The plus-minus method

Seismic Data Analysis

Series	Investigations in Geophysics
Author	Öz Yilmaz
DOI	http://dx.doi.org/10.1190/1.9781560801580
ISBN	ISBN 978-1-56080-094-1
Store	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 x_c (Figure 3.4-11a) for a reliable estimate of the weathering velocity or thickness. In most cases, v_w cannot be measured and a reasonable value is assumed for it.

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

$t_{+}=t_{ABCD}+t_{DEFG}-t_{ABFG}$

(47a)

and

$t_{-}=t_{ABCD}-t_{DEFG}+t_{ABFG}.$

(47b)

• 

  Figure 3.4-11  (a) Geometry for refracted arrivals. Here, v_w = weathering velocity, v_b = bedrock velocity, z_w = depth to the refractor equivalent to the base of the weathering layer, θ_c = critical angle, and x_c = crossover distance. The direct wave arrival has a slope equal to 1/v_w and the refracted wave arrival has a slope equal to 1/v_b. (b) A shot record that exhibits the direct wave and the refracted wave depicted in (a). (c) Geometry for a dipping refractor with forward traveltime profile associated with the direct wave and refracted wave arrivals, and (d) with both forward and reverse traveltime profiles. See text for details.

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  Figure 3.4-12  (a) Geometry for the plus-minus method. (b) Geometry for the generalized reciprocal method. Here, z_w 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, v_w is the weathering velocity, and θ_c is the critical angle of refraction.

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)

$t_{+}={\frac {2z_{w}{\sqrt {v_{b}^{2}-v_{w}^{2}}}}{v_{b}v_{w}}}.$

(48a)

Rewrite equation (41a)

$z_{w}={\frac {v_{b}v_{w}t_{i}}{2{\sqrt {v_{b}^{2}-v_{w}^{2}}}}}.$

(41a)

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

(48b)

and note that the plus time t₊ in equation (48a) is identical to the intercept time t_i in equation (48b). Hence, instead of measuring t_i directly from the shot record, Hagedoorn’s method suggests estimating t_i 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 v_b by

$t_{-}=t_{+}+{\frac {2x}{v_{b}}},$

(48c)

where x is the source-receiver separation AD.

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 t_i (equations 48a and 48b) and bedrock velocity v_b (equation 48c),
4. Assuming a value for weathering velocity v_w,
5. Computing the depth z_D to bedrock below station D (Figure 3.4-12a) from equation (41a), and
6. Computing the shot-receiver static shift Δτ_D at that station by

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

(49)

where E_S and E_D are the surface and datum elevations at station D (Figure 3.4-12a). If there is a shot at station D, Δτ_D 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.

See also

• First breaks
• Field statics corrections
• Flat refractor
• Dipping refractor
• The generalized reciprocal method
• The least-squares method
• Processing sequence for statics corrections
• Model experiments
• Field data examples
• Exercises
• Topics in moveout and statics corrections

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