Trace the strike-slip fault along the southwest–northeast valley, approximately through $(u,v)=(0.12,0.83),(0.28,0.75),(0.44,0.68),(0.53,0.62),(0.65,0.42),(0.83,0.20),(0.97,0.04)$. The long oblique strand is interpreted as left-lateral strike-slip faulting: draw the southeastern side moving northeast and the northwestern side moving southwest, parallel to the trace.

This sense is consistent with horizontal north–south maximum compression: for an approximately northeast-striking fault, the resolved shear stress moves the southeastern side northeastward relative to the northwestern side. The deflected central drainage supports that interpretation. The regional stress direction supplies a consistency check rather than a uniquely precise trace location; follow the geomorphic break rather than drawing a single straight line through every coordinate.

A reverse fault can create a sharp mountain front by raising its hanging wall beside a lower footwall. Drainages entering the rising block may be deflected along the front or cut narrow gorges across it. In cross-section, draw a dipping fault with the hanging wall displaced up-dip and a surface-height contrast above the trace.

In Attachment A, mark the approximately east–west to west-northwest–east-southeast break between the southwestern mountain block and the central basin. Its approximate image coordinates run from $(0.03,0.56)$ through $(0.18,0.61)$ and $(0.34,0.65)$ to $(0.45,0.69)$. The abrupt topographic change and diversion of drainage along this front support the reverse faulting interpretation. It is the southern margin of the central basin, not the much straighter oblique valley farther northeast. The original interpretive figure gives a schematic section.

A displaced drainage forms when repeated horizontal fault slip offsets a channel and its upstream and downstream reaches no longer line up. In plan view, draw two initially continuous channel reaches crossing the fault, translate one side parallel to the fault strike, and connect the reaches by a short fault-parallel reach. A linear valley develops where erosion follows damaged rock along a straight fault zone.

In Attachment A, use the oblique, relatively straight valley running southwest–northeast through the lower centre towards the upper right. For an identifiable example, follow the central approximately north–south drainage to its bend near the southern edge of the central basin, about $(u,v)=(0.46,0.67)$, where $u$ is fraction from the left image edge and $v$ fraction from the top. Mark its deflected reach and the adjacent straight valley. The positional precision is at the level of the supplied image; establishing a numerical fault offset would require correlated piercing points and field checks.

For $T_n=A\cos (k_{n}x)e^{-\kappa k_n^{2}t}$, direct differentiation again gives ${\partial }_t{T}_n=-\kappa k_n^2{T}_n$ and ${\partial }_x^2{T}_n=-k_n^2{T}_n$. Thus it satisfies the heat equation. Moreover,vanishes at both endpoints for integer $n$, proving the homogeneous Neumann boundary conditions. Now $n=0$ gives the nonzero constant mode $T=A$, which does not decay. It represents conserved mean temperature in an insulated plate.

Put $k_n=n\pi /L$ and ${\lambda }_n=\kappa k_n^2$. Direct differentiation of $T_n=A\sin (k_{n}x)e^{-{\lambda }_{n}t}$ givesso ${\partial }_t{T}_n=\kappa {\partial }_x^2{T}_n$, the heat equation. At $x=0$, the sine factor vanishes; at $x=L$, it is $\sin (n\pi )=0$ for integer $n$. Thus the homogeneous Dirichlet boundary conditions hold. The nonzero, nonredundant spatial modes use $n=1,2,\dots$; $n=0$ is the zero function and negative $n$ only change the amplitude sign.

By Fourier's law, $q_x=-k{T}_x$. With nonzero thermal conductivity $k$, zero outward heat flux at both endpoints gives homogeneous Neumann boundary conditions:The outward normal has opposite signs at the two ends, but a zero-flux condition gives zero derivative at each.

For $t>0$, fixed zero endpoint temperatures are homogeneous Dirichlet boundary conditions:An initial condition is also needed to determine a particular solution of the heat equation.

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.

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.

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.

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]

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.

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.

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.

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.

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.

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.

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.

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 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.