Alex Hobé , Ari Tryggvason

Because geophysical inversion is used in many vital societal applications, it is unfortunate that some aspects of inverse methods are so abstract. The difficulty of identifying fundamental behaviors is exacerbated when investigating large non-linear problems which combine multiple datasets into a single model, or which produce multiple models with constraints between them. In this first of multiple papers, we investigate and visualize fundamental behaviors of these abstract methods beyond what has been described previously by using simple problems. Instead of using the common resolution description, we use the concepts of the Null Space and Image Space. After providing readers with an intuitive sense of the behaviors of simpler inverse methods, we investigate cases of Joint, Constrained, and Time-Dependent inversion without errors, before moving on to the influence of errors. We then extract the fundamental behaviors of these complex methods from the presented best and worst cases. These new insights allow us to propose four avenues to improve inversion results (including two novel methods), which we present with similar simple problems. Overall, we show the benefits of producing multiple estimated models using constraints over combining the inverse problems into a single model, and, the benefit of visualizing simple problems to uncover deep insights into the fundamentals of our everyday methods.