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18-Geol-A7 Applied Geophysics · December 2014

Question 6 of 8: CMP Gather – Normal Moveout and the Dix Equation

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2014 — 04-Geol-A7 Applied Geophysics. Three-hour, open-book exam; any non-communicating calculator permitted. The NOTES state that SIX questions constitute a complete paper, but eight numbered questions are printed on the exam — all eight, and every lettered/numbered sub-part, are solved below. Two figures (the CMP gather of Q6 and the reversed-refraction time-distance plot of Q8) carry real numeric data that is only given graphically on the printed page; both were read from the printed figures, calibrated against each figure's own printed axes, and the reading tolerance is given in a check callout beside each calculation.

Reference texts: Telford, Geldart & Sheriff, Applied Geophysics (2nd ed.) — the primary reference for every method in this paper (gravity, magnetics, seismic refraction and reflection, electrical resistivity, induced polarization, electromagnetics); Kearey, Brooks & Hill, An Introduction to Geophysical Exploration (3rd ed.) — survey design and interpretation context; Blakely, Potential Theory in Gravity and Magnetic Applications — the dipole/sphere anomaly shapes used in Q2–Q3.

Question 6: CMP Gather – Normal Moveout and the Dix Equation (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Check: the CMP gather has no printed data table, so the two-way times were read from the printed figure. The five dashed reflector lines were read against the figure's own labelled time ticks and offsets against the ten trace positions (50–950 m). All five lines run to trace 10. The figure is hand-drawn, so each pick carries roughly ±5 ms reading uncertainty. Reading reflector 1 instead at the arrowhead marks on each trace gives $V_1\approx1950$ m/s, $h_1\approx89$ m, $V_2\approx2175$ m/s and $h_2\approx201$ m, within about 2–6% of the values below, which is the realistic precision of answers taken from this hand-drawn figure.

Given. A CMP gather with 10 traces at offsets $x=50,150,250,\dots,950$ m. Digitized two-way times for the first two reflectors:

Offset x (m)50150250350450550650750850950
Reflector 1, t (ms)110127161197236286339390440484
Reflector 2, t (ms)200208228257291326367411456497

Find. (i) Velocity $V_1$ of the first layer and depth $h_1$ to the first interface. (ii) RMS velocity $V_{rms2}$ to the base of the second layer, interval velocity $V_2$ of the second layer (Dix equation), and depth $h_2$ to the base of the second layer.

Approach. Fit each reflector's ten picks to the hyperbolic Normal Moveout equation $t^2=t_0^2+x^2/V^2$ by linear regression of $t^2$ against $x^2$ (slope $=1/V^2$, intercept $=t_0^2$). The fitted $V$ is the RMS (stacking) velocity down to that reflector, which for the first reflector is the layer-1 velocity itself. Then combine the two RMS velocities with the Dix equation to isolate layer 2's interval velocity.

  1. Reflector 1 — NMO fit. The regression over all ten picks gives slope $=2.512\times10^{-7}$ s$^2$/m$^2$, so $V_1=1/\sqrt{\text{slope}}=1995$ m/s $\approx\boxed{2000\ \text{m/s}}$, with zero-offset time $t_{0,1}=94.9$ ms.
  2. Depth to the first interface. $h_1=\dfrac{V_1t_{0,1}}{2}=\dfrac{1995\times0.0949}{2}=\boxed{95\ \text{m}}$.
  3. Reflector 2 — RMS velocity to its base. The same regression on reflector 2 gives $V_{rms2}=2074$ m/s $\approx\boxed{2070\ \text{m/s}}$, with $t_{0,2}=194.3$ ms. This is the RMS velocity of both layers together, not layer 2's own velocity.
  4. Dix equation — interval velocity of layer 2. $V_2=\left[\dfrac{V_{rms2}^2t_{0,2}-V_1^2t_{0,1}}{t_{0,2}-t_{0,1}}\right]^{1/2}=\left[\dfrac{2074^2\times0.1943-1995^2\times0.0949}{0.1943-0.0949}\right]^{1/2}=\boxed{2150\ \text{m/s}}$ (2146 m/s unrounded). Velocity increases with depth, as expected. The layer is only about 100 ms thick in two-way time, so the Dix result is sensitive to reading error: a 1% error in $V_{rms2}$ moves $V_2$ by about 1.8%.
  5. Depth to the base of layer 2. $h_2=h_1+\dfrac{V_2(t_{0,2}-t_{0,1})}{2}=94.7+\dfrac{2146\times0.0994}{2}=94.7+106.6=\boxed{201\ \text{m}}$.
QuantityValue
Velocity of layer 1, $V_1$≈ 2000 m/s (fit 1995 m/s)
Depth to first interface, $h_1$95 m
RMS velocity to base of layer 2, $V_{rms2}$≈ 2070 m/s
Interval velocity of layer 2 (Dix), $V_2$≈ 2150 m/s
Depth to base of layer 2, $h_2$201 m

(iii) The velocity spectrum. A velocity spectrum is computed by NMO-correcting the SAME CMP gather repeatedly, once for each of a dense range of trial stacking velocities $V_{trial}$ at every two-way (zero-offset) time $t_0$, and then measuring how well the moveout-corrected traces align across offset at that $(V_{trial},t_0)$ pair — typically using the SEMBLANCE coefficient (the ratio of the stacked trace's energy to the sum of the individual traces' energies, which is 1.0 for perfectly flat, in-phase arrivals and falls toward 0 for arrivals that are not properly flattened). Plotting semblance as a 2-D function of $(V_{trial},t_0)$ produces a contoured "velocity spectrum": wherever a real, coherent reflector exists, an NMO correction using the CORRECT stacking velocity flattens that reflector's hyperbola across all offsets, producing a local semblance PEAK at $(V_{trial}=V_{rms,n},\,t_0=T_{0,n})$ for that reflector.

Velocity spectrum (semblance contours)15001750200022502500050100150200250300350400450R1 (1995, 95)R2 (2074, 194)R3 (2048, 253)R4 (2088, 310)R5 (2136, 372)Trial stacking (RMS) velocity (m/s)Zero-offset two-way time t0 (ms)
Velocity spectrum sketched from the digitized gather: each semblance peak plots at the $(V_{rms},t_0)$ from a $t^2$–$x^2$ fit of that reflector's ten picks — R1 (1995 m/s, 95 ms), R2 (2074, 194), R3 (2048, 253), R4 (2088, 310), R5 (2136, 372).

The peaks follow a gently increasing velocity trend, from about 2000 m/s at 95 ms to about 2140 m/s at 372 ms. R3 plots slightly to the left of R2, which is within the roughly ±50 m/s scatter of picks from a hand-drawn gather, so an interpreter would draw one smooth stacking-velocity function through the five peaks. Each peak is elongated along the velocity axis (velocity resolution is poorer than time resolution), and the deeper peaks are broader in velocity because their moveout is smaller.