Towards Combining Probabilistic, Interval, Fuzzy Uncertainty, and Constraints: On the Example of Inverse Problem in Geophysics

In many real-life situations, we have several types of uncertainty: measurement uncertainty can lead to probabilistic and/or interval uncertainty, expert estimates come with interval and/or fuzzy uncertainty, etc. In many situations, in addition to measurement uncertainty, we have prior knowledge coming from prior data processing, prior knowledge coming from prior interval constraints. In this paper, on the example of the seismic inverse problem, we show how to combine these different types of uncertainty. 1 Seismic Inverse Problem: A Brief Description In evaluations of natural resources and in the search for natural resources, it is very important to determine Earth structure. Our civilization greatly depends on the things we extract from the Earth, such as fossil fuels (oil, coal, natural gas), minerals, and water. Our need for these commodities is constantly growing, and because of this growth, they are being exhausted. Even under the best conservation policies, there is (and there will be) a constant need to find new sources of minerals, fuels, and water. The only sure-proof way to guarantee that there are resources such as minerals at a certain location is to actually drill a borehole and analyze the materials extracted. However, exploration for natural resources using indirect means began in earnest during the first half of the 20th century. The result was the discovery of many large relatively easy to locate resources such as the oil in the Middle East. However, nowadays, most easy-to-access mineral resources have already been discovered. For example, new oil fields are mainly discovered either at large depths, or under water, or in very remote areas – in short, in the areas where drilling is very expensive. It is therefore desirable to predict the presence of resources as accurately as possible before we invest in drilling. From previous exploration experiences, we usually have a good idea of what type of structures are symptomatic for a particular region. For example, oil and gas tend to concentrate near the top of natural underground domal structures. So, to be able to distinguish between more promising and less promising locations, it is desirable to determine the structure of the Earth at these locations. To be more precise, we want to know the structure at different depths z at different locations (x, y). Data that we can use to determine the Earth structure. In general, to determine the Earth structure, we can use different measurement results that can be obtained without actually drilling the boreholes: e.g., gravity and magnetic measurements, analyzing the travel-times and paths of seismic ways as they propagate through the earth, etc. To get a better understanding of the Earth structure, we must rely on active seismic data – in other words, we must make artificial explosions, place sensors around them, and measure how the resulting seismic waves propagate. The most important information about the seismic wave is the travel-time ti, i.e., the time that it takes for the wave to travel from its source to the sensor. to determine the geophysical structure of a region, we measure seismic travel times and reconstruct velocities at different depths from these data. The problem of reconstructing this structure is called the seismic inverse problem. 2 Known Algorithms for Solving the Seismic Inverse Problem: Description, Successes, Limitations We want to find the values of the velocity v(x) at different 3-D points x. Based on the finite number of measurements, we can only reconstruct a finite number of parameters. So, we take a rectangular grid and to reconstruct the velocities vj at different grid points. Algorithm for the forward problem: brief description. Once we know the velocities vj in each cell j, we can then determine the paths which seismic waves take. Seismic waves travel along the shortest path – shortest in terms of time. It can be easily determined that for such paths, within each cell, the path is a straight line, and on the border between the two cells with velocities v and v′, the direction of the path changes in accordance with Snell’s law sin(φ) v = sin(φ′) v′ , where φ and φ′ are the angles between the paths and the line orthogonal to the border between the cells. (If this formula requires sin(φ′) > 1, this means that this wave cannot penetrate into the neighboring cell at all; instead, it bounces back into the original cell with the same angle φ. ) In particular, we can thus determine the paths from the source to each sensor. The travel-time ti along i-th path can then be determined as the sum of traveltimes in different cells j through which this path passes: ti = ∑ j `ij vj , where `ij denotes the length of the part of i-th path within cell j. This formula becomes linear if we replace the original unknowns – velocities vj – by their inverses sj def = 1 vj , called slownesses. In terms of slownesses, the formula for the travel-time takes the simpler form ti = ∑