jährlicher Regenereignisse. In: Gnauck A. (Hrsg.) Modellierung und Simulation von Ökosystemen. Shaker Verlag, Reihe Umweltinformatik, Aachen.

Bogdan L., Studzinski J. (2006D) Kriging approximation: algorithms, program and calculation results. In: Studziński J., Hryniewicz O. (Eds.) Eco-Info and Systems Research. PAS SRI, Series Systems Research, 52, Warsaw.

Bogdan L., Studzinski J. (2007) Modeling of water pressure distribution in water nets using the kriging algorithms. In: Industrial Simulation Conference ISC’2007 (J. Ottjes and H. Vecke, eds.) Delft, TU Delft Netherlands, 52-56.

Krige D.G. (1951) A statistical approach to some basic mine valuation problems on the Witwaterland. Journal Chem. Metali. Min. Sc. South Africa, 52, 119-139.

Matheron G. (1962) Traîté de géostatistique. Appliqué, tome 1. Memoires de Bureau de Recherches Geologiques et Minieres, vol. 1. Editions Techn., Paris.


Sichel H.S. (1952) New methods in statistical evaluation of mine sampling data. Trans. Inst. Min. Metali, vol. 61, part 6, Londyn.

Straubel R., Holznagel B. (1998) Mehrkriteriale Optimierungen für Planung und Steuerung von Trink- und Abwasser-Verbundsystemen. Proceeding of the Conference on Problems in monitoring and automation of wastewater purification plants, Ustronie Morskie / Poland, 30-42.

Studzinski J. (2006) Entwicklung von Modellen und Algorithmen zur Simulation und Optimierung von komplexen Wassernetzen. In: Modellierung und Simulation von Ökosystemen. Workshop Kölpinsee 2004 (A. Gnauck, Hrsg.) Shaker Verlag, Aachen, 1-12.

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