Exercise 2: Linear Tikhonov Inversion

In this investigation you can explore the components of a 1D synthetic inversion example, starting with model building, defining the kernel function, then forward modelling the data, adding noise, and finally inverting the data to recover a model. Specific instances of these components have been shown throughout the text above, and can be reproduced in the Jupyter notebook LinearTikhonovInversion_Notebook.ipynb. The corresponding notebook LinearTikhonovInversion_App.ipynb contains the interactive widgets that allow you to adjust parameters and view results for the following.

1Forward Problem¶

The concepts about forward modelling can be explored individually in Steps 1-3, or in the composite widget. There are many aspects to explore but one idea can be readily investigated is that a datum is an inner product of the kernel function with the model. Equivalently, a datum is a volumetric “average” of the model with the “averaging function” being the kernel function. This is a useful mental model and forms the basis of resolution analysis, data importance and nonuniqueness that will be discussed throughout this module. For each forward modelling concept, consider the following questions while adjusting parameters:

• Step 1 - Create a model
  • How does changing the structure of model change the data?
  • How does the data changed when a constant background is added to the model?
• Step 2 - Generate a sensitivity matrix
  • How does changing the number of data $N$change your insights from the data?
    • Is there a number of data $N$ where you no longer see changes in the data plot?
  • Adjust the parameters controlling the decay rate - $p_{min}, p_{max}$
    • How do the rows of $\mathbf{G}$ change?
    • How does the data change?
  • Adjust the parameters controlling the frequency - $q_{min}, q_{max}$
    • How do the rows of $\mathbf{G}$ change?
    • How does the data change?
• Step 3 - Simulate data and add noise
  • Consider different noise levels
    • When is the data dominated by the noise rather than signal from the model?

2Inverse Problem¶

The numerical details regarding how we obtain a solution to the inverse problem are postponed until the next chapter, but using Step 4 we can investigate the essential elements and understanding of linear Tikhonov inversion. By adjusting both the parameters in the Forward and Inverse Problems, consider the following scenarios and questions:

• Clean data - run the inversion with clean data (no noise added)
  • Can the true model be recovered from the inversion exactly? Why or why not?
  • How does changing the misfit change the recovered model in this case (the misfit value must be non-zero)?
• Noise estimation - run the inversion with noise contaminated data
  • It is typical in field surveys to have only an estimate of the noise level - What happens if the estimated noise in the misfit is different from what was added in the Forward Problem? Too high? Too low?
• Noise level - run the inversion with different levels of noise contaminated data
  • How does changing the noise level of the data in the Forward Problem change the recovered model?
• Number of data - run the inversion with different values of $N$
  • Does more data always improve the recovered model?
    • Consider a large number of data with a higher noise level
    • Consider a smaller number of data with a lower noise level
• Reference model - run the inversion with variations of the reference model
  • Recall $m_{ref}$ is an adjustable constant and is effective only in the smallest model component - the derivative of a constant is zero
  • How much influence does the reference model $m_{ref}$ have on the recovered model?
    • Consider a zero reference model
    • Consider a reference model equal to the background model
    • Consider an incorrect reference model
• Model norms - run the inversion with varying values of $\alpha_s$ and $\alpha_x$
  • Recall the smallest model and flattest model norms are not on the same scale
  • How does the model change with only the smallest model norm included? $(\alpha_x =0)$
  • How does the model change with only the flattest model norm is included? $(\alpha_s =0)$
    • Does this scenario make sense?
  • Does the same balance of these terms work for different true models?
• Model components
  • Which components of the true model are best recovered by the inversion? Why?
    • Consider the location, amplitude, and shape of both the boxcar and Gaussian, the background value
    • Future consideration - If you knew something about the expected model, how could you influence the model to recover those details?
    • Does changing the kernel function parameters ($p, q$) change the recovered model components?
• Trade-off Parameter β - run the inversion with varying β ranges
  • Do you get the same model with different ranges of β values?
  • What happens the range of β values is too small? - try several small ranges
  • What happens if are only a few values of β over a large range?
  • _Does the optimal value \beta^_ stay the same or similar when any of the changes in this investigation are made?*
  • In Explore mode view the $\phi_d~\text{vs}~\phi_m$ plot and adjust $\beta_i$ - similar to the examples in 2.7. Objective Function for the Inverse Problem-Figure 4 and Figure 6
  • How does the model change when a β that is too large is chosen? (overfitting the data)
    • How does the model change when a β that is too small is chosen? (underfitting the data)
• Target Misfit - run the inversion with varying values of $\chi_{fact}$
  • Recall the target data misfit value is $\phi_d^*=\chi_{fact}N$, thus when $\chi_{fact}=1$ the expected value of the misfit is the number of data $N$ ((3))
  • How does the model change when the inversion overfits the data $(\chi_{fact}<<1)$?
  • How does the model change when the inversion underfits the data $(\chi_{fact}>>1)$?

The Jupyter notebooks associated with this investigation can be downloaded and run locally or run through the Binder link at the top of the notebooks. More details on this can be found in Getting Started - Jupyter Notebook Apps.