Resolution analysis of geophysical images: Comparison between point spread function and region of data influence measures

Practical decisions are often made based on the subsurface images obtained by inverting geophysical data. Therefore it is important to understand the resolution of the image, which is a function of several factors, including the underlying geophysical experiment, noise in the data, prior information and the ability to model the physics appropriately. An important step towards interpreting the image is to quantify how much of the solution is required to satisfy the data observations and how much exists solely due to the prior information used to stabilize the solution. A procedure to identify the regions that are not constrained by the data would help when interpreting the image. For linear inverse problems this procedure is well established, but for non-linear problems the procedure is more complicated. In this paper we compare two different approaches to resolution analysis of geophysical images: the region of data influence index and a resolution spread computed using point spread functions. The region of data influence method is a fully non-linear approach, while the point spread function analysis is a linearized approach. An approximate relationship between the region of data influence and the resolution matrix is derived, which suggests that the region of data influence is connected with the rows of the resolution matrix. The point-spread-function spread measure is connected with the columns of the resolution matrix, and therefore the point-spread-function spread and the region of data influence are fundamentally different resolution measures. From a practical point of view, if two different approaches indicate similar interpretations on post-inversion images, the confidence in the interpretation is enhanced. We demonstrate the use of the two approaches on a linear synthetic example and a non-linear synthetic example, and apply them to a non-linear electromagnetic field data example.