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