MODELING OF SEISMIC DATA PROCESSING USING INTEGRAL GEOMETRY PROBLEMS ON A FAMILY OF BROKEN LINES

THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 190 MODELING OF SEISMIC DATA PROCESSING USING INTEGRAL GEOMETRY PROBLEMS ON A FAMILY OF BROKEN LINES A.K.Seidulleav1,2, S.S.Nizamatdinova1 1Karakalpak State University, Nukus, Uzbekistan 2V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan https://doi.org/10.5281/zenodo.17740297 Integral geometry problems provide a mathematical framework extensively applied in seismological data processing. It involves the transformation of seismic waveforms into alternative domains to enhance analytical tractability and interpretative clarity. This approach is particularly effective for detecting and characterizing linear features within seismic datasets, which are indicative of constant ray parameters and play a critical role in subsurface imaging. The Radon transform, commonly referred to as the slant stack, is a mathematical technique employed to reproject seismic data from the conventional time-offset domain into the p  − domain, where  denotes the intercept time and p represents the ray parameter [1]. This transformation facilitates the alignment of seismic traces along trajectories of constant slowness, thereby enhancing the interpretability of wavefield characteristics. It is particularly advantageous for analyzing high-resolution reflection and refraction datasets, especially those acquired from linear source geometries in horizontally layered media. The method proves effective in isolating and interpreting near-vertical reflection events, contributing to improved subsurface imaging and velocity model estimation [2]. This study addresses the inverse problem of reconstructing an unknown function ( , )u x y , which represents the spatial distribution of a wave field from observed seismic measurements ( , )f x y . The problem is formulated within the framework of integral geometry, specifically over a family of broken-line trajectories. An analytical relationship is derived that connects the integral transforms of the unknown wave field function ( , )u x y with the measured data ( , )f x y . Furthermore, it is demonstrated that the solution can be expressed through the application of the Laplace operator to the known function ( , )f x y , providing a pathway for efficient computational recovery. We investigate a class of integral geometry problems characterized by the equation ( ) ( ) ( ) ( ) , ,, xy y u ds f x y     −=  where ( ) ,u  is the unknown function representing the wave field distribution, and ( , )f x y is a known function derived from seismic measurements. The integration is performed along the curve ( , )xy  , defined by the condition: ( )   , , ,0x y x y x R y H     = − = −     where ds denotes the differential arc length along the curve. This formulation models the propagation of seismic waves along broken-line trajectories and serves as a foundation for THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 191 reconstructing the underlying wave field from integral measurements. Theorem. Let ( , )f x y be a prescribed function defined on the domain [0, ) H LH= , and let ( , )u x y denote an unknown function that is finite on H L and satisfies the integral equation ( , ) ( ) ( , ) ( , ), xy y u ds f x y     −=  where ( , )xy  represents a specific curve in the domain, and $ ds $ denotes the differential arc length along this curve. Under the assumption that 2 0 ( , ) ( ) H u x y C L the solution to this inverse problem is unique and can be explicitly expressed in terms of the known function ( , )f x y via the relation 2 22 22 2 ( , ) ( , ), 4 u x y f x y xy    = −    where 22 22 xy   = +  denotes the Laplace operator. This result provides a direct analytical mechanism for recovering the wave field distribution from integral measurements along broken-line trajectories. Numerical experience. To validate the theorem numerically, we considered a smooth test function ( ) ,u x y on a periodic domain  ) ) 0, 0, xy LL   with 2 x L= and 1 y L= , sampled on a 192 192 grid. The chosen function was a combination of Fourier modes: ( ) 2 4 4 6 , sin sin 0.6cos sin real x y x y x y x y u x y L L L L             =+                     . Using spectral differentiation, the Laplacian in Fourier space is: ( ) ( ) ( ) 22 ˆ ,, x y x y x y u k k k k u k k = − + , where 22 , xy xy mn kk LL  == for integer modes ,mn . Using FFT-based spectral differentiation, we computed u and then recovered ( ) ,f x y from the identity ( ) 2 22 xy u c f =  − where 2 4 c= . In Fourier space, the operator ( ) 2 22 xy  − becomes ( ) 2 22 yx kk− . So: ( ) ( ) ( ) 2 22 , ˆ,xy xy yx u k k f k k c k k  =− , for ( ) 2 22 yx kk  − , where  is a small cutoff to avoid division by zero. Next, we reconstructed explicit u by applying the explicit formula: Compute ( ) ( ) 2 22 ,xy g x y c f=  −  , in Fourier space THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 192 ( ) ( ) ( ) 2 22 ˆ ˆ,, x y y x x y g k k c k k f k k=− . Then we solve explicit ug= , so ( ) ( ) ( ) explicit 22 ˆ, ˆ,xy xy xy g k k u k k kk =−+ , setting the zero-mode to zero (zero-mean solution). We observe the numerical visualization of the aforementioned formulas using numerical methods as follows: Fig1. Heatmap of ( ) real ,u x y (zero mean) Fig2. Heatmap of ( ) explicit ,u x y Fig3. Error map real explicit uu− Fig4. Profile at 2 y yL= We subtract the mean from both fields and compute: ( ) 2 real explicit 1 RMSE u u N =−  , real explicit 22 real 2 Relative uu Lu − = , ( ) real explicit real explicit , Correlation Cov u u  = . The numerical experiment confirms the theoretical result with high accuracy. Starting from a prescribed smooth test function ( ) ,u x y , we computed the Laplacian u and ( ) ,f x y . THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 193 We then reconstructed explicit u by applying the operator ( ) 2 22 xy c − to f and solving the Poisson equation ug= spectrally. The comparison between the original real u and the reconstructed explicit u yielded an RMSE of approximately 13 1.4 10−  , a relative 2 L error of 13 2.4 10−  , and a correlation coefficient of nearly 1.0, indicating agreement to machine precision. Visualizations of both fields and their difference confirmed that the explicit formula provides an accurate and stable mechanism for recovering the unknown function from integral measurements. This demonstrates that the theoretical approach is numerically feasible and highly reliable when implemented with spectral methods on smooth periodic data. REFERENCES 1. Munadi, S. (2022). The hyperbolic radon transform and some of its application in seismic data processing. Scientific Contributions Oil and Gas, 15(1), 12–19. https://doi.org/10.29017/scog.15.1.1113 2. Chapman, C. H. (1981). Generalized Radon transforms and slant stacks. Geophysical Journal International, 66(2), 445–453. https://doi.org/10.1111/j.1365-246x.1981.tb05966.x