Plus–minus method

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°.

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°. t^+ = t_{AX} + t_{BX} - t_{AB} t^- = t_{AX} - t_{BX} - t_{AB} where t_{AB} is the traveltime from A to B, t_{AX} the traveltime from A to X and 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 ( t^+ corresponds to the intercept time in classic refraction analysis and the minus time t^- can be expressed as: t^- = t^+ + \frac{2x}{v_2} where x is the offset between A and X and v_2 is the velocity of the halfspace. Therefore, the slope of the minus time \triangle t^-/\triangle x can be used to estimate the velocity of the halfspace v_2 : v_2(x) = 2 \frac{\triangle x}{\triangle t^-} The interval \triangle x over which the slope is estimated should be chosen according to data quality. A larger \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^+ : 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) 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 \triangle \tau(x) : \triangle \tau(x) = - \frac{z(x)}{v_1(x)} + \frac{E_X - E_S + z(x)}{v_2(x)} where E_X is the datum elevation and E_S the surface elevation at station X.

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