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Following the determination of the refractor velocity, the next step in defining undulating refractors is the formation of generalized time-depth functions at each geophone location. The generalized time-depth function in refraction interpretation corresponds with (but is not identical to) the one-way traveltimedepth function in reflection methods.

Using the symbols of Figure 2, the generalized time-depth tG(hereafter referred to as “time-depth”) at G is defined by the equation

Several special cases of the generalized time-depth can be derived, depending upon the XY separation used.

For XY equal to zero, the conventional time-depth (Hagiwara and Omote, 1939, p. 127; Hawkins, 1961, p. 807, eq. 3; Dobrin, 1976, p. 218, eqs. 7-35, 7-36) is obtained. It is similar to the plus term in the plus-minus method (Hagedoorn, 1959; Hawkins, 1961, p. 814) and to a term in the method of differences (Heiland, 1963, p. 549, eq. 9-68).

For the calculation of the conventional time-depth, no knowledge of the refractor velocity is required.

For XY selected such that the forward and reverse rays emerge from near the same point on the refractor, a result similar to the mean of the migrated forward and reverse delay times, as defined by Barry (1967, p. 348), is obtained.

The delay time method was first described by Gardner (1939, 1967), and has been developed by many others (Barthelmes, 1946; Wyrobek, 1956; Bernabini, 1965; Layat, 1967; Peraldi and Clement, 1972). Although all theoretical derivations assume negligible dip angles, the method is generally considered valid for dips less than 10 degrees.

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