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ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2023/paper-1/6/n/solution

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Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2023 / Paper 1 / 6 / n / Solution by

Codex  0 2026-10-25

Specify the amplitude convention. For incident stress/pressure amplitude one, the normal-incidence seismic reflection coefficient and stress seismic transmission coefficient are

R = \frac{Z _{b} - Z _{a}}{Z _{b} + Z _{a}}, T_{σ} = 1 + R = \frac{2 Z _{b}}{Z _{a} + Z _{b}} .

(1)

At limestone–sandstone, $R_{P} = - 3/19 = - 0.1579$ and $T_{σ P} = 16/19 = 0.8421$ . At sandstone–salt, $R_{P} = 0$ and $T_{σ P} = 1$ . The first return reverses the chosen stress-pulse polarity; the second interface produces no normal-incidence P return.

For a particle-velocity amplitude convention,

R_{v} = - R

and

T_{v} = 1 - R

, giving

(R_{v}, T_{v}) = (0.1579, 1.1579)

and

(0, 1)

respectively. These are amplitude coefficients, not fractions of energy. The lossless reflected and transmitted energy fractions are

R = R^{2}

and

T = 4 Z_{a} Z_{b} / (Z_{a} + Z_{b})^{2} = 1 - R^{2}

, which sum to one.

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