Minimum of Cost Function with Nonzero First Derivative

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Time dependent Model Output Sensitivity
Inverse Analysis of Soil Moisture from
Observations of Atmospheric Temperature and
Humidity
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Outline of the Model
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Outline of the Model
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Outline of the Model
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Outline of the Model
Question How do observations depend on control
variables hi?
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Sensitivity of model output may depend on
integration time
At time tp/2 all information about initial q is
contained in velocity v!
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Output
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• Derivatives of atmospheric humidity at 2100 with
  respect to prior
• Solid Prior upward heat fluxes in the soil
  
• Dotted Prior heat fluxes from the soil to
  the atmosphere

Time (Hour of the Day)
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Parameter Optimisation
(Adjoint model more efficient than linear model)
Example Algae Growth in the River Elbe
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Green Algae
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Diatoms
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Parameter
Radiation Water Temperature Discharge
Model WAMPUM
Simulated Chlorophyll-a
Data
Model Output
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A) Evaluation of the model fit to data (Cost
Function)
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A) Evaluation of the model fit to data (Cost
Function)
Model Results
Data
Data Chlorophyll-a Oxygen
Scaling Observation Error
Model Parameters
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Parameter Uncertainties
(Linear model more efficient than adjoint model)
Example Algae Growth in the River Elbe
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Uncertainty when combining two observations with
observation errors s1 and s2, respectively
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Evaluation of the model fit to data
Tangent Linear model more efficient than adjoint
model
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Posterior Standard Deviations
For individual parameters
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Posterior Standard Deviations
For individual parameters
Two Species Green algae, Diatoms

• Demand of Light K_light, K_light
• Growth Rate K_growth, K_growth
• Fraction of Mass being Silica F_silica
• Physical Parameter Depth

• 2 light parameters depend on prior knowledge
• other 4 parameters controlled by data??

In the direction of eigenvectors
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Spectrum of eigenvalues
1. eigenvector
2. eigenvector
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Isolines of Cost Function
Projecting Monte Carlo Samples onto Eigenvectors
of the Posterior Covariance Matrix
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Non-Differentiable Cost Functions
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Minimum of Cost Function with Non-zero First
Derivative
Minimum
Cost Function

• Problem
• First derivative is discontinuous
• Second derivative in the minimum (Hessian) cannot
  be analysed.

• Possible Reasons
• Unproper parameterisations
• Problems in numerical integration

• Adjoint Modelling
• Helps to detect and analyse this situation.

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Convergence to Minimum with non-zero Gradient
IMAS Output
J ? J
grad J step size 1 1.0982E03
0.0E00 1.1E02 2.4E01 2
1.0420E03 -5.6E01 6.3E01 2.4E01
3 9.1270E02 -1.3E02 8.8E01
2.4E01 4 8.8356E02 -2.9E01
2.2E02 2.4E01 ........ 10
6.4632E02 -8.5E00 2.0E01 7.4E00
11 6.4283E02 -3.5E00 1.2E01
1.5E01 12 6.4045E02 -2.4E00
9.6E00 2.7E00 13 6.3033E02
-1.0E01 4.1E01 2.2E00 14
6.1719E02 -1.3E01 1.9E01 5.7E00
15 6.1295E02 -4.2E00 1.4E01
9.8E00 ......... 38 6.0103E02
-1.6E-10 2.8E-02 3.3E-03 39
6.0103E02 -1.0E-10 2.8E-02 4.4E-03
40 6.0103E02 -9.4E-11 8.7E-02
1.0E-05 41 6.0103E02 -1.7E-12
2.8E-02 8.9E-06 42 6.0103E02
-1.2E-11 2.8E-02 2.8E-05 43
6.0103E02 -5.0E-12 2.8E-02 4.5E-05
convergence at delta x
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Result of Adjoint Model Run (Gradient)
type (control_var_list_type) Results of
adjoint model no_variables 6
no_components 6 6 active, 0 passive
variables no type a d sigma size
scaling name
value 1 1 T T 0.87 1
e.87 k_light_green -
1.722188E-02 2 1 T T 0.43
1 e.43 k_light_diatom -
5.807458E-08 3 1 T T 0.29E-01 1
e.29E-01 k_growth_green 1.073324E-02
4 1 T T 0.29E-01 1 e.29E-01
k_growth_green 1.430435E-06 5 1 T
T 0.87E-02 1 e.87E-02 frac_si
- 2.437749E-07 6 1 T T
0.87E-01 1 e.87E-01 depth
- 1.954283E-02 end type (control_var_list_t
ype)
Conclusion Some problem in the module of
the program dealing with green algae.
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Zoomed Cost Function
Before Correction
601.0258
601.0256
601.0254
601.0252
1.6995
1.6996
1.6997
1.6998
1.6999
1.7000
water depth
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Acknowledgements
Fred Kucharski Soil Moisture
Analysis Andreas Rhodin IMAS, Soil
Moisture Analysis Gerd Blöcker Data (Water
Quality) Mirco Scharfe Water Quality,
Uncertainty Analysis Friedhelm Schroeder Model
WAMPUM