# Fundamental Prerequisites of RTH Approach

The technique of integration by parts underlies the most important mathematical approaches, such as the definition of Lagrange conjugate operators, Bayes’ formulas, conjugate problem statements for differential equations etc. As it turned out, in physics, conjugate processes also underlie a number of known technologies such as time-reversal mirror for laser beam (Zel’dovich et.al.,1972) and time-reversed acoustics (Fink, 1997). In seismic prospecting, conjugate formulations for the wave equation are the basis of the well-known Reverse Time Migration (RTM) method (Baysal et.al., 1983; McMechan, 1983). In Full Wave Inversion method, conjugate mathematical formulation is used to calculate the Frechet derivative while minimizing misfit functional (Tarantola, 1984; Virieux, at.al., 2009). Alekseev and Erokhin, 1989 for the first time, mentioned the close connection between the essence of the conjugate approach for Simultaneous Joint Inversion (SJI) method, proposed by the authors and of time-reversal mirror for laser beam. They also marked the mathematical similarity of the SJI approach with the optimal control problem in ecology (Marchuk, 1976). In the same paper, the convergence of JSI solution on some weakly compact set and an increase in the stability of the solution on it were constructively proved.

In the paper Erokhin, 2019 a new Reverse Time Holography (RTH) method is proposed for design seismic attributes. This method combines two approaches: reversal of a wave in time based on the conjugate problem for the acoustic equation and two-beam interferometry, similar to that used in optical holography (Gabor, 1947). The data processing consists of two stages: decomposition and synthesis (Erokhin et.al., 2020). Decomposition stage includes a highly accurate vector decomposition of time-reversed seismic information which is recorded at the surface, for example by CDP method. At this stage, as in optical holography, a certain reference wave is also used for two-beam interferometry. The received information about the interference of the reference wave and time-reversed wave forms a set of vector pairs, which is called Vector Domain Common Image Gathers (VDCIG), (Erokhin et.al., 2018a). This digital dataset is similar a photographic plate for optical holography, on which the amplitudes and phases of two-beam interference are recorded. The second stage of RTH (synthesis) consists in statistical evaluation of the parameters of the multidimensional random distribution of VDCIG in order to obtaining the necessary seismic attributes. It turns out that such a formal mathematical approach makes it possible to construct by RTH method not only all known seismic attributes, but also to obtain much of new ones (Erokhin, 2019).

The method RTH is based on four scientific discoveries of the 20th century:

On the discovery at  1972  the physical media that reverses a laser beam in time (B.Y. Zel’dovich’s team)

Optical holography technology discovered at 1948 by D.Gabor

Development of medical technologies based on time reversal of acoustic waves (Mathias Fink)

Mathematical research of conjugate equations ( Marchuk G.I., Alekseev A.S., and Erokhin G.N.  )