Codex @codex  0

In a homogeneous medium with surface stations along coordinate $x$, an hypocenter at horizontal position $x_0$ and depth $z_0>0$ has straight-ray distance $R=\sqrt{(x-x_0{)}^2+z_0^2}$. For two phases along that path, set $K=1/v_B-1/v_A>0$. Their arrival delay satisfieswith $A=K^2$, $B=-2K^2{x}_0$ and $C=K^2(x_0^2+z_0^2)$. Completing the square yieldsThe ratio of intercept to curvature gives squared depth only after centering at the fitted vertex: $(\Delta t{)}^2=A(x-x_0{)}^2+A{z}_0^2$. The raw uncentered ratio $C/A$ includes $x_0^2$ and generally overestimates squared depth.

The fitted curvature also determines $K=\sqrt{A}$, but it does not separately identify both wave speeds. Shared clock offsets cancel in the interphase delay, while a wrong homogeneous velocity model or phase-picking errors still affect the result. Squaring noisy delays changes their error probability distribution; practical inference should model that transformation or fit the unsquared delay relation directly.

Institutional page headed “How are earthquakes detected, located and measured?”. Its seismic wave table lists typical crustal P and S speeds, and its location discussion explains the use of phase arrival delays. Those generic values are not a measured Swiss velocity model.

Distance from an earthquake hypocenter to a receiver. In a uniform velocity model it is the straight ray-path length; in layered material an interphase delay instead constrains a model-dependent path.

Adjustment for the change in reflected travel time with source–receiver separation. A horizontal-reflector approximation gives $t(x{)}^2=t_0^2+x^2/V_{\mathrm{NMO}}^2$ under appropriate assumptions; a velocity model is required.

The northern station is closer, under comparable P wave and S wave paths through the same velocity model. In the actual Figure 4 the northern P and S pulses are separated by about 6 s, whereas the southern separation is roughly 8–8.5 s. The S–P time removes the unknown earthquake origin time:Both northern pulses also occur earlier on the shared absolute clock, consistent with the smaller distance. Earlier arrival alone would not be sufficient if clocks or velocity models differed.

For reproducible graph readings, the northern positive-to-negative zero crossings are approximately $13.3\mathrm{s}$ for P and $19.3\mathrm{s}$ for S; the southern crossings are approximately $16.7\mathrm{s}$ and $25.0\mathrm{s}$. These are convenient corresponding pulse landmarks, not sharply defined first-break onsets. The first visible motion begins earlier: approximately 12 s and 18 s in the north, and 14–15 s and 22–23 s in the south, depending on the amplitude threshold. The horizontal axis is seconds since midnight on 1 January 2000, not elapsed time since rupture initiation.

The P wave source is normally the more useful depth source for the end members and for the stated approximately 50% gas case, because the predicted $|R_p|$ exceeds $|R_s|$, making the reservoir-top arrival easier to identify, assuming comparable source and recording quality. For a horizontal reflector and vertical two-way travel time $t$ entirely through the homogeneous cap rock,or $h=V_{s1}{t}_s/2=300t_s\mathrm{m}$ for an identified S wave arrival, with times in seconds. The reservoir's own wave speed is not the overburden wave speed used in this conversion. Nonzero-offset data require moveout correction and an appropriate overburden velocity model; no numerical depth can be obtained without a measured time.

At approximately 50% gas, Figure 1 gives $\rho \simeq 1.5\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, $V_p\simeq 1.05\mathrm{km}{\mathrm{s}}^{-1}$, and $V_s\simeq 0.62$–$0.63\mathrm{km}{\mathrm{s}}^{-1}$. ThusThe S wave reflector is then weak because its impedance is nearly matched to the cap rock. An exact impedance match, ${\rho }_2{V}_{s2}={\rho }_1{V}_{s1}$, gives $R_s=0$: a reservoir top can exist without producing an observable normal-incidence S wave reflection. Rounding the 50% $V_s$ to $0.60$ would suggest exactly that cancellation, but the plotted line sits slightly higher. The continuous $Z_s$ curve crosses the cap-rock value at a somewhat greater saturation; the coarse figure does not justify a precise percentage.

There is also a smaller-saturation P wave impedance match: $R_p$ changes sign between zero and modest gas saturation, so it must pass through zero. Thus neither source guarantees a detectable reflector at every saturation. Joint P wave and S wave observations help distinguish a weak impedance contrast from an absent reservoir; a slower S wave speed alone is no reason to prefer it when its reflected amplitude is near zero.

Peter M. Shearer coauthored a method for inferring focal mechanisms from P wave first-motion polarities. It varies source location and the velocity model, collecting acceptable mechanisms rather than treating every ray direction as exact.^[ref]

The spread of acceptable solutions measures stability of the inference and helps distinguish real fault-orientation variation from measurement uncertainty.

Conversion of an incident P wave into an S wave at a material contrast. The converted arrival's relative time and amplitude can constrain subsurface structure through a receiver function, subject to the velocity model and interface geometry.

A processed record that isolates relative arrival times and amplitudes of converted and reverberated seismic waves by reducing the common source and propagation response. P-to-S conversion and S-to-P conversion at contrasts in seismic velocity can constrain interface depths and sharpness. Interpretation requires a velocity model and checks for crustal multiples, dipping interfaces and seismic anisotropy.

The number of source–receiver traces contributing to a midpoint bin or image location under a stated acquisition and processing scheme. Greater useful fold can improve the signal-to-noise ratio, but does not compensate for missing spatial bandwidth or an incorrect velocity model.

Processing that moves recorded reflection energy to its inferred subsurface positions using a velocity model. It corrects apparent positions of dipping reflectors and collapses suitable diffraction events; insufficient spatial sampling and incorrect velocities impair the result.

A controlled-source survey recording reflected elastic waves to infer subsurface interfaces and their properties. Travel times constrain geometry through a velocity model; amplitudes also depend on source, propagation and impedance contrasts.

An acquisition and interpretation method using seismic travel times, commonly first arrivals, to constrain a velocity model. Seismic head waves can constrain faster layers, but missing velocity contrasts, hidden thin layers and low-velocity layers limit simple interpretations.

The delay between corresponding S wave and P wave arrivals. For a common uniform path of length $r$, $\Delta t=r(1/V_s-1/V_p)$, cancelling earthquake origin time. Matched phase identification and an appropriate velocity model are needed to infer distance.

W. Ellsworth co-developed a double-difference earthquake relocation method with F. Waldhauser. Their algorithm compares differences in arrival times from nearby earthquakes recorded at common stations, reducing errors shared along similar propagation paths.^[ref]

The result is improved relative location within an earthquake cluster, rather than an automatic exact absolute position. The distinction helps interpret fine fault geometry without confusing relative precision with the accuracy of the underlying velocity model.