The Prediction of Debris Flow Distribution on Merapi - Bandung Arry Sanjoyo SCCSCE 2014

8/16/2019 The Prediction of Debris Flow Distribution on Merapi - Bandung Arry Sanjoyo SCCSCE 2014

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The Prediction of Debris Flow Distribution on Merapi

Volcano in Central Java which Involves

Measurements at Several Locations Through The

Ensemble Kalman FilterBandung Arry Sanjoyo, Mochammad Hariadi, Mauridhi Hery Purnomo

Departement of Electrical Engineering

Institut Technology of Sepuluh Nopember

Surabaya, Indonesia [email protected], [email protected], [email protected]

Abstract — Debris flow occurs in the downhill area of Merapi in

Central Java is very dangerous threat that threatens human life,

and destroys infrastructure facilities. Modeling of the debris flow

in that area can be approximated by Eulerian continuous flow

equations and discretized into dynamics systems model. This

paper present the dynamic systems model and a strategy for

estimate the distribution of the debris flow by Ensemble Kalman

Filter (EnKF) that is combine with measurement data. The

number of measurement points obtained by applying EnKF on

several PDE which are part of Navier-Stokes equation. The EnkF

method is prepare for prediction of debris flow distribution on

Merapi Volcano downhill in Central Java using the combination

of one and two dimension model of debris flow.

Keywords — debris flow, Kalman filter, dynamic systems.

I. I NTRODUCTION

One of the natural disasters that often occur in volcanicregions is a flood that carries sediments material of sand and

mud mixed with gravel and stones of various sizes. Sediments

flow has high concentrations and a very large destructive

power moving by gravity called debris flow. Debris flow is one

of the very dangerous threats that threatens human life,animals, plants, and destroys infrastructure facilities. This kind

of debris flow often occurs in the downhill area of Merapi in

Central Java. This region becomes one of the most seriously

affected by debris flow disaster in Java.

The damage caused by debris flow can be minimize by

knowing the characteristics of the flow, the amount of

sediment, flow velocity and flow depth. Information of

characteristics of debris flow in the volcanic area is very

important to predict the flow distribution and to determine the

flow areas.

The debris flow can be approximated by formulation in the

form of Eulerian continuous fluid equation [1]. In [2] has been

done a simulation of prediction of debris flow distribution in

one side of Merapi downhill using finite difference approach.

But this finite difference results needs to be fixed or updated

with including the field measurement result, which can not be

done with finite difference approach. So, this paper want to

consider another approach for estimate nonlinear model called

Ensemble Kalman Filter (EnKF). EnKF has been used to

forecast nonlinear weather model which has high order,uncertainty initial state, and big measurements [4].

This research is trying to use EnKF method to predict

debris flow with adding measurements result as finitedifference calculation’s correction to minimize the covariance

error. Discrete scheme is chosen to transform debris flow

equation to dynamical system model that compatible to EnKF

System. Firstly, we simulate EnKF for some partial differential

equation (PDE) that is part of debris flow equations to

determine how much minimum points of measurement neededfor EnKF to work well.

II. DEBRIS FLOW MODEL AND ENSEMBLE KALMAN

FILTER A. Debris Flow Model

Debris flow is a fluid flow mixture of sediment and water

are driven by gravity. The term sediment means all the

particulate substances from clay to huge boulders. In case of

finding macro behavior of flow motion, such as the depth and

the velocity of flow, the debris flow can be treat as a

continuum material flow. So, the characteristics of the flow can

be analyzed by Eulerian equations.

The one dimensional model of debris flow down the valley

was presented in many references has equation as in Eq. 1.

(1)

Where h : depth of` flow, u : velocity, M =uh, g : gravitational

acceleration, H =h+ Z b, c : sediment concentration, ρ m : mass

density of flow, , S T : erosion velocity, Z b:

slope height and σ b : shear stress.

2014 IEEE International Conference on Control System, Computing and Engineering, 28 - 30 November 2014, Penang, Malaysia

978-1-4799-5686-9/14/$31.00 ©2014 IEEE 136



Two dimensional debris flow model contained in [3,5]

consists of three equations. Equation 2 and 3 are respectively

express the movement of the flow to the x and y axis, while theEq. 4 is the continuity equation.

X direction:

(2)

Y direction:

(3)

Continuity Equation:

0 (4)

where u and v are respectively state for flow velocity on the x

and y-axis.Variable M and N respectively state for mass flux

on the x and y axis, and are expressed with M = uh and N =

vh. Symbols τbx and τby are shear forces on the edge of the flow.The analysis of debris flow in a mountainous area, can be

divided into: a steep area and a more sloping areas. Steep area

can be analyzed using a one-dimensional model of debris,

while the more sloping areas can be analyzed using a two-dimensional model of debris flow [3]. This model has been

applied in the southern area of downhill of Merapi Volcano,

along the Gendol river, using a finite difference method

without involving the measurement data [2]. Qualitatively, the

results are already close to the real events of debris flow in the

Gendol river area. Also, the performance of the model in Eq. 1was solved using finite volume compare with experiment [5].

Reference [6] try to couple Eq. (2)-(4) with basal entrainment

applied in debris flow over erodible beds in Wenchuan-China

using finite difference method. On the other hand, the above

two-dimensional model has been developed into a two-phase

flow, such as in [7], and it is solved using a finite volume and it

can be used as a debris flow hazard assessment.

B. Ensemble Kalman Filter

EnKF is a recursive filter suitable for problems with a large

number of variables, such as discretizations of PDE in

geophysical models [9]. EnKF can be interpreted as an

estimator that using a Monte Carlo method with the mean of

ensemble as the best estimate and the error variance is the

spreading of the ensemble [10]. Another ensemble is used to

representing the observation and the mean of the ensemble asthe actual measurements.

Let’s consider nonlinear dynamic systems in Eq. (5).

, (5)

and measurement model as in Eq. (6).

(6)

where , , , and , . is

assumed zero mean white noise with covariance matrices

and is assumed zero mean white noise with covariance

matrices . The formulation of Kalman filter are expressed in

two stage:

1. Forecasting step: generate value of prediction state f

k x̂

and value of covariance error

. EnKF algorithm

initialize ensemble matrix … . With is state variable sized n and N is the number of

ensemble. Matrix X is called with prior ensemble. It will

be calculated mean and covariance of matrix X .The mean ensemble of X is Eq. 7.

∑ (7)

and the covariance ensemble is Eq. 8.

(8)

where … .2. Correction / measurement step: this stage produce

correction values based on measurement, z k , to produce

estimating valuea

k x̂ and covariance errora

k P . If the

measurement data z k is not known, the best estimation for

state xk is mean value k x with 0 x is given [11]. Intial value

0ˆ x is calculated from ensemble mean [9].

Replicate measurement datam R∈ into matrix

… , where

, 1,… , (9)

coloumn filled with data from vector z added with

random vector from normal distribution 0,.

Step 1 and 2 are processed repeatedly until we find a best

estimationa

k x̂ that minimizes the error covariancea

k P .

III. DEBRIS FLOW MODEL AS A DISCRETE DYNAMICS

MODEL AND FUTURE WORK

EnKF will be used to solve the debris flow models with

improving the results using measurement data. Before that, theminimum number of measurement points needed to be known.

To estimate the amount of measurement data, firstly: EnKF

will be used to solve several PDE which is part of the Navier-

Stokes equations, i.e. Burger's equation and the convection

equation. The second: the number of measurement data will be

used to place the measuring instrument for debris flow along

the river.


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A. Estimating the number of Measurement

1) Ensemble Kalman Filter for One Dimensional Burgers’

EquationEquation 10 is one dimensional Burgers’ equation which

is part of debris flow equation in one dimension. The using ofEnKF will be applied to solve (10).

(10)

The analytical solution of Burgers’ equation is Eq. 11.

2

4 (11)

where exp exp

.

Forward difference in time and backward difference in

space scheme used to discretize Eq. 10 and arranged it to

become dynamical systems as in Eq. 12.

∆∆ + ∆

∆ 2 (12)

where 1, 2, … , and is measurement value. We will

determined the minimum number points of measurement that

ensure good results. Equation 12 is stable for ∆∆ 1, or

for small ∆. Supposed we have boundary conditions

0, , 0 for

0, and initial conditions

, 0 sin, for 0 1.The value of state variable can be estimated with using

EnKF. Let … , for 0 obtained the

initial value … 0 1 1 … 1 2 2 … 2 1 1 1 … 1 0. The results of running program are shown in Figure 1and Figure 2.

Figure 1. Solution of 1-D Berger’s Equation using EnKF with

40 grid points and 2 number location measurements.

Figure 2. Solution of 1-D Berger’s Equation using EnKF with40 grid points and 20 number location measurements.

The results of running program shows that the solution

using finite difference and EnKF have similar patterns. Graphs

of the solution using EnKF method close to the value of the

measurement data, and close to the solution using finite

difference methods. The number of measurement points 2 or 3

in the EnKF method have solution close to the solution by

using half the number of the grid.

2) Ensemble Kalman Filter for Two Dimensional

ConvectionEquation 13 is two dimensional convection equation. This

equation is also part of debris flow equation. The EnKF

method will be tested to solve Eq. 13.

0 (13)

c is constant transport velocity. Equation 13 will be

transformed into dynamical systems as in Eq. 14.

, , , , , , (14)

where ∆∆ and ∆

∆ . The initial conditions are chosen as follow.

2, for 0.5 1 and 0.5 1

1, everywehre else in 0,2 0,2

and the boundary condition are 0 if 0.2 and 0.2.

Discretization of 2D domain with index , can be

converted into linear index 1 as illustrated in

Figure 3.


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Figure 3. Discretization of 2D domain with index ,

convert into linear index 1 .

Let , , … , , , , … , , … , ,, … ,. For k=0, we obtained the initial

value

1 1 1…1 2 2…2 1 1 1…1 1. The results

of running program of 2D convection using finite difference

shown in Figure 4, while the result running program of 2D

convection with EnKF is depicted in Figure 5. Both of the

results are similar in pattern and very close in value.

Figure 4. Solution of 2D Convection using finite difference

Figure 5. Solution of 2-D Convection using EnKF

B. Debris Flow as a Dynamics System Model

1) One Dimensional Model of Debris Flow

Reference [2] has done a discretization model of debris

flow in one and two dimensions. But its discretization results

can not be brought into the form of dynamic models.

Therefore, in this research did discretization by selecting the

appropriate schema, so the results can be brought into the

form of a dynamic system. The discretization of one

dimensional debris flow model can be expressed in Eq. 15.

∆

∆

∆ (15)

∆2 ∆ ∆

Where .

Equation 15 can be written in non linear dynamic system as

in Eq. 16.

, (16)

where

- , ,

- , ∆ ∆ ,

∆

- , is measurement value,

The value of state variable can be estimated using

EnKF. Model (15) and (16) will be used to simulate one-

dimensional debris flow upstream Kali Putih on the west sideof the upper slopes of Merapi, which has a very high

steepness. Merapi eruptions that occurred over the years,

always flowing to the west side. The future activities are

taking the DEM (digital elevation model) data of the upstream

Kali Putih area and simulate the distribution of debris flow.

2) Two Dimensional Model

Physical interpretation of finite difference formulations fordebris flow is illustrated in Figure 6. The input of one-

dimensional simulation of debris flow are flow properties in

boundary conditions. Output of one-dimensional simulation of

debris flow is used as input for debris flow simulation in two

dimensions.


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Figure 6. Discretization for debris flow in 1D and 2D.

The equation of motion in the x-axis and y-axis direction

can be discretized into Eq. 17 and 18.

,

,

∆ ∆ ,

, ,

,

∆ ∆ , , , , ,

∆ ∆ , , ,

, ∆ (17)

,

,

∆ ∆ , , , ,

∆ ∆ , , , , ,

∆ ∆ ,

, ,

,

∆ (18)

Discretization of the continuity equation produce Eq. 19.

, , 2∆/, /,∆ ,/ ,/∆ (19)Similar to one dimensional case, Eq. 17, 18, and 19 can be

written in nonlinear dynamic systems as in Eq. 20.

,

,

, (20)

In the future work, the EnKF method will be applied to

simulate and visualize the distribution of debris flow in Merapi

downhill west area, especially in the Kali Putih river area using

schematic as in Figure 6. The upper segment of the river will

be simulated using one-dimensional model and the downstream

area of the river will be simulated using two-dimensionalmodels of debris flow. At each upper segment and downstream

area are installed two measurement tools.

IV. CONCLUSIONS

The performance of EnKF used to solve part of Navier-

Stokes equation is close enough to the results of finite

difference methods, both of them have the same pattern. And

the result of EnKF method follow the given measurementdata. The solution of EnKF method with two points

measurement is close to the solution of EnKF by using a half

number of the grid. Thus, the future real application in thearea of the Kali Putih river will use two or three measurement

points.

R EFERENCES

[1]

Takahashi T., Debris Flow Mechanics, Prediction and Counter

measures. Taylor & Francis Group, London, UK, 2007.

[2] Adzkiya, A., Sanjoyo, B.A., “One and two dimensional debrisflow simulation using finite difference method”, InternationalConference on Mathematical Applications in Engineering(ICMAE’10), Kuala Lumpur, Malaysia, 3-4 August 2010.

[3] Kuncoro, D.A., Numerical Simulation for Prediction of DebrisFlow Scale, Ministry of Settlement and Infrastructure Region,Indonesia, 2004.

[4]

S. Gillijns, O. Barrero Mendoza, J. Chandrasekar, B. L. R. De

Moor, D. S. Bernstein, and A. Ridley, “What is the ensemble

Kalman Filter and how well does it work?”, Proceedings of the2006 American Control Conference, Minneapolis, Minnesota,USA, 2006.

[5]

A. D’Aniello, L. Cozzolino, L. Cimorelli, C. Covelli, R. D.

Morte, and D. Pianese, “One-dimensional Simulation of Debris-

flow Inception and Propagation,” Procedia Earth Planet. Sci.,vol. 9, pp. 112–121, 2014.

[6] C. Ouyang, S. He, and C. Tang, “Numerical analysis of

dynamics of debris flow over erodible beds in Wenchuanearthquake-induced area,” Eng. Geol., 2014.

[7] A. Armanini, L. Fraccarollo, and G. Rosatti, “Two-dimensional

simulation of debris flows in erodible channels,” Comput.Geosci., vol. 35, no. 5, pp. 993–1006, May 2009.

[8] Chen, H. and Lee, C.F., “Numerical simulation of debris flow”,Canadian Geotechnical Journal, 37: 146–160, 2000.

[9] Mandel, J., “A brief tutorial on the ensemble Kalman Filter”,

2007.

[10] Evensen, G.,“The Ensemble Kalman Filter- theoreticalformulation and practical implementation”, Ocean Dynamics 53:343–367, 2003.

[11]

Lewis, F. L., Xie, L., and Popa. D.,Optimal and Robust

Estimation With an Introduction to Stochastic Control Theory,

CRC Press. 2nd Editon, 2008.


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The Prediction of Debris Flow Distribution on Merapi - Bandung Arry Sanjoyo SCCSCE 2014