# inverse-modeling-PPT

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```					PARAMETER OPTIMIZATION USING
WHAT IS INVERSE METHOD ? ? ?
INVERSE MODELING (soil hydraulic
properties and root water uptake)                                                                                                                                                                                                                   Solution of an inverse problem entails determining
unknown causes, based on observation of their effects
This is in contrast to the corresponding direct problem,
whose solution involves finding effects based on the
0.00

-0.20
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-0.20
complete description of their causes.
-0.40                                                                                                              -0.40

EXAMPLES:
-0.60                                                                                                              -0.60
S o il D e p th [ m ]

-0.80                                                                                                              -0.80

-1.00                                                                                                              -1.00
0.00   0.20   0.40   0.60   0.80       1.00     1.20    1.40      1.60     1.80       2.00
0.00   0.20   0.40   0.60   0.80   1.00   1.20   1.40   1.60   1.80   2.00

0.00
0.00                                                                                            o Computerized Tomography (CT)
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-0.20

o Boundary inverse problem and
-0.40
-0.40

-0.60
-0.60

backward problem (to find initial
-0.80
-0.80

-1.00
-1.00

conditions)
0.00   0.20   0.40   0.60   0.80   1.00   1.20   1.40   1.60   1.80   2.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00

Surface drip                                                                                                 Subsurface drip

Radial distance trunk tree [m]                                                                                                                                          o Parameter Estimation (pde
parameters)

PARAMETER ESTIMATION                                                                                                                                                                                                                                      For example, fit parameters to
by inverse modeling                                                                                                                                                                                                                                   Convection-Dispersion Equation
(Parameter estimation)
Forward Problem
Soil                                                                                       Out = ?
Problem:              ∂c         ∂ 2c    ∂c
=D         −v
“White-Box” Technique”
In                                         parameters                                                                                                                                                                               R
∂t         ∂x 2    ∂x
Inverse Problem
“Black-Box” Technique”                                                                                ?                                                                                                                                                                                         vx
Rx − vt ⎤ 1 D        Rx + vt ⎤
= erfc ⎡           + e erfc ⎡
c 1
Solution:                ⎢ 4 DRt ⎥ 2          ⎢ 4 DRt ⎥
- 100
c
o
2     ⎣         ⎦          ⎣         ⎦
- 90
0-250, direct
- 80
0-250, inverse
- 70

- 60

Parameter Estimation                                                                   - 50

- 40

Difference: Solution is explicitly known, whereas
- 30

- 20
(a)
-10

“Gray-Box” Technique”                                                                     0
0 .0 5     0.1     0 .15      0.2     0 .2 5        0.3   0.3 5   0.4

in inverse modeling, solution can be only obtained
Wa t er C o nte nt [ - ]

by numerical modeling

1
WHY NEED FOR MEASUREMENT OF SOIL                                       Flowchart of Inverse Modeling (IM)
HYDRAULIC PARAMETERS ?                                               START                                     Parametric model
Boundary &
initial                    for soil hydraulic
functions
conditions
Multi-step
As input to water flow and contaminant                     Outflow                         Numerical
transport models;                                          Experiment                      Simulation
Initial
To characterize soil physical                                                                                        parameters

characteristics, including their spatial and                                        Cumulative Outflow and
Input data files
Soil water matric potential
temporal variability;
Nonlinear
To correlate with other, more easily to                                     Optimization
New
parameters
measure soil physical properties, e.g.
texture.                                                                          ok?
no

yes
Stop

Parameter estimation of soil hydraulic
properties, (Methods of Soil Analysis, 2002)                             Miniature tensiometer for multi-
step experiments

⎡            ⎤
m
⎢
Se = ⎢       1    ⎥                     ⎡             m⎤    2

n⎥
⎢ 1 + ( αh ) ⎥
l
K r = Se ⎢1- ⎛ 1-S1/m ⎞ ⎥
⎢ ⎜      e ⎟ ⎥
⎢
⎣            ⎥
⎦                        ⎝        ⎠
⎣                ⎦

2
MULTI-STEP OUTFLOW EXPERIMENT
2
porous
quick                    ceramic
disconnect               cups      1-psi differential
air                    fitting                            pressure
inlet                                                     transducer

Experiment:
3.5 cm

Multi-step outflow, with tensiometric
measurements inside soil core;
port for

Apply a sequence of air pressure steps to initially
flushing
bubbles
15-psi gauge
pressure
transducers            near-saturated soil core;
water

quick
outlet
porous                        Monitor cumulative drainage volume and
tensiometer pressure with pressure transducers;
nylon
disconnect               membrane
fittings

Measure boundary and initial conditions

1-psi gauge
pressure
transducer

Interpretation of tensiometric
Experimental Setup
measurements

3
Data prep variables             Correction to outflow transducer
as caused by flushing
3.05

3.00

2.95

mV
2.90

2.85

2.80
1790 1800 1810 1820 1830 1840 1850 1860 1870
Time                 3.05

3.00

2.95

mV
2.90

2.85

2.80
1790   1800   1810   1820   1830   1840   1850   1860   1870
Time

SIMULATION MODEL:

Correction with dataprep                              HYDRUS-2D for Windows
Simulating Two-Dimensional Water Flow,
Heat and Solute
Movement in Variably Saturated Media

4
MODELING:                                                                                                   PARAMETER OPTIMIZATION

Use van Genuchten—Mualem or Kosugi
Compare measured with simulated flow
lognormal pore size distribution model to                                                                          variables;
describe soil water retention and
Find optimum parameter set, so as to
unsaturated hydraulic conductivity functions;
minimize differences between measured
Assume initial parameter values for these                                                                          and simulated flow variables: I.e. soil
functions;
water matric potential and cumulative
Solve transient one-dimensional water flow                                                                         drainage from soil sample
model with known initial and boundary
conditions to solve for cumulative drainage                                                                        Use Levenberg-Marquardt method,
and soil water matric potential                                                                                    Simplex method or Genetic Algorithm

Optimization – Minimize Objective
Function                                                                                            COMPARISON OF MEASURED WITH
SIMULATED OUTFLOW AND MATRIC
POTENTIAL

Objective                                                                                                  1                                                                  0
Function
Value, Φ                                    Other parameter                                                         M4+                                                                         Tensiometer 1
Cumulative Outflow [cm]

-100
value                                                         0.8                                                                                   Fitted 1

-200                Tensiometer 2
0.6                                                                                   Fitted 2
-300
0.4
Parameter                                                                                                                                           -400
value                                                                             0.2                            Measured
-500
Fitted
N
Φ ( b ) = W Q ∑ {ω i [ Q ex p ( t i ) − Q sim ( t i , b )]}
0                                                                -600
2
0         20            40          60                             0   20         40            60
i =1                                                                                                       Time [hours]                                                 Time [hours]
M
+ W hm    ∑ {ω
j =1
j   [ h m ,ex p ( t j ) − h m , sim ( t j , b )]} 2

5
Multi-step outflow method to                                                                                                                                                                       Example of multi-dimensional root water uptake
indirectly estimate soil water retention                                                                                                                                                             In Press: Vrugt et al., SSSAJ and Water Resour Res.
and unsaturated hydraulic conductivity
functions
Lincoln sand
Solve unsaturated water flow equation
(Wildenschild et al., 2001)
- 10 0
for multi-dimensional soil domain, with
root water uptake term (S)
1.E+01
Hydraulic Conductivity [cm/hour]

-90
0-250, direct
1.E+00           (a)
-80
0-250, inverse
-70                                                                                                                       1.E-01

∂θ
-60
1.E-02

= ∇ ⋅ [K∇(h − z )] − S ( x, y, z , t )
-50

-40                                                                                                                       1.E-03

∂t
0-250, inverse
-30
(a)                                                                                                 1.E-04                                            0-250, K-3
-20
0-250, K-23
- 10                                                                                                                      1.E-05
0-250, K-32
0
1.E-06
0 .0 5   0 .1   0 .15    0 .2     0 .2 5        0 .3   0 .3 5   0 .4
Wa t e r C o nt e nt [ -]
0.05     0.1    0.15       0.2   0.25        0.3    0.35      0.4
Water Content

MULTI-DIMENSIONAL ROOT WATER                                                                                                                                                                                  HOW TO FIND MULTI-
UPTAKE MODEL                                                                                                                                                                                   DIMENSIONAL ROOT WATER UPTAKE
⎛ px              py                pz         ⎞                PARAMETERS?
⎛     x ⎞⎛     y ⎞⎛     z ⎞ -⎜ Xm x -x + Ym y -y + Zm z -z ⎟
*                *                 *

β ( x, y, z ) = ⎜ 1 -   ⎟ ⎜1 -   ⎟ ⎜1 -   ⎟e
⎝                             ⎠

⎝ Xm ⎠ ⎝ Ym ⎠ ⎝ Zm ⎠                                                                                                                                                                     Conduct multi-dimensional experiment
Construct a multi-dimensional root water uptake
X Ym β ( x, y, z )                                                                                              model
RDFWi ( x, y, z ) =                           m
Xm Ym Zm
Integrate root uptake into multi-dimensional water
∫ ∫ ∫ β ( x, y, z ) dxdydz
0   0                                      0                                                                                   flow model
Compare experimental with numerical data
S max,i = Tpot RDFWi                                                                                                                                                                             Minimize their residuals

α (ψ m ) =
1                                                                                           Si (ψ m ) = α (ψ m ) Smax,i
⎡ ⎛ ψ ( x, y , z , t ) ⎞ p ⎤
⎢1 + ⎜ m
⎢ ⎝
⎣
⎜   ψ m,50
⎟ ⎥
⎟ ⎥
⎠ ⎦
PARAMETER OPTIMIZATION BY
INVERSE MODELING

6
EXPERIMENTAL LAYOUT OF THREE-
PARAMETER OPTIMIZATON OF THREE-
DIMENSIONAL SOIL MOISTURE                                                  DIMENSIONAL ROOT WATER UPTAKE MODEL
MEASUREMENTS (ALMOND TREE)

R
Obtain field measurements of multi-
dimensional soil moisture during a 2-week
monitoring period;
Simulate multi-dimensional soil moisture
distribution, using transient HYDRUS-3D
model;

Z
Optimize root water uptake parameters
using inverse modeling, minimizing residuals
Y                        of measured and simulated water contents
X

Optimization – Minimize Objective
Flowchart of Inverse Modeling (IM)
Function
START
Boundary &
initial                       Root uptake
conditions                         model
Water Content
Measurements
Objective
In Root Domain                           Numerical                             Function
Simulation                            Value, Φ                       Other parameter
Initial                                    value
parameters
Input data files
3-Dim Water Content
Parameter
Nonlinear                                New
parameters                         value
Optimization

2
no                                                         n
ok?
Φ(p,θ ) = w1 ∑ ⎡θ * (x, ti ) −θ (x, ti , p)]⎤
⎣                            ⎦
yes
i =1
Stop

7
Flexibility of two-dimsional root water
Parameterize root water uptake                                                                                                   uptake model
model
⎛p                       pr * ⎞
⎡⎛       z ⎞ ⎤ ⎡⎛      r ⎞ ⎤ -⎜ z m
z
z* - z +         r -r ⎟
β ( r, z ) = ⎢ ⎜ 1 -                       ⎝                                                                                      rm      ⎠
⎛ p         py *     p        ⎞                                                                                              ⎟ ⎥ ⎢⎜ 1 -   ⎟⎥ e
⎛     x ⎞⎛     y ⎞⎛     z ⎞                                         -⎜ x x* -x +    y -y + z z* -z ⎟                                                                                  ⎣ ⎝     z m ⎠ ⎦ ⎣ ⎝ rm ⎠ ⎦
β ( x, y, z ) = ⎜ 1 -                                                                ⎝ Xm        Ym       Zm       ⎠
⎟⎜ 1 -   ⎟⎜ 1 -   ⎟e
⎝ Xm ⎠⎝ Ym ⎠⎝ Zm ⎠                                                                                                              0.00

-0.20
0.00

-0.20

-0.40                                                                                                  -0.40

-0.60                                                                                                  -0.60

Soil D epth [-m ]
-0.80                                                                                                  -0.80

-1.00

Xm Ym β ( x, y, z ) Tpot
0.00      0.20     0.40     0.60     0.80     1.00    1.20    1.40    1.60    1.80    2.00         -1.00
0.00   0.20    0.40   0.60   0.80   1.00   1.20   1.40   1.60    1.80   2.00

Sm ( x, y, z ) =      Xm Ym Zm
0.00

-0.20
0.00

-0.20

∫∫∫            β ( x, y, z ) dxdydz
-0.40
-0.40

-0.60
-0.60

-0.80

0    0     0
-0.80

-1.00
-1.00                                                                                                   0.00   0.20    0.40   0.60   0.80   1.00   1.20   1.40   1.60    1.80   2.00
0.00     0.20     0.40     0.60     0.80    1.00    1.20    1.40    1.60    1.80   2.00

Surface drip                                                                                  Subsurface drip
Unstressed Normalized Root
Water Uptake, at a point                                                                                                                                                   Radial distance trunk tree [m]

Computation of Tpot                                                                                Water Uptake under Water-
Stressed Conditions
Tpot ,almond = ETalmond − Es
ETalmond = K c ET0                                                            Water Stress                                                                                                      α ( hm ) =
1
⎡ ⎛ h (x, y, z, t) ⎞p ⎤
Response                                                                                                                                     ⎢1 + ⎜ m           ⎟ ⎥
Function                                                                                                                                    ⎢ ⎜      h m,50    ⎟ ⎥
⎣ ⎝                ⎠ ⎦
5
Evaporation or transpiration

ET0
]
-1

4
[mm d

3
ETalmond                                                                                                                                                                                        Actual Plant Transpiration:
2                                                                                 Actual Water Uptake:
Xm Ym Zm
1                    Es                                                            S(h,x,y,z)=α( h) Sm(x,y,z)                                                                                                                 1
0
Ta,alm =
XmYm                  ∫ ∫ ∫ S(h,x,y,z)dxdydz
0 0 0
0          100         200           300
Time [hours]

8
Optimization minimizes residuals of
THREE-DIMENSIONAL SOIL                                                                                                                                                   measured and simulated water
MOISTURE OBSERVATIONS                                                                                                                                                            content values
(Vrugt et al., 2001)                                                                                                                                      HYDRUS-3D
0.30

Simulated water content
0.12                                                             RMSE = 0.0180
0.25      R2 =0.92
θ [m3 m-3]
2.40                                                                                                                                                           0.08
2.40
1.40                                                                                                                                                          2.40
y [m]                                                                         1.40
0                         0.40                                                                   y [m]                                                     1.40
0.90                                            0.40                                                                            y [m]   0.04

[m3 m-3]
1.40                                                      0.40
x [m]
1.90
2.40
0.90
1.40
1.90
2.40
0.40
0.20
x [m]

0.15

0.10

0.05

0.00
0.00    0.05     0.10     0.15      0.20   0.25        0.30

Measured water content [m3 m-3]

SIMULATED THREE-DIMENSIONAL SOIL                                                                                                                                                         RESPONSE SURFACE ANALYSIS:
MOISTURE AND ROOT WATER UPTAKE
DISTRIBUTION
0.2

0.16
θ [m3 m-3 ]
0.12                                                                                                     Objective
Function
Other parameter
0.08

2.40
Value, Φ
value
1.40                                                                                  2.40
0.04
0.40           y [m]                                                                   1.40
0.40      0.90                                          0.40           y [m]
1.40     1.90
x [m]                   2.40

8.0

6.0
Parameter
value
4.0

S m [m3 m-3 h-1 ]
X 10-4
2.0

2.40
1.40 y [m]                                                                    1.40
2.40
0.0
o Objective function contains residuals;
0.40                  0.40     0.90                                           0.40           y [m]

o Shows local and global minima;
1.40      1.90
x [m]                   2.40

o Analysis of uniqueness of inverse problem

9
RESPONSE SURFACE ANALYSIS:                                                                                                                                                                                                                                                              Well-posed inverse problems:
Can also be used to investigate parameter
sensitivity and parameter correlation                                                                                                                                                                                                            Test for global and local minima
2.00
Test for unique solutions
Saturated Hydraulic Conductivity [cm/d[

2.20

Independently measure parameters that are
0.08
Residual Water Content [-]
1.50

not sensitive to solution
2.00
0.06
n [-]

1.80                                                                                                                 1.00

Do not estimate highly correlated
0.04

parameters
1.60
0.50
0.02

1.40

Include independently-measured information
0.00                                                                                              0.00
0.002      0.006        0.010     0.014    0.018                                                                      0.002   0.006        0.010       0.014      0.018                                                 0.002   0.006        0.010        0.014          0.018

Alpha [1/cm]                                                                                                                                                                                                         Alpha [1/cm]

to objective function
Alpha [1/cm]

2.00

Minimize number of optimized parameters
Saturated Hydrualic Conductivity [cm/d]

0.08                                                                                                                                                                                                                    0.08
Residual Water Content [-]

Residual Water Content [-]

1.50

Minimize measurement errors
0.06                                                                                                                                                                                                                    0.06

1.00

Estimate model error
0.04                                                                                                                                                                                                                    0.04

0.50
0.02                                                                                                                                                                                                                    0.02

Compare uncertainties of optimized
parameters
0.00                                                                                                                  0.00                                                                                              0.00
0.50          1.00        1.50      2.00                                                                       1.40   1.60        1.80       2.00      2.20                                                     1.40    1.60        1.80       2.00       2.20
Saturated Hydraulic Conductivity [cm/d]                                                                                                   n [-]                                                                                            n [-]

OTHER APPLICATIONS OF INVERSE
MODELING:                                                                                                                                                                                                                                                                                                             LIMITATIONS:
Non-uniqueness
Instability
Other soil hydraulic properties techniques,
such as evaporation method, suction
infiltrometer method and instantaneous profile
method;                                                                                                                                                                                                                                                                        Inverse problems are not
Estimation of solute and heat transport
necessarily well-posed;
properties;                                                                                                                                                                                                                                                                    Selection of weighting factors;
Estimation of root water and nutrient uptake
parameters;
Parameter estimates are valid for
experimental range only;
Effective field soil properties, and in multi-
layered systems;                                                                                                                                                                                                                                                               Method requires a lot of experience
. . . . . . . . . . . . . . . . . . . .

10
BENEFITS:

Mandates marriage of experimentation with
numerical modeling;
Method is consistent, I.e. estimated hydraulic
functions are used in model predictions;
Uses transient measurements, as in real world;
Relatively fast method, and lends itself for
automation

11

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