39 2. Refraction surveying Picked times of direct and reverse shot points (Figure 2.3) give access to the t + and t − curves which allow the computation of the refractor velocity analysis function, tV and the generalized time-depth or delay time, tG respectively. The analysis of the travel time data proceeds in two stages, with the computation of seismic velocities followed by the depth computations. Formulae for the computation of the velocity analysis function tV and time-depths tG are given in Chapter 8 of “Refraction seismics”, (Palmer, 1986). The symbols used are defined in Figure 2.3. The refractor velocity analysis function, tV at position G (Figure 2.3-a), is defined by the equation: t t t t t V G AY BX AB = = − + ( ) − 1 2 1 2 (2.3) This function is computed for each pair of forward and reverse arrival times, tAY and tBX and the reciprocal time, tAB. The value of the function is referenced to G, which is midway between X and Y, and it is plotted as a function of the distance AG. Equation (2.3) is a linear relation between tV and the distance AG. Considering a multi-layer model with a plane dipping interface, the slope or gradient of this equation is taken as the inverse of an apparent velocity, ′ Vn, where: d dx t V V n ⋅ = ′ 1/ If dip angles are reasonable, and appear planar over a Fresnel zone, the relation between the true refractor velocity Vn and the apparent velocity, ′ Vn, is: V V n n n ≈ ′ − cosθ 1 with θn−1 the true dip angle of layer n–1 In general, ′ Vn is usually taken as the true refractor velocity. The generalized time-depth, tG at position G (Figure A1-b), is defined by: t t t t t XY V G G AY BX AB n = = + − + ′ ( ) ( ) + 1 2 1 2 / (2.4) The relationship between layer thicknesses and the generalized time-depth is: t Z V G j n jG jn jn j = + ( ) = − ∑1 1 2 cos cos / α β (2.5) where αjn, βjn are the ray path angles of incidence at interface j, Vj is the interval velocity of layer j, and ZjG is the thickness of layer j at surface position G. The depth conversion can be conveniently approximated with the zero-dip expression: t Z V G jG jn j j n = ( ) = − ∑ cos / φ 1 1 with sin / φjn j n V V ( ) = (2.6) where φjn are the ray path angles of incidence at interface j if the dip angle θj of layer j is 0.
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