# Linear Tikhonov Inversion

## Introduction¶

In this chapter we present the basic elements for how an inverse problem can be formulated and solved using optimization theory. The quantity to be minimized is a weighted sum of misfit and regularization terms with their relative importance controlled by an adjustable Tikhonov parameter.

The inverse problem has many elements and a solution is best achieved by adhering to the workflow shown in Figure 1 below. Throughout this chapter we investigate each of these steps and illustrate the concepts with a simple linear problem. Jupyter notebooks are provided so that the concepts can be explored (LinearTikhonovInversion_App.ipynb) and all the corresponding figures in this article can be reproduced using LinearTikhonovInversion_Notebook.ipynb. The formative material for this chapter is extracted from the tutorial paper Inversion for Applied Geophysics: A Tutorial Oldenburg & Li, 2005.

**Key Points**

- Forward Problem - model, mesh, kernel function, data, noise
- Inverse Problem - data misfit, regularization function, norms
- Optimization - choice of Tikhonov parameter

## Contents¶

**2.1. Forward Problem:** Defining the model, kernel function, data and noise for the forward problem. We also define the mesh, model parameters and mapping for discrete forward problems that can be solved numerically.

**2.2. Defining the Inverse Problem:**

**2.3. Data Misfit:**

**2.4. Non-Uniquness:**

**2.5. Model Norm:**

**2.6. Objective Function:**

**2.7. Summary:** An overview of what was learned in this chapter.

- Oldenburg, D. W., & Li, Y. (2005). 5. Inversion for Applied Geophysics: A Tutorial. In
*Near-Surface Geophysics*(pp. 89–150). Society of Exploration Geophysicists. 10.1190/1.9781560801719.ch5