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ASCEM : Parameter Estimation (PE)

  1. ASCEM
  2. Tutorials

ASCEM : Parameter Estimation (PE)

Introduction

Objective

The objective of this tutorial is to demonstrate a parameter estimation (PE) analysis. The objective of the PE is to estimate parameters that are consistent with the measured data.

Toolsets/modules exercised

The following Akuna modules are being exercised:

  1. PE using Levenberg-Marguqardt algorithm.
    1. Input parameter selection
    2. Observation definition
    3. Analysis options
  2. Job submission and monitoring.
  3. Visualization of PE output

The following Amanzi modules are being exercised:
  1. Variably saturated flow using Richards equation
    1. Steady-state
    2. Transient
  2. Transient transport

Pre-requisite

Although it is advised to run a Single Run (SR) before launching a PE, it is not required. This tutorial uses the conceptual model Richard-1D-transport_exa located in the tutorial directory under the shared projects directory. If this simulation has not yet been generated, create a new model by copying the Richard-1D-transport_exa model from tutorial folder.

  1. Right click on projects > tutorial > Richard-1D-transport_exa and choose copy.
  2. Right-click on your user folder and choose paste.
  3. The following window will appear:

image0

Figure 1. Select tools to import when copying an existing model

  1. Click the radio button for single-run to import the single-run example. If selecting the option of “Copy Outputs”, the simulated results from single-run will be copied. However, it is not required in this tutorial. Press OK.
  2. A new model will appear under the user’s working folder.

Problem Description

The tutorial problem is a one-dimensional, 3 layer, unsaturated flow and transport problem, where

  1. steady-state unsaturated flow is calculated with a constant infiltration rate, providing the initial conditions for year 1950;
  2. discharge of water and 99Tc occurs at the top of the ground surface between 1950.1-1950.26 y; and
  3. continuing infiltration drives the 99Tc plume downward, to the water table, which is represented by the bottom boundary of the domain.

The parameter estimation will provide effective values of permeabilities that can be considered consistent with the measured data.

Initialize PE Toolset

To start a PE Toolset:

  1. On the main window (Figure 2), click to highlight the Richard-1D-transport conceptual model |image1| icon that was created in the Model Setup tutorial or copied from projects > tutorial > Richard-1D-transport_exa; it will be highlighted as shown in the figure below.
  2. Click image2 to launch the Parameter Estimation Toolset. The item PE is not clickable unless a model is selected.

image3

Figure 2. Create a new PE.


3.  A window (Figure 3) that prompts for a PE Name will appear. Give the PE a name and click OK.


image4
Figure 3. Provide a name to the PE

This action launches the Parameter Estimation Toolset window (Figure 4). The tabs are the same as the other toolsets (e.g., see SR, SA, UQ tutorials), except for the Analysis Options tab.

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Figure 4. PE Toolset window

Setup PE Toolset

Akuna simplifies the setup by providing the user with the ability to import data from an existing analysis or alternatively all fields can be populated manually. The following provides guidance on importing data from the SR Toolset to the  PE Toolset.

  1. Click image6 on the upper left corner of the PE Toolset window.

  2. Select all sections except for the Visualization and Checkpoint to be imported, as shown in Figure 5. These sections are not imported because plots will be generated at certain locations in the domain, and spatial visualizations will not be needed. Press OK.

  3. Navigate to an existing SR (created in the SR tutorial or copied from projects > tutorial > Richard-1D-transport_exa) as shown in Figure 6. Press OK and all the tabs will be filled with the data used in the SR (Figure 7).


image7

Figure 5. Selecting sections for import

image8

Figure 6. Import from a SR run that was previously created.

image9

Figure 7. PE Toolset window after importing information from an existing SR.

Parameter Setup

The parameters of interest in the parameter estimation are setup as follows:

  1. Click the Parameters section and then click image10 to add a parameter, as shown in Figure 8.

image11

Figure 8. Parameters section


2.  The dialog box shown in Figure 9 will appear. Click on image12 to expand the list of parameters that can be chosen.


image13

Figure 9. Select Parameters from Model Setup window.

  1.  For the first parameter, choose
Material Properties > Facies1 > Permeability > x; Material Properties

> Facies1 > Permeability > y; Material Properties > Facies1 > Permeability > z | simultaneously while holding down the Shift key. These three variables will be treated as a single parameter. Click Add.

4.  For the second and third parameters, repeat the previous step with Facies2 and Facies3.

 5.  Close the window.

6.  Double click on any of the entries in the table to define properties related to each parameter and change the entries to those shown in Figure 10.  All the 3 parameters are examined in the logarithm space and thus the Transformation Type (column 5) is changed to Logarithm. As a result the distributions of the permeabilities are log-normal when the Distribution Type (column 6) is Normal.

Make sure the ranges of the Distribution Parameters are properly defined. Notice that this tutorial describes a synthetic data set, so that the original parameter estimates can be identified.

image14

Figure 10. Edit the entries of the Parameter table based on entries shown above.

Define Observations

Observations are defined under the Observations tab. Observations have been prefilled since they were defined during model setup and in the existing SR (Figure 11). If not, observations can be imported from existing SR by clicking image15 . Observations can be added, edited, or deleted in four tabs: Observations, Measured data, Regions and Time sets.

image16

Figure 11. Add Observations for PE analysis.

For calibration, simulated results from previous SR will be imported and used as measured data. Click Measured dataImport from file to import a pre-existing .csv file (Figure 12). The instructions to import measured data are listed on the panel. Then click image17 to located the .csv file from the local machine (e.g. Figure 13).

image18

Figure 12. Import measured data for Observations in PE analysis.


image19

Figure 13. Select measured data (.csv file) for Observations in PE analysis.

The window for selecting measured data will appear (Figure 14). The header line (Row 1) shows the names of the variables. And each column lists the values of each variable. By clicking image20 , unused variables (columns) will be ignored. For example, to import measured data of Aqueous concentration at Well 1, make sure to define X, Y, Z (Well1: Z), Time and Aqueous concentration: Tc-99. All the other columns need to be ignored. Then click Apply and Close.

image21

Figure 14. Measured data imported from .csv files.

The measured data will then be plotted in the Observations tab (Figure 15). Import measured data from the same .csv file for the other three observations. And the measured data will all be plotted as shown in Figures 16-18.

image22

Figure 15. Measured data (Well 1 - Aqueous concentration: Tc-99) imported from .csv file (measuredData.csv) for PE analysis.


image23

Figure 16. Measured data (Well 1 - Volumetric water content) imported from .csv file (measuredData.csv) for PE analysis.


image24

Figure 17. Measured data (Well 2 - Aqueous concetration: Tc-99) imported from .csv file (measuredData.csv) for PE analysis.


image25
Figure 18. Measured data (Well 2 - Volumetric water content) imported

from .csv file (measuredData.csv) for PE analysis. | Imported data can be checked and modified by clicking Measured data – Edit measurements and weights. Measurement weights can also be assigned in this window (Figure 19).

image26

Figure 19. Check/modify measured data in Observations tab for PE

analysis. |

Specify Analysis Options

The analysis options are set in the Analysis Options tab:

  1. Choose Lease squares (sum of the squared weighted residuals) for Objective Function.
  2. Choose Levenberg-Marquardt for Minimization Algorithm.
  3. In Sub-Options, choose standard algorithm for Algorithm options. Set the Initial Levenberg Parameter to 1.0 and press ENTER. Set the Marquardt Parameter to 2.5 and press ENTER.
  4. In Derivatives, choose Numerical for Type. Set the Perturbation Factor to 0.1 and press ENTER. Choose relative for Perturbation Type. Choose Backward for Finite Difference.
  5. In Stopping Criteria, set the Maximum Number of Iterations to 5 and press ENTER. Set the Maximum Number of Simulation Runs to 5 and press ENTER. Set the Maximum Number of Unsuccessful Steps to 2 and press ENTER. Set the Parameter Step Tolerance to 1.0E-12 and press ENTER. Set the Objective Function Tolerance to 1.0E-6 and press ENTER. Set the Gradient Tolerance to 1.0E-12 and press ENTER. The Analysis Options are summarized in Figure 20.

image27

Figure 20. Analysis options for PE analysis.

Job Configuration

The job configuration process is similar to job configuration demonstrated in the SR Tutorial (Figure 21). The only additional step involves specifying the number of Processors and Processors per Tasks. In the SR Tutorial, only a single simulation is executed, the number of Processors and Processors per Tasks are the same. In the PE, three simulations will be executed simultaneously, with each simulation using 2 processors. Because each node contains 24 processors and only one job per node is permitted on Hopper, the total number of processors specified here is 72. Therefore, set Processors to 72 and Processors per Tasks to 2. Also set Max Simultaneous Simulations to 3.

image28

Figure 21. Job configuration of PE simulations.

Save the model by clicking image29 at the top left corner. The image30 button will turn green, indicating that all information needed to run the analysis has been provided. Two .xml files (agni.xml and amanzi.xml) are generated in the Inputs folder as the input files for Agni and Amanzi. The measuredData.csv file is also stored in the Inputs folder (Figure 22).

image31

Figure 22. Input files of PE simulations.

Submit PE Simulations

Click image32 to submit the job (credential required). The remaining steps for submission and monitoring of jobs follows the SR Tutorial.

Analyze PE Results

During the simulation, the information of each run can be monitored in Outputs folder. When the PE simulations are completed (Success), the Agni and Amanzi output files will be generated in the Outputs folder (Figure 23). For example, open Agni.out in the preview pane (Figure 24), it shows the estimated parameters, computational parameters, residuals during iteration and other information.

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Figure 23. Output files in the Outputs folder upon successful simulation.

image34

Figure 24. View Agni.out in the preview pane.

Upon successful completion of the PE simulation, the Plot button is activated, as shown in Figure 25. The PE Summary and General Information are also shown in the Summary panel.

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Figure 25. Plot button activation in Summary Panel.

  1. To generate plots, click on the PE to highlight, and then click the Plot button in the Summary pane, or at the top of the data browser window.
  2. There are two automatically-generated summaries in PE: Parameter Summary (Figure 26) and Objective Summary (Figure 27). The Parameter Summary shows the initial and best guess of each estimated parameter, along with the error estimation (standard deviation). And in Objective Summary, the value of the objective function is decreasing with number of iterations. This shows that the least squares summation of differences between the simulated and measured data decreases only after a few iterations.

image36

Figure 26. Automatically-generated plots in the PE simulation: Parameter Summary.


image37

Figure 27. Automatically-generated plots in the PE simulation: Objective Summary.

3.  To create additional plots, the Guided Control in the left column is used. Three plot types are available: histogram, scatter plot and line plot. To visualize differences between simulated and measured data, two line plots are generated at Well 1 and Well 2 respectively. Select line plot in the guided control, followed by the entries shown in Figure 28 (for Well 1). Press Generate, and the line plot will be shown on the right plotting area, as shown in Figure 29.

image38

Figure 28. Entries under Guided Controls.

4.  For line plot at Well 2, repeat the same procedure except for choosing Select pointsWell2. Figure 30 shows the comparison between measured and simulated Aqueous concentration (Tc-99) at Well 2.

 5.  Select Options on the right panel and select Show legend, the legend of measured and simulated data will be shown at the bottom of the figure. In both line plots (Figures 29-30), image39 represents simulated aqueous concentration (Tc-99) and image40 represents measured data. And it is clear that the simulated data are in excellent agreement with the measured data, which proves the parameter estimation is successful.


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Figure 29. Line plots for comparisons between simulated and measured data at Well 1.


image42

Figure 30. Line plots for comparisons between simulated and measured data at Well 2.

  1. Right-click the mouse to edit the plot (Properties, Save as, Print, Zoom In, Auto Range, etc).
  2. Click image43 in the upper left corner to save the results.