Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2022 / Paper 1 / 6

Section C: Geophysical Methods.

For an approximately isotropic elastic solid,where $K$ is bulk modulus, $\mu$ is shear modulus and $\rho$ is mass density. Five relevant controls are:

These controls interact: a dense rock is not automatically faster if its moduli are also different.

The reflected P wave remains in the upper layer. Conservation of horizontal seismic ray parameter gives $\sin i_r/V_{P1}=\sin 30^{\circ }/V_{P1}$, so the reflected P angle is $30^{\circ }$ from vertical. This is specular reflection at the horizontal interface.

For the converted reflected S wave, Snell law for elastic waves givesThus $i_{S1}=17.46^{\circ }$ from vertical, in the upper layer. The converted in-plane mode is SV; an isotropic planar P incidence does not generate an independent SH mode.

The required transmitted P wave angle would satisfyTherefore there is no real transmitted P-ray angle. The incidence exceeds the P critical angle $\arcsin (1500/3500)=25.38^{\circ }$; the transmitted P contribution is evanescent away from the interface rather than a propagating body ray. Reporting $90^{\circ }$ would describe critical incidence, not the stated $30^{\circ }$ incidence.

The transmitted converted S wave obeysHence $i_{S2}=41.81^{\circ }$ from vertical. A propagating transmitted SV mode can exist even though the transmitted P mode is already supercritical.

At P critical incidence the transmitted P wave travels along the interface. The common horizontal seismic ray parameter is $1/V_{P2}$, givingThe resulting seismic head wave radiates back into the slower upper layer.

For a converted lower-layer S wave head wave, set the transmitted S angle to $90^{\circ }$:This is possible because $V_{S2}>V_{P1}$. It is the incident P angle needed for the converted SV branch to become tangential, not an S incident angle in layer 1.

The normal-incidence acoustic impedance is $Z=\rho V_P$. ThereforeThese are $2.25$ and $7.00$ MRayl. They combine the material’s density and wave speed; using $V_S$ would give shear rather than acoustic P impedance.

Specify the amplitude convention. For normal-stress/pressure amplitudes, the seismic reflection coefficient and seismic transmission coefficient areFor particle displacement or velocity resolved in one fixed normal coordinate, the reflected amplitude changes sign and the transmitted ratio is insteadThese are alternative conventions, not conflicting energy predictions. The reflected and transmitted energy flux fractions arewhich sum to one for the ideal lossless interface. Thus an amplitude transmission coefficient greater than one does not violate conservation of energy.

An explosive seismic source gives an impulsive, potentially high-energy broad-band signal and can couple effectively to competent material. It can be useful where strong penetration is needed and heavy vibrating vehicles cannot reach the site. Disadvantages include destructive ground disturbance, permitting and specialist handling, variable source coupling and signature, and the cost of repeated source preparation. The comparison does not require a prescription for charge size or emplacement.

Vibroseis spreads controlled energy over a longer frequency sweep. It is repeatable, permits tailored bandwidth and repeated sweeps, and its known signal supports correlation processing and rejection of unrelated noise. It avoids the impulsive ground disruption of an explosive source. Disadvantages include vehicle access and terrain restrictions, dependence on baseplate coupling, limited force and low-frequency performance, possible harmonic distortion, and strong near-surface waves. For a fractured shallow geothermal site, test which source produces usable coherent reflections or arrivals in the desired band; nominal source energy alone does not establish data quality.

At each source position, couple the vibrator to the ground, transmit a known frequency sweep $s(t)$ and record the ground response on synchronized geophones. Record a pilot or measured source-force reference and sufficient listening time for delayed arrivals. Repeat sweeps where beneficial and retain geometry and timing information.

If the Earth’s impulse response is $h(t)$, a linear model is $d=s\ast h+n$. Vibroseis correlation cross-correlates the record with the known sweep:The sweep’s autocorrelation $R_{ss}$ is a compact, approximately zero-phase wavelet, so the long chirp is compressed into identifiable delayed events. In the frequency domain this multiplies $D(\omega )$ by $\overline{S(\omega )}$, yielding $H(\omega )|S(\omega ){|}^2$ plus noise. It is pulse compression, not an exact recovery of $h$ at frequencies never transmitted. Spectral shaping or stabilized deconvolution can further compensate the finite source bandwidth and sidelobes.^[ref]

Choose using target depth, scale, expected contrasts, overburden, noise and required result. A seismic reflection survey uses reflected travel times and amplitudes to image interfaces; it needs useful acoustic impedance contrasts, sufficient bandwidth, suitable offsets and dense coverage for moveout correction, stacking and seismic migration. A large clay-cap/igneous contrast gives a potentially strong reflection here, but fractures, scattering and attenuation can obscure coherent events. At $100$ Hz the nominal quarter-wavelength vertical seismic resolution is about $3.75$ m in the claystone and $8.75$ m in the igneous rock, if that frequency reaches the target.

A seismic refraction survey uses first arrivals and critically refracted waves to estimate velocities and interface structure. The increase from $1500$ to $3500\mathrm{m}{\mathrm{s}}^{-1}$ is favorable for a cap-base head-wave branch, provided the spread reaches sufficient offsets. Refraction/tomography may better constrain lateral velocity variations and fractures, but simple head-wave inversion can miss a low-velocity layer or a thin hidden layer and often has poorer boundary detail.

Use expected reflector continuity, velocity inversions, required aperture, access, near-surface complexity and cost to choose or combine the surveys. A pilot gather and travel-time modelling can test which phases will be measurable; the supplied velocities alone do not specify the reservoir depth or the best acquisition geometry.

The Nyquist–Shannon sampling theorem gives a Nyquist rate of $200$ samples per second for a $100$ Hz upper frequency. In practice choose $f_s>200$ Hz, for example $500$ Hz with $\Delta t=2$ ms, and use an anti-aliasing filter whose passband retains the required signal below the Nyquist frequency. Exactly $200$ Hz leaves no filter transition band and can entirely miss a $100$ Hz sinusoid at an unfortunate phase; it is a limiting rate, not a robust acquisition choice.

Choose shot and receiver intervals to sample the shortest relevant spatial wavelength and steepest event moveout without spatial aliasing. A simple bound isusing the smallest apparent velocity of the events one needs to retain or separate. Using $1500\mathrm{m}{\mathrm{s}}^{-1}$ and $100$ Hz alone suggests $7.5$ m, but slower surface waves or dipping events can demand finer spacing. It is not a universal final design interval.

Also consider target depth and lateral resolution, reflector dip, near-surface heterogeneity, source strength and coupling, maximum offset and aperture, common-midpoint bin size and seismic acquisition fold, refraction crossover distances, access and budget. Receiver spacing and source spacing jointly determine coverage and cannot be selected from temporal sampling alone.

First measure the generator’s spectrum, timing and propagation pattern on quiet records. If practicable, coordinate shutdowns or quiet acquisition windows; otherwise place source/receiver lines to reduce coupling to the noise source and use good ground coupling and adequate instrument dynamic range. Saturated channels cannot be repaired by a later filter.

Separate noise using its measured frequency and moveout: a band-pass/high-pass filter can reject frequencies below the useful seismic band, and an appropriate narrow notch can reject a narrow coherent component. Avoid deleting wanted low-frequency arrivals along with the generator noise. Dense spatial sampling enables frequency–wavenumber filtering or array methods to separate low-apparent-velocity surface noise from target arrivals. A reference sensor near the generator can support coherent-noise subtraction when the transfer is stable.

Repeated shots/sweeps and stacking improve the signal-to-noise ratio for uncorrelated noise; persistent coherent generator noise does not necessarily average down. Known-sweep Vibroseis correlation also helps reject unrelated noise. Verify the final processing against test gathers to ensure that real arrivals have not been mistaken for noise.