In the last lecture, we discussed the problems of application of seismic tomography for different geological stories, and hopefully you are convinced that the seismic tomography is able to provide very colorful and very nice pictures of the processes that occur inside the earth. Probably some of you want to learn how to use this tomography algorithms to build your own models like those that were shown previously, and the in this series of lectures, in this module of lectures, I will present some theoretical basis of seismic tomography algorithms and then in the next group of lectures, we will learn some practical algorithms that are used to build such tomography models. In this lecture I would like to start from the beginning, what is seismic tomography? What is the definition of seismic tomography? What is the input of seismic tomography and the output of seismic tomography? What are the main problems and challenges when you work with seismic tomography? Let's start from the input and output, so the seismic tomography it's a black box which is built by some programmers and scientists and usually people do not enter inside many peoples are using this programs as Blackbaud box. They have some input data and they have some output models. What kind of data are used as the input data for seismic tomography? First of all, in most cases scientists and explorers use a travel times of seismic raise that propagate from source to receivers. In the case if you have earthquakes, you don't know the time in the source so that the origin time of earthquakes is a priori unknown. In this case, the input data is arrival time in hours, minutes and days and so on. Besides the travel time data of body waves, you can use also the attenuation parameters like amplitudes that can be used to estimate the attenuation, the quality factor inside the earth. Besides volume body waves, you can use also the surface wave, which is a very important part of seismic tomography studies are based on the surface wave data. Also, what is important is the input data you can use also the just continuous noise. Actually, these methods becomes more and more popular now and in the future I will present the whole lecture about using the seismic noise as a source of input data for tomography. What we have as the output of this tomography processing. First of all, in maybe 90-95 percent of results is a distribution of seismic velocities, we have a P-wave velocity and the S-wave velocities which are together very important information to reveal some processes inside the earth. We can also add the here as a attenuation parameters or the quality factor its opposite, its reverse values. We can use a scattering parameters or we can use some emission parameters, in which a study volume emit some seismic signal and we can restore the distribution of this microseismicity which is also part of tomography problem. In some cases, we can also investigate the geometry of interfaces also inside the study volume although this is not a typical part of seismic tomography algorithm. We have a variety of output and the input data, how can we define the seismic tomography? Seismic tomography is a method for the determination of continuous, one-dimensional, two-dimensional, three-dimensional, or even four-dimensional in case if you have a time variation of our parameters. We have a continuous distribution of seismic parameters based on the parameters of seismic wave transmission through the study of volume, so we have different kind of parameters of seismic waves it might be attenuation or travel time that are transmitted through the study of volumes, study area, and we use these parameters to get the continuous model of the earth. This is a definition of seismic tomography. I think it would be useful to compare seismic tomography verse for example, medical tomography. Everybody of you knows medical tomography and maybe some of you passed the examination of using medical tomography and remembers this huge tube which is surrounding your body and you are illuminated from all direction by this tube and in this tube the x-rays emission is used, so the x-rays are the straight lines that are not affected by the density inside your body so you know exactly it's a path of rays. Because you have continuous distribution of rays around your body, you can use some sophisticated methods of the integral analysis that provides very fast and the high quality images of your bodies. But in seismic tomography, unfortunately is much more complicated. First of all, you should keep in mind that all your stations you can deploy only on the surface and your target areas allocated at some depths. You cannot put your stations, your receivers underneath this subject bodies. In this case you are very limited by this condition, and most important is that the ray path that you use in your case a seismic ray path, very strongly dependent on the velocity which is the unknown parameter of your tomography problem, so it means that you need to reconstruct the velocity distribution based on the unknown distribution of seismic rays which are dependent on this unknown velocity. Moreover,if you use it for example, earthquakes for your tomography. In this case, the location of earthquakes is also known, and this location is strongly dependent on the distribution of the velocity that you should recover, so there are too many unknowns in your case. If you come up with this problem to mathematician and say what we know we have accurate coordinates of seismic stations. This is more or less known the coordinates of seismic stations, and we have arrival times of seismic raise in absolute travel times in days, hours, minutes and seconds, so we have two types of data. We need to reconstruct the location of sources, the ray paths, which remains unknown and we need to reconstruct the P and S-velocity distribution three-dimensional, and of course the classical mathematician will say is that you are crazy and it is impossible to recover so many unknown from so few data. But nevertheless, such kind of problems are solved. There are some algorithm that in step-by-step, I will show how to do it by this course and finally we will get to a stage when we will use our algorithms to recover all these unknown parameters, and the maker high-quality tomography models despite of this problems.