# The generalized reciprocal method

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Figure 3.4-12  (a) Geometry for the plus-minus method. (b) Geometry for the generalized reciprocal method. Here, zw is the depth to the refractor at the surface station where the plus-minus times as for (a) and intercept times as for (b) are to be estimated, vw is the weathering velocity, and θc is the critical angle of refraction.

In practice, raypaths that are suitable for first-break picking and are coincident at station D are not always found. [1] generalized Hagedoorn’s method of using raypaths (Figure 3.4-12b). Palmer’s technique, the generalized reciprocal method (GRM), takes into account offset separation D1D2 when computing the plus time

 ${\displaystyle t_{+}=t_{ABCD_{2}}+t_{D_{1}EFG}-t_{ABFG}-{\frac {D_{1}D_{2}}{v_{b}}}.}$ (50a)

The definition of the minus time remains the same as in equation (47b), except for accounting for the raypath geometry in Figure (3.4-12b):

 ${\displaystyle t_{-}=t_{ABCD}-t_{DEFG}+t_{ABFG}.}$ (47b)

 ${\displaystyle t_{-}=t_{ABCD_{2}}-t_{D_{1}EFG}+t_{ABFG}.}$ (50b)

Note that more than one combination of raypaths associated with different separations of D1D2 can be used to measure (pick) the traveltimes on the right sides of equations (50a, 50b). Consequently, there is more than one estimate of the plus-minus times at a given (shot-receiver) station D. By carefully editing the first breaks, these estimates can be refined and reduced to a single estimate for each station.

To derive the near-surface model, the generalized reciprocal method uses the observed traveltimes from refracted arrivals that are assumed to be associated with the base of weathering. A problem arises when a nearsurface model with more than one layer needs to be defined. This is the case in areas covered with glacial tills and sand dunes. Several specialized techniques based on generalized linear inversion (GLI) have been devised for these problems [2]; [3]. The GLI technique is an iterative, model-based approach that provides flexibility in defining a near-surface model consisting of arbitrarily parameterized multilayers. The process begins by computing the refracted arrival times from an assumed initial near-surface model. These computed traveltimes then are compared with the actual first-break picks (observed traveltimes). The procedure tries to minimize the difference between the computed and observed traveltimes by iteratively modifying model parameters for the near surface (such as velocities and thicknesses). A GLI method applicable to a single-layer near-surface model is presented next.