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 and/or 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.
Seismic Inverse Problem: A Brief DescriptionIn 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 nd 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 rst 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 elds 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 arti cial 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 t i , i.e., the time that it takes for the wave to travel from its source to ...