Backprojection versus backpropagation in multidimensional linearized inversion

Geophysics ◽  
1989 ◽  
Vol 54 (7) ◽  
pp. 921-926 ◽  
Author(s):  
Cengiz Esmersoy ◽  
Douglas Miller
Keyword(s):  
Wave Field ◽  
Image Point ◽  
Common Tangent ◽  
Seismic Migration ◽  
Reflection Time ◽  
Digital Methods ◽  
Data Point ◽  
Surface Passing

Seismic migration can be viewed as either backprojection (diffraction‐stack) or backpropagation (wave‐field extrapolation) (e.g., Gazdag and Sguazzero, 1984). Migration by backprojection was the view supporting the first digital methods—the diffraction and common tangent stacks of what is now called classical or statistical migration (Lindsey and Hermann, 1970; Rockwell, 1971; Schneider, 1971; Johnson and French, 1982). In this approach, each data point is associated with an isochron surface passing through the scattering object. Data values are then interpreted as projections of reflectivity over the associated isochrons. Dually, each image point is associated with a reflection‐time surface passing through the data traces. The migrated image at that point is obtained as a weighted stack of data lying on the reflection‐time surface (Rockwell, 1971; Schneider, 1971). This amounts to a weighted backprojection in which each data point contributes to image points lying on its associated isochron.

Principle of prestack migration based on the full elastic two‐way wave equation

Geophysics ◽  
1987 ◽  
Vol 52 (2) ◽  
pp. 151-173 ◽  
Author(s):  
C. P. A. Wapenaar ◽  
N. A. Kinneging ◽  
A. J. Berkhout
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Field ◽  
Particle Velocity ◽  
Synthetic Data ◽  
Matrix Operator ◽  
Shear Stresses ◽  
Elastic Wave Equation ◽  
Seismic Migration ◽  
Prestack Migration

The acoustic approximation in seismic migration is not allowed when the effects of wave conversion cannot be neglected, as is often the case in data with large offsets. Hence, seismic migration should ideally be founded on the full elastic wave equation, which describes compressional as well as shear waves in solid media (such as rock layers, in which shear stresses may play an important role). In order to cope with conversions between those wave types, the full elastic wave equation should be expressed in terms of the particle velocity and the traction, because these field quantities are continuous across layer boundaries where the main interaction takes place. Therefore, the full elastic wave equation should be expressed as a matrix differential equation, in which a matrix operator acts on a full wave vector which contains both the particle velocity and the traction. The solution of this equation yields another matrix operator. This full elastic two‐way wave field extrapolation operator describes the relation between the total (two‐way) wave fields (in terms of the particle velocity and the traction) at two different depth levels. Therefore it can be used in prestack migration to perform recursive downward extrapolation of the surface data into the subsurface (at a “traction‐free” surface, the total wave field can be described in terms of the detected particle velocity and the source traction). Results from synthetic data for a simplified subsurface configuration show that a multiple‐free image of the subsurface can be obtained, from which the angle‐dependent P-P and P-SV reflection functions can be recovered independently. For more complicated subsurface configurations, full elastic migration is possible in principle, but it becomes computationally complex. Nevertheless, particularly for the 3-D case, our proposal has improved the feasibility of full elastic migration significantly compared with other proposed full elastic migration or inversion schemes, because our method is carried out per shot record and per frequency component.

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