Seismic Refraction Over a Low-Velocity Layer — Ray Paths, Snell's Law and Interpretation: 18-Geol-A7 Applied Geophysics, December 2018, Q2

Question 2: Seismic Refraction Over a Low-Velocity Layer — Ray Paths, Snell's Law and Interpretation (16.66% of paper)

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.

Survey layout and ray paths. A single source (hammer/plate, weight-drop, or a small explosive charge) is placed at one end of a straight line of geophones spaced at regular intervals; the shot is fired and every geophone records the ground motion arrival time. For a two-layer model (velocity $V_1$ in the surface layer, thickness $h_1$, over a higher-velocity half-space $V_2>V_1$), three ray paths matter: the direct wave, travelling along the surface from source to receiver at velocity $V_1$; the reflected wave, bouncing off the layer-2 interface; and the head wave (critically refracted wave), which travels down to the interface at the critical angle, along the interface at the higher velocity $V_2$, then back up to the surface at the critical angle — it is this head wave whose first-arrival times are inverted for $V_1$, $V_2$ and $h_1$.

Two-layer refraction geometry: direct wave (dashed, along the surface at V1), wide-angle reflection off the interface (green dashed) and head wave (down at the critical angle θc, along the interface at V2, back up at θc) recorded by geophones G1–G4.

Snell's law and the critical angle. At an interface separating velocities $V_1$ and $V_2$, an incident ray of angle $\theta_1$ (from the normal) and a transmitted ray of angle $\theta_2$ obey

$$\dfrac{\sin\theta_1}{V_1}=\dfrac{\sin\theta_2}{V_2}$$

The head wave exists only when $V_2>V_1$, and only along the special raypath for which the refracted ray travels exactly along the interface, i.e. $\theta_2=90^\circ$, $\sin\theta_2=1$. Substituting into Snell's law gives the critical angle directly:

$$\sin\theta_c=\dfrac{V_1}{V_2}\quad\Rightarrow\quad\theta_c=\sin^{-1}\!\left(\dfrac{V_1}{V_2}\right)$$

Any ray leaving the source at exactly $\theta_c$ is refracted along the interface, continuously radiating energy back up into layer 1 at the same angle $\theta_c$ (Huygens' principle) — this radiated energy is the head wave recorded at surface.

Arrival-time curves and interpretation. The direct wave arrives at $t_{direct}=x/V_1$ — a straight line through the origin of slope $1/V_1$. The head wave arrives at

$$t_{head}=\dfrac{x}{V_2}+\dfrac{2h_1\cos\theta_c}{V_1}$$

a straight line of slope $1/V_2$ with a non-zero intercept time $t_i=2h_1\cos\theta_c/V_1$ at $x=0$. On a time-distance plot the two lines cross at the crossover distance $x_{cross}$, beyond which the head wave (higher apparent velocity, shallower slope) overtakes the direct wave as the first arrival.

Time-distance plot: direct-wave and head-wave first-arrival branches, crossing at the crossover distance.

Two independent, cross-checking routes recover the model: (1) fit straight lines to the two branches by least squares; the reciprocals of their slopes give $V_1$ and $V_2$ directly, and the head-wave intercept time gives $h_1=t_iV_1/(2\cos\theta_c)$; (2) the crossover distance obeys $x_{cross}=2h_1\sqrt{(V_2+V_1)/(V_2-V_1)}$, which can be solved for $h_1$ once $V_1,V_2$ are known from the slopes, as an independent check on the intercept-time result.

Circumstances and purpose. Refraction surveying requires velocity to increase monotonically with depth (a hidden low-velocity layer beneath a higher-velocity layer produces no head wave and is invisible — the velocity-inversion or "hidden layer" problem; a layer too thin to give a first arrival is likewise missed, the related "blind zone" problem), a condition well satisfied by the common near-surface case of soil/overburden over bedrock. It is routinely used in geotechnical practice to map depth to bedrock or the water table along a proposed road, dam or foundation alignment, to estimate rippability from the recovered rock velocity, and for shallow groundwater/aquifer-base mapping, because it is fast, inexpensive, and well suited to the simple 1–2 layer geometries typical of engineering-scale site investigation.

Concept & Theory

Topic. Two-layer seismic refraction: ray geometry, Snell's-law critical angle, and the intercept-time/crossover-distance interpretation of a refraction profile.

Key relations. Snell's law $\sin\theta_1/V_1=\sin\theta_2/V_2$; critical angle $\sin\theta_c=V_1/V_2$; head-wave arrival $t_{head}=x/V_2+2h_1\cos\theta_c/V_1$; crossover distance $x_{cross}=2h_1\sqrt{(V_2+V_1)/(V_2-V_1)}$.

Why this works. Beyond the crossover distance the head wave, though it travels a longer physical path, arrives first because most of that path is covered at the much higher velocity $V_2$ — the same reason critically-refracted energy, not the direct or reflected wave, defines the far-offset first-arrival branch that the refraction method actually interprets.

Common pitfall. Assuming refraction "sees" every layer present — a layer that is too thin, or whose velocity is not itself an increase over the layer above (the low-velocity-layer/blind-zone problem), never produces a first-arrival head wave and is silently missed by a straightforward slope-intercept interpretation.

Source. Telford, Geldart & Sheriff, Applied Geophysics, Ch. 4 (seismic refraction); Kearey, Brooks & Hill, Introduction to Geophysical Exploration, Ch. 5.

Question 2 of 10: Seismic Refraction Over a Low-Velocity Layer — Ray Paths, Snell's Law and Interpretation

Question 2: Seismic Refraction Over a Low-Velocity Layer — Ray Paths, Snell's Law and Interpretation (16.66% of paper)