Usage of Particle Filter for Exact Estimation of Constant Head Boundaries in Unconfined Aquifer

Abstract

Having the exact values of boundary conditions is one of the effective way to develop precise groundwater models. In the present study, the exact value of constant head boundaries in Birjand aquifer is specified with the usage of particle filter linked to meshless groundwater model. Particle filter known as one of the common data assimilation methods applies to dynamic systems in order to improve performance. Meshless model, one of the numerical models that do not mesh the problem domain, enforces the governed equation to the nodes. Birjand aquifer with an almost 269 km 2 area, has 190 extraction and 10 observation wells. There are also nine inflow and one outflow regions with constant head boundary conditions, which include 105 boundary nodes. In this research after determination of the lower and upper bounds of groundwater head for each node, the exact values of this parameter is computed. Finally, the simulated groundwater head compared with observation data. The closeness of the achieved results to the observation data showed the performance of the engaged method, as the results indicated a significant decrease in RMSE occurs just with the usage of particle filter linked to meshless model. RMSE value reduced to 0.386 m as its previous value was around 0.757 m. Results also showed that the model was more accurate when the number of particles in particle filter were increased. The RMSE value for 500, 700 and 1000 particles were 0.484, 0.401 and 0.386m respectively.

Full text

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Usage of Particle Filter for Exact Estimation of
Constant Head Boundaries in Unconfined Aquifer
Ali Mohtashami1, Seyed Arman Hashemi Monfared2
*
, Gholamreza Azizyan3, Abolfazl Akbarpour4
1PhD Student of Civil Engineering Department, University of Sistan and Baluchestan, Zahedan, Iran
2,*Associate Professor of Civil Engineering Department, University of Sistan and Baluchestan, Zahedan, Iran
3 Associate Professor of Civil Engineering Department, University of Sistan and Baluchestan, Zahedan, Iran
4 Associate Professor of Civil Engineering Department, University of Birjand, Birjand, Iran
ABSTRACT
Having the exact values of boundary conditions is one of the effective way to develop precise groundwater
models. In the present study, the exact value of constant head boundaries in Birjand aquifer is specified with the
usage of particle filter linked to meshless groundwater model. Particle filter known as one of the common data
assimilation methods applies to dynamic systems in order to improve performance. Meshless model, one of the
numerical models that do not mesh the problem domain, enforces the governed equation to the nodes. Birjand
aquifer with an almost 269 km2 area, has 190 extraction and 10 observation wells. There are also nine inflow and
one outflow regions with constant head boundary conditions, which include 105 boundary nodes. In this research
after determination of the lower and upper bounds of groundwater head for each node, the exact values of this
parameter is computed. Finally, the simulated groundwater head compared with observation data. The closeness
of the achieved results to the observation data showed the performance of the engaged method, as the results
indicated a significant decrease in RMSE occurs just with the usage of particle filter linked to meshless model.
RMSE value reduced to 0.386 m as its previous value was around 0.757 m. Results also showed that the model
was more accurate when the number of particles in particle filter were increased. The RMSE value for 500, 700
and 1000 particles were 0.484, 0.401 and 0.386m respectively.
KEYWORDS
Birjand aquifer, RMSE, Constant Head Boundary Condition, Particle Filter, Meshless Groundwater Flow Model.
*
Corresponding Author: Email: hashemi@eng.usb.ac.ir
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1. Introduction
Groundwater is one of the main sources of fresh water in
all regions and the only source of water supply in arid and
semi-arid regions of the world. Nowadays, due to the
excessive extraction of groundwater, these resources
endanger intensely [1]. Therefore, Groundwater flow
simulation in aquifers is the best way to recognize its
behavior undoubtedly. Many numerical methods have
been used for this aim. Meshless local Petrov-Galerkin
(MLPG) categorized as a weak form method is used in
this study. The independency of this method removes the
drawbacks of mesh based methods e.g. finite difference
method (FDM) and finite element method (FEM) [2].
Besides of the groundwater simulation procedure,
uncertainty assessment must be considered as well. Many
researches is carried out with the purpose of uncertainty
assessment. Hamraz et al. (2015), Abedini et al. (2017)
and Du et al. (2018) used GMS software and GLUE
method in order to simulate groundwater flow and
assessment the uncertain parameters.
In the present study, a new method i.e. particle filter
known as the online calibration method is linked to the
meshless local Petrov-Galerkin simulation model to find
the optimal values of constant head boundaries. The
purpose of this study is to improve the accuracy of
simulation results. This model is used for the first time in
this field.
2. Methodology
2.1 Particle Filter
A particle filter known as a powerful estimation method
computes the probability density function of a random
process and also estimates the exact state of the object in
the future time based on the states and observations of
previous times. The particle filter makes some estimates
for the state of the object to select the best one [3].
To this end, in the initial step, particles are scattered
in the space which its dimension equals to the number of
parameters that must be estimated. Each particle is
assigned with a weight value. This weight value in the
first step is equal for all particles around (1
𝑁) (N is the
number of the particles scattered in state space) [4]. In the
next step, the weight value depending on the position of
the particle is updated. Once the weight of each particle
is determined, to prevent from degeneracy occurrence,
re-sampling method is carried out.
2.2 Meshless Local Petrov-Galerkin (MLPG)
MLPG is a weak form of meshless methods presented in
1998 by Atluri and Zhu (1998) to solve the potential
equation. This method is generally used in fluid
mechanics and involves two functions: a weight function
(cubic spline) and a moving kriging.
2.3 Groundwater flow equation in unconfined aquifer
Based on the Dupouit assumption, the governed
equation of groundwater flow in a transient condition is
stated in Eq. 1 [5]:
(1)
  
y
xy
ww
SH
HH
k H k H
x x y y t
Q x x y y q



   




    


    
Here, 𝐻 represents groundwater head [L], 𝑘 stands for
hydraulic conductivity coefficient [L/T], 𝑄 denotes the
discharge (+) or recharge (-) rate [L/T], and q stands for
the distributed flow, e.g. precipitation and evaporation.
𝑆𝑦 is specific yield.
3. Discussion and Results
Solving groundwater partial differential equation
requires precise values of constant head boundaries.
Therefore, determination of the exact values of these
boundaries is one of the fundamentals steps in
groundwater studies. Particle filter method linked to the
MLPG flow model that is calibrated and verified by the
authors in the previous studies [6].
Finally, the optimal values of constant head
boundaries are obtained, and, the groundwater table is
computed using the achieved optimal values. Figure 1.
shows the results of PF-MLPG, MLPG, FDM and
observation data. In figure 1, the graphs show the high
correspondence of PF-MLPG method (yellow line) to the
observation data (black line). This fact clearly indicate
the high accuracy of PF-MLPG method due to the usage
of optimal values for constant head boundaries.
(a)
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Groundwater head (m)
Time(month)
MLPG PF_MLPG
OBS FDM
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(b)
Figure 1. Comparison of results in different methods
In order to evaluate the performance of PF-MLPG
method, mean, absolute mean, root mean square error
calculated by equation (2-4):
(2)
 
11
mn
os
ji
hh
ME mn
 


(3)
11
mn
os
ji
hh
MAE mn



(4)
 
2
11
mn
os
ji
hh
RMSE mn
 


where ℎ𝑜𝑖
𝑡, ℎ𝑠𝑖
𝑡, ℎ𝑜



are the level of observed groundwater
and the simulated and mean of the observed values,
respectively, n and m are the number of piezometers and
the number of time steps. Table 1 shows the achieved
results.
Table 1. Calculation of errors
PF-MLPG
(1000 particles)
(m)
FDM(m)
MLPG
(m)
Mean error (m)
-0.061
0.159
-0.12
Mean Absolute
error (m)
0.298
1.434
0.573
Root mean
square error (m)
0.386
1.197
0.757
RMSE is the main index for evaluation of accuracy [9].
Based on the table 1 results of PF-MLPG is more
accurate than FDM and MLPG methods due to its lower
value of RMSE. Also the performance of PF-MLPG
method with different number of particles is investigated
in Table 2. RMSE value decreases while the number of
particles increases. The model also is ran for 2000
particles. However, its results are as the same of 1000
particles.
Table 2- Calculation of errors
PF-MLPG (500
particles) (m)
PF-MLPG (700
particles) (m)
Mean error (m)
-0.101
-0.083
Mean Absolute
error (m)
0.416
0.332
Root mean square
error (m)
0.484
0.401
4. Conclusions
Particle filter known as one of the data assimilation
method, is linked to the meshless local Petrov-Galerkin
flow model to find the best values of constant head
boundaries of a real aquifer. In the first step, a set of
particles with the same weight values (1
𝑁) are generated
in the state space. The dimension of state space are equal
to the number of uncertain parameters. The studied
region is Birjand unconfined aquifer which is located in
South-Khorasan province. Finally, the optimal values of
constant head boundaries for boundary nodes are
obtained. Mean, mean absolute and root mean square
error indices is calculated for PF-MLPG, MLPG and
FDM method. The RMS error are 0.386, 0.757 and
1.197m for PF-MLPG, MLPG and FDM respectively.
The results also reveal that with increasing the number of
particles, the RMS error decreases.
5. References
[1]
Nayak, P., SatyajiRao Y., Sudheer, K., 2006.
"Groundwater level forcasting in a shallow aquifer
using artificial," Water Resources Management, 20,
pp. 77-90.
[2]
Liu, G. R., Gu Y. T., 2005, An introduction to
Meshfree Methods and Their Programming,
Singapore: Springer.
[3]
Hamraz, B. S., Akbarpour, A., Pourreza Bilondi M.,
Sadeghi Tabas, S., 2015. "On the assessment of
ground water parameter uncertainty over an arid
aquifer," Arabian Journal of Geosciences, 8, pp.
10759-10773.
[4]
Abedini, M., Ziai, A. N., Shafiei, M., Ghahraman, B.,
Ansari H, Meshkini, J., 2017. "Uncertainty
Assessment of Groundwater Flow Modeling by Using
Generalized Likelihood Uncertainty Estimation
Method (Case Study: Bojnourd)," Iranian Journal of
Irrigation and Drainage, 10(6), pp. 755-769.
[5]
Du, X., Lu, X., Hou J., Ye X., 2018. "Improving the
Reliability of Numerical Groundwater Modeling in a
Data-Sparse Region," Water, 10(3), pp. 289-304.
[6]
Arulampalam, S., Maskell, S., Gordon N., Clapp, T.,
2002. "tutorialon particle filters for Online
nonlinear/nongaussian Bayesian tracking," IEEE
Transaction Signal Process, 50(2), pp. 174-188.
[7]
Fearnhead P., Kuensch, H. R., 2018. "Particle Filters
and Data Assimilation," Annual Review of Statistics
and Its Application, 5(1), pp. 421-449.
[8]
Dupouit, J., 1863. Estudes Theoriques et Pratiques sur
le Mouvement desEaux, Paris: Dunod.
[9]
Mohtashami, A., Akbarpour A., Mollazadeh, M.,
2017. "Development of two dimensional groundwater
flow simulation model using meshless method based
on MLS approximation function in unconfined aquifer
in transient state," Journal of Hydroinformatics, 19(5),
pp. 640-652.
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