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
                                                                                                                                                                     0.00



                                                                                                                                                                     -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)
                                                                                                                                                                     -0.20
                                                     -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
                                                                                                                                                          Matric Head [cm]




                                                                                                                                                                             -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




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