The plus–minus method, also known as CRM (conventional reciprocal method), is a geophysical method to analyze seismic refraction data developed by J. G. Hagedoorn. It can be used to calculate the depth and velocity variations of an undulating layer boundary for slope angles less than ~10°.
Theory
In the plus–minus method, the near surface is modeled as a layer above a halfspace where both the layer and the halfspace are allowed to have varying velocities. The method is based on the analysis of the so-called 'plus time' t + {\displaystyle t^{+}} and 'minus time' t − {\displaystyle t^{-}} that are given by:
t + = t A X + t B X − t A B {\displaystyle t^{+}=t_{AX}+t_{BX}-t_{AB}}
t − = t A X − t B X − t A B {\displaystyle t^{-}=t_{AX}-t_{BX}-t_{AB}}
where t A B {\displaystyle t_{AB}} is the traveltime from A to B, t A X {\displaystyle t_{AX}} the traveltime from A to X and t B X {\displaystyle t_{BX}} the traveltime from B to X. Assuming that the layer boundary is planar between A'' and B'' and that the dip is small (<10°), the plus time t + {\displaystyle t^{+}} corresponds to the intercept time in classic refraction analysis and the minus time t − {\displaystyle t^{-}} can be expressed as:
t − = t + + 2 x v 2 {\displaystyle t^{-}=t^{+}+{\frac {2x}{v_{2}}}}
where x {\displaystyle x} is the offset between A and X and v 2 {\displaystyle v_{2}} is the velocity of the halfspace. Therefore, the slope of the minus time △ t − / △ x {\displaystyle \triangle t^{-}/\triangle x} can be used to estimate the velocity of the halfspace v 2 {\displaystyle v_{2}} :
v 2 ( x ) = 2 △ x △ t − {\displaystyle v_{2}(x)=2{\frac {\triangle x}{\triangle t^{-}}}}
The interval △ x {\displaystyle \triangle x} over which the slope is estimated should be chosen according to data quality. A larger △ x {\displaystyle \triangle x} results in more stable velocity estimates but also introduces stronger smoothing. Like in classical refraction analysis, the thickness of the upper layer can be derived from the intercept time t + {\displaystyle t^{+}} :
z ( x ) = t + v 1 ( x ) v 2 ( x ) 2 v 2 2 − v 1 2 {\displaystyle z(x)={\frac {t^{+}v_{1}(x)v_{2}(x)}{2{\sqrt {v_{2}^{2}-v_{1}^{2}}}}}}
This requires an estimation of the velocity of the upper layer v 1 ( x ) {\displaystyle v_{1}(x)} which can be obtained from the direct wave in the traveltime diagram. Furthermore, the results of the plus–minus method can be used to calculate the shot-receiver static shift △ τ ( x ) {\displaystyle \triangle \tau (x)} :
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