By the end of this lecture…

• [LO-7.1] Derive the $N$-layer travel-time generalization and the dipping-interface equations
• [LO-7.2] Explain why low-velocity zones and thin layers are invisible to refraction surveys
• [LO-7.3] Apply the delay-time method to map irregular refractors from reversed profiles
• [LO-7.4] Enumerate principal sources of data uncertainty and their effect on depth estimates
• [LO-7.5] Implement a forward model predicting $T$-$x$ curves for layered and dipping geometries

Where We Left Off

Single-layer horizontal model (Lecture 6):

$$t_2(x) = \frac{x}{V_2} + \frac{2h_1 \cos\theta_{ic}}{V_1}, \quad \theta_{ic} = \sin^{-1}\!\frac{V_1}{V_2}$$

• Slope $\Rightarrow$ velocity; intercept $\Rightarrow$ depth
• Works when: horizontal layers, monotonically increasing velocity

Today: What happens when these assumptions fail?

Multi-Layer Generalization

For $N$ horizontal layers ($V_1 < V_2 < \cdots < V_N$):

$$t_n(x) = \frac{x}{V_n} + \frac{2}{V_n} \sum_{i=1}^{n-1} h_i \frac{\sqrt{V_n^2 - V_i^2}}{V_i}$$

• Each head wave yields one $T$-$x$ segment with slope $1/V_n$
• Intercept times solved sequentially: $h_1$ from $t_{i_2}$, then $h_2$ from $t_{i_3}$ using known $h_1$, etc.
• Layer thicknesses are not independent: every deeper estimate depends on all shallower ones

Multi-Layer $T$-$x$ Diagram

Three layers, three slope segments. The first head wave from each interface is the first arrival only beyond its crossover distance.
[Python-generated: assets/scripts/fig_multilayer_traveltime.py]

Complication 1: Low-Velocity Zone

What if $V_2 < V_1$?

• $\sin\theta_{ic} = V_1/V_2 > 1$ → no critical angle exists
• No head wave from the $V_1$–$V_2$ interface
• The intermediate layer is invisible to refraction

Consequence: The $T$-$x$ diagram looks like a simple two-layer Earth. The interpreted depth to $V_3$ is too large. There is no warning in the data.

Common cause: saturated clays over indurated bedrock; gas-bearing sands; weathered zones

LVZ: The Hidden Layer

No $1/V_2$ segment appears. P-wave first-arrival refraction alone cannot detect the LVZ.
[Python-generated: assets/scripts/fig_lvz_traveltime.py]

Detecting the LVZ: What Works?

P-wave first-arrival refraction cannot detect an LVZ — but these methods can:

Reflection is the direct remedy. MASW is the most powerful for velocity inversions.

Complication 2: Thin Intermediate Layer

Even when $V_1 < V_2 < V_3$, a thin layer may be undetectable.

The $V_2$ head wave is first arrival only over a limited offset window:

$$x_{c,1} = 2h_1\sqrt{\frac{V_2+V_1}{V_2-V_1}}$$

Rule of thumb: Layer $n$ is detectable only if the window width $\gtrsim \Delta x_{station}$.

For typical ratios: station spacing $\lesssim 0.6\, h_n$.

Complication 3: Dipping Interface

Dipping layers still produce head waves — but apparent velocity depends on shooting direction.

Down-dip: $t_d(x) = \dfrac{x}{V_1}\sin(\theta_{ic}+\delta) + t_{id}$

Up-dip: $t_u(x) = \dfrac{x}{V_1}\sin(\theta_{ic}-\delta) + t_{iu}$

• Down-dip: $\alpha_d < V_2$ (underestimates true refractor velocity)
• Up-dip: $\alpha_u > V_2$ (overestimates true refractor velocity)

Solution: shoot from both ends (reversed profile)

Dipping Interface: Reversed Profile

Reversed profiling resolves the ambiguity between dip and velocity.
[Python-generated: assets/scripts/fig_dipping_interface_reversed.py]

Recovering True Velocity and Dip

From apparent velocities $\alpha_d$ (down-dip) and $\alpha_u$ (up-dip):

$$\delta = \frac{1}{2}\left[\sin^{-1}\!\frac{V_1}{\alpha_d} - \sin^{-1}\!\frac{V_1}{\alpha_u}\right]$$

For small dips ($\delta \lesssim 15$–$20°$):

$$\frac{1}{V_2} \approx \frac{1}{2}\left(\frac{1}{\alpha_d} + \frac{1}{\alpha_u}\right)$$

Reciprocal time check: Travel time from source $A$ to far receiver must equal travel time from source $B$ to near receiver. Failure indicates timing error or lateral velocity variation.

The Delay-Time Method

For irregular refractors, the delay time at geophone $G$ is:

$$\delta t_G = \frac{h_G \cos\theta_{ic}}{V_1}$$

Depth from both forward and reverse delay times:

$$h_G = \frac{V_1 V_2}{2\sqrt{V_2^2 - V_1^2}}\left[\delta t_{F,G} + \delta t_{R,G}\right]$$

Result: a point-by-point refractor profile beneath every geophone position.

Delay-Time Method: The Tangent-Arc Construction

The refractor surface is the common tangent to all depth arcs.
[Python-generated: assets/scripts/fig_delay_time_method.py]

Sources of Uncertainty

Non-uniqueness: different model combinations can fit the same T-x data within noise — always integrate with borehole and independent geophysical data.

Worked Example: Puget Lowland

From field $T$-$x$ slopes and intercepts: $h_1 = 1.1$ m, $h_2 = 14.6$ m

Bedrock at ~15.7 m depth — but is there a LVZ in the gravel? Is the bedrock dipping toward the Seattle Fault?

→ Reversed profiling and borehole control needed for reliable site characterization.

Applications in the Pacific Northwest

• Liquefaction hazard: Depth to water table and bearing material in Holocene sediments beneath Seattle
• Debris flow characterization: Bedrock-colluvium interface on Cascades volcano flanks
• Transportation: WSDOT/Sound Transit tunnel and light-rail corridor characterization
• Seattle Basin: Sediment-bedrock interface controls site amplification for Cascadia megathrust events

Concept Check

1. A $T$-$x$ diagram shows only two linear segments even though a borehole shows three velocity units. Name two geological explanations and describe how you would distinguish them.

2. A reversed refraction profile gives apparent velocities $\alpha_d = 1163$ m/s and $\alpha_u = 2146$ m/s with overburden $V_1 = 500$ m/s. Calculate the refractor dip angle $\delta$ and true velocity $V_2$.

3. A geophone array has 3 m spacing. A thin gravel layer with $V_2/V_1 = 2$ is suspected. What is the minimum thickness you could confidently detect?