Consider a seismic source located at the Earth's surface. A vertical well is located at a distance of 1 km away from the seismic source, and a geophone is suspended inside the well at a depth of 800 m, as shown in the figure below. The travel time of the primary reflected wave from a horizontal reflector located at a depth of 2 km is ___________ s (rounded off to three decimal places).
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Use the image-source (mirror) method for a horizontal reflector: reflect the source (or receiver) across the reflector depth and take a straight-line path to the actual receiver (or source), then divide by velocity.
The reflector is horizontal and the medium above it has a single constant velocity, \(V=3\ \text{km/s}\) (read off the figure). For this geometry the fastest way to get the travel time of the primary (single-bounce) reflection to an offset receiver is the classical image-source (mirror-source) method used throughout reflection seismology.
Step 1: Set up coordinates. Let the source S sit at the surface at \(x=0,\ z=0\). The reflector is horizontal at depth \(z=2\) km. The well is 1 km away (horizontally) from the source, and the geophone inside it sits at depth 800 m = 0.8 km. So the receiver R is at \((x,z)=(1,\,0.8)\) km.
Step 2: Construct the image source. For a flat horizontal reflector, the reflected ray path from S to any receiver has exactly the same length as a straight (direct) ray from an 'image' source \(S^*\), obtained by mirroring S across the reflector plane: \(z_{S^*}=2z_{reflector}-z_S=2(2)-0=4\) km, at the same horizontal position, so \(S^*=(0,4)\) km.
Step 3: Compute the straight-line distance from the image source to the receiver. \[ L=\sqrt{(x_R-x_{S^*})^2+(z_R-z_{S^*})^2}=\sqrt{(1-0)^2+(0.8-4)^2}=\sqrt{1+10.24}=\sqrt{11.24}\approx3.3526\ \text{km} \]
Step 4: Divide by velocity to get the travel time. \[ t=\dfrac{L}{V}=\dfrac{3.3526}{3}\approx1.1175\ \text{s} \]
\[ \boxed{t\approx1.118\ \text{s}} \]
This lies inside the official 1.116–1.120 s window.
