# The plus-minus method

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

It often is difficult to use first breaks to estimate the intercept time and velocities for the weathering layer and bedrock. This is primarily because the base of weathering typically is an undulating surface, which makes traveltime plots difficult to interpret. Traveltime plots also are affected by severe elevation changes. Additionally, a typical field cable layout does not provide a sufficient number of channels inside the crossover distance xc (Figure 3.4-11a) for a reliable estimate of the weathering velocity or thickness. In most cases, vw cannot be measured and a reasonable value is assumed for it.

[1] formulated a method to indirectly estimate intercept time and bedrock velocity. The method still requires picking the first breaks. However, it does not require interpreting the traveltime profile (Figure 3.4-11a). (Interpretation means drawing the linear segments for the direct and refracted waves.) Figure 3.4-12a shows three raypaths associated with shot-receiver pairs AD, DG, and AG. The basis of Hage-doorn’s method involves computing two time values, the plus and minus times given by

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

and

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

The times given on the right side of these equations are the measured (picked) values from the first breaks for the three raypaths shown in Figure 3.4-12a. From the raypath configuration, we find that (Section C.7)

 ${\displaystyle t_{+}={\frac {2z_{w}{\sqrt {v_{b}^{2}-v_{w}^{2}}}}{v_{b}v_{w}}}.}$ (48a)

Rewrite equation (41a)

 ${\displaystyle z_{w}={\frac {v_{b}v_{w}t_{i}}{2{\sqrt {v_{b}^{2}-v_{w}^{2}}}}}.}$ (41a)

 ${\displaystyle t_{i}={\frac {2z_{w}{\sqrt {v_{b}^{2}-v_{w}^{2}}}}{v_{b}v_{w}}},}$ (48b)

and note that the plus time t+ in equation (48a) is identical to the intercept time ti in equation (48b). Hence, instead of measuring ti directly from the shot record, Hagedoorn’s method suggests estimating ti from the first break picks given by the right-hand terms in equation (47a).

By applying algebra (Section C.7), we find that minus time t is related to bedrock velocity vb by

 ${\displaystyle t_{-}=t_{+}+{\frac {2x}{v_{b}}},}$ (48c)

Thus, Hagedoorn’s plus-minus method involves:

1. Picking the first breaks,
2. Computing the plus-minus times, t and t+ (equations 47a and 47b),
3. Deriving from the plus-minus times the intercept time ti (equations 48a and 48b) and bedrock velocity vb (equation 48c),
4. Assuming a value for weathering velocity vw,
5. Computing the depth zD to bedrock below station D (Figure 3.4-12a) from equation (41a), and
6. Computing the shot-receiver static shift ΔτD at that station by

 ${\displaystyle \Delta \tau _{D}=-{\frac {z_{D}}{v_{w}}}+{\frac {E_{D}-E_{S}+z_{D}}{v_{b}}},}$ (49)

where ES and ED are the surface and datum elevations at station D (Figure 3.4-12a). If there is a shot at station D, ΔτD represents the shot static, and if there is a receiver at station D, it represents the receiver static. Again, uphole and elevation corrections are needed before making the plus-minus statics corrections.