Beyond normal incidence

1Four results, restated

Everything this set takes as already established

The resolution modules built a chain from a column of rock to a seismic wiggle. Four of its results are used constantly from here on, and they are restated below so that the settings used later have a starting point.

Impedance is velocity times density. One number per layer, and at normal incidence it is the only rock property a reflection responds to.

A reflection measures a contrast. The reflection coefficient is the difference in impedance across the boundary divided by the sum. Equal impedances give no reflection, whatever else is different about the two rocks.

The sign is the polarity. Going into a higher impedance gives one sign, into a lower impedance the other, and which one appears as a peak on a display is a convention rather than a fact about the rock.

The trace is a sum. Each boundary returns a copy of the wavelet, and what is interpreted is the total. When two boundaries are close together their wavelets overlap and the amplitude that gets measured belongs to neither of them alone.

1 Velocity × density, for both rocks and the contrast between them

2 The reflection, with the wavelet applied at normal incidence

Impedance above—

Impedance below—

Contrast across the boundary—

Reflection coefficient—

Polarity—

What this number is a property of—

These are the resolution modules' results and they are not re-derived here. The module that builds them is Where a seismic trace comes from.

2One assumption ran through all of it

The sound went straight down and straight back

Every trace in the resolution modules was built for a wave arriving perpendicular to the boundary. That case has a name: normal incidence. It is the simplest one, the arithmetic is a single fraction, and it is where the impedance result comes from.

A real survey does not record that case, or rather it records it only once. A source is fired at one point and a line of receivers stretches away from it, so most of the energy that returns took a slanted path down and a slanted path back. The angle at which the wave meets the boundary is the incidence angle, and across a single recording it takes a range of values rather than one.

The resolution modules state this exclusion rather than hide it. Their trace module says it is normal incidence only, with no offset, no AVO and no mode conversion. The wedge module lists offset among the things left out because each deserves separate treatment. This set is that separate treatment.

The two controls under where the angle comes from set the distance between the source and the receiver, which is called the offset, and the depth of the boundary. For a ray that travels in a straight line and reflects at the midpoint, the half-offset and the depth are the two short sides of a right triangle, so the angle is the arctangent of one divided by the other. Offset and angle are related but not the same number: the same offset gives a wide angle at a shallow boundary and a narrow one at a deep boundary.

One shot does not produce one ray. The source releases energy as a wave that spreads outward in all directions at once, so it meets the boundary everywhere beneath the spread, not at a single point. A line of receivers — geophones on land, often called nodes when each one records independently — is laid out along the ground, and each of them picks up energy that reflected at its own point and arrived at its own angle. The third panel draws the whole spread at once. One shot into that line yields a set of recordings covering a range of angles, and every one of them is a measurement of the same boundary.

How those recordings are organized, and what happens once velocity increases with depth and the rays stop traveling in straight lines, is the subject of module 04. What matters here is that the range of angles exists and that the resolution modules covered one end of it.

1 The case that has been covered zero offset, straight down and back

2 The offset you set drawn to the same scale

3 One shot, one line of receivers the wave spreads, and every node records a different angle

Offset, source to receiver—

Depth to the boundary—

Half-offset ÷ depth—

Incidence angle that gives—

Receivers in the spread—

Angle range one shot into that line records—

The rays are straight here. Velocity generally increases with depth, which bends a real ray and makes the true angle at the boundary larger than this triangle gives, by an amount that grows with offset. Module 04 works that out.

3The reflection coefficient is not one number

The observation this set is about

Once the angle is allowed to vary, the reflection coefficient varies with it. The left-hand panel shows that for two rocks: a shale over a sand filled with brine, and the same shale over the same sand filled with gas. Both curves are computed from the full reflection equations rather than from the impedance fraction.

The picture has two separate features. The curves start at different heights, which is the impedance contrast from step 1 and is already familiar. They also have different shapes, and that is new: the gas curve falls away from its starting value much faster than the brine curve does.

The right-hand panel plots the gap between the two curves. It widens with angle. A measurement made only at normal incidence sees the difference at the left-hand edge of that plot; a measurement that uses the whole range sees the difference grow. That is the observation the rest of this set examines.

A wave arriving perpendicular to a boundary moves the rock particles straight into it and straight back out. The rock is compressed and nothing slides, so the reflection depends only on resistance to compression, which impedance describes completely. A wave arriving at an angle moves the particles along a direction that is no longer perpendicular, so part of the motion is parallel to the boundary and the rock either side is also sheared. Resistance to shearing is a separate property with its own velocity — the shear velocity, Vs, alongside the Vp the earlier modules used. The wider the angle, the more of the motion is parallel and the more Vs matters, which is why one number becomes a curve.

The third panel shows what swapping the pore fluid does to each property of the sand, and the pattern in it is the reason the two curves have different shapes: Vp drops a long way, Vs barely moves, and Vp/Vs falls further than either. Where Vp and Vs come from, and why a fluid acts on one and not the other, is module 01.

1 Reflection coefficient against angle the same sand, two pore fluids

2 The gap between the two curves how much the fluid changes the reflection

3 What the fluid changes, property by property the same sand with brine, then with gas

Vp of the sand, brine → gas—

Vs of the sand, brine → gas—

Vp/Vs, brine → gas—

Reflection at 0°, brine / gas—

Gap between them, 0° → this angle—

What the extra angles added—

At 30% porosity the swap from brine to gas takes Vp down 16.5%, takes density down 11.7%, and puts Vs up 6.4% — the only fluid property Vs responds to is the mass it has to move. Vp/Vs falls 21.5%, further than any single velocity moved. The two rocks here are computed from a rock physics model rather than set by the sliders, because their behavior with angle needs the shear velocity and the sliders carry only Vp and density.

WWhy it matters

What the extra angles are used for

Two measurements instead of one. A normal-incidence amplitude is a single number, and a single number cannot separate two unknowns. The variation with angle supplies a second number from the same recording, which is the basis of everything this set covers.

Fluid discrimination. The main practical use is distinguishing a sand filled with gas from the same sand filled with brine. The panels in step 3 show why that is possible in principle. How reliable it is in practice is what modules 05 through 08 examine.

What stacking costs. The image most interpreters work from is a stack, which averages the angle range into one trace. It is a cleaner picture and it discards the variation shown here. Both statements are true at once, and module 04 covers the trade.

The resolution limits still apply. Nothing in this set repeals tuning. An amplitude measured on a bed thinner than the resolution limit belongs to two merged reflections, and that is as true at 30° as at 0°.

Exercises

Each answer is a number read off this page at the settings described.

Separate offset from angle. Set the offset to 2400 m and the depth to 2000 m and read the incidence angle. Then leave the offset alone and take the depth to 3000 m.
Answer
The angle falls from about 31° to about 21.8°, on an unchanged offset. In the third panel every ray in the spread flattens together, and the whole range the shot records narrows from 4.3°–36.9° to 2.9°–26.6°. The angle is set by the half-offset divided by the depth, so one receiver line reaches wide angles on a shallow target and narrow ones on a deep target. A statement about which offsets to include in an analysis does not transfer between targets at different depths, and a deep enough target may not reach a usable angle range at all.
Confirm that the impedance result still holds. Set the rock below to 2280 m/s and 1.91 g/cm³ with the rock above at 2700 m/s and 2.23 g/cm³, then find a different pair of numbers for the rock below that gives the same reflection coefficient.
Answer
Any pair with the same product works, because the coefficient depends on velocity and density only through impedance. Roughly 2600 m/s at 1.67 g/cm³, or 1950 m/s at 2.23 g/cm³, land in the same place. This is the ambiguity the resolution modules already established, and it is the starting condition for this set rather than something this set fixes.
Make the reflection disappear, then check what step 3 does. Match the impedances across the boundary so the trace goes flat, and note that step 3 is unaffected.
Answer
Steps 1 and 2 use the four sliders; step 3 uses a modeled sand and shale instead. They are deliberately not connected. The sliders carry Vp and density, which is enough for a normal-incidence reflection and not enough for a reflection at an angle, because that also needs the shear velocity. Separating them is the cleanest way to show which properties each case requires.
Reach the widest angle the geometry allows. Put the offset at 3000 m and the depth at 1500 m, read the angle, and compare the two curves there against their values at 0°.
Answer
A half-offset of 1500 m over a depth of 1500 m gives a tangent of exactly 1, so the angle is 45°. There the brine sand reads about −0.099 and the gas sand about −0.315, a gap of roughly 0.216 against 0.151 at normal incidence — about 1.4 times as large. The extra angles did not repeat the normal-incidence measurement more times; they carried information it did not have.
Reduce the porosity and watch both curves. Take porosity from 35% down to 18% and follow the gap panel, then look at where the two curves start.
Answer
At normal incidence the gap shrinks steadily, from about 0.172 at 35% porosity to about 0.100 at 18%, because a tight sand has little pore space and its fluid has correspondingly little influence on the rock as a whole. At 45° the gap barely moves at all: it stays near 0.21 across the whole range, dipping slightly around 24%. So what the wide angles add over normal incidence grows as the sand tightens — about 1.3 times at 35% porosity and about 2.2 times at 18%. The angle information counts for most where the normal-incidence measurement is weakest. Watch the brine curve as well: its value at 0° crosses zero just below 30% porosity, because a tighter sand becomes harder than the shale above it rather than softer, and that sign change has nothing to do with the fluid.

Key points

This set assumes the resolution modules. Impedance, the reflection coefficient, polarity, the wavelet and tuning are established there and used here without re-derivation.
Every one of those results assumed normal incidence — a wave meeting the boundary perpendicularly. It is the simplest case and the only one they covered.
The source releases a wave, not a ray. Energy spreads out in every direction, meets the boundary everywhere beneath the spread, and is picked up by a line of receivers — so one shot yields a set of recordings covering a range of angles, all of the same boundary.
The reflection coefficient changes with angle, and two interfaces with different pore fluids change differently. That is the observation this set examines.
The variation supplies a second measurement from the same recording, which is what makes separating two unknowns possible in principle.
Offset and angle are related but not equal. The angle comes from the half-offset divided by the depth, so the same offset reaches a wide angle on a shallow boundary and a narrow one on a deep boundary.
The variation exists because an angled wave shears the rock. At normal incidence the motion is perpendicular to the boundary and only compression matters. At an angle part of the motion is parallel to it, so the shear velocities on both sides enter the reflection.
A pore fluid moves Vp and leaves Vs almost alone, so replacing brine with gas drops Vp/Vs sharply, and it is the Vp/Vs contrast across the boundary that sets how steeply a curve falls away. Where Vp and Vs come from is module 01.

MMethod, simplifications and references

What is computed, and what is deferred

Steps 1 and 2 use the four sliders directly. Impedance is velocity times density, the reflection coefficient is the impedance difference over the sum, and the trace places a 30 Hz Ricker wavelet at the reflection. Nothing there needs a shear velocity, which is why the sliders do not carry one.

Step 3 does not use the sliders, apart from porosity. Its two rocks are a shale from Castagna's mudrock line and a sand built from a Hertz-Mindlin dry frame with the pore fluid inserted by Gassmann's relation, at 23 MPa and 64 °C, with brine and then with gas. The curves are the exact Zoeppritz P-wave reflection coefficients for those two rock pairs. Using the slider values there instead would require inventing a shear velocity for them, and the usual way of doing that is a shale trend that does not apply to sands.

Deferred rather than simplified:

Straight rays. Step 2 computes the angle as arctan of the half-offset over the depth, which is exact only for constant velocity. Velocity generally rises with depth, so a real ray curves and arrives at the boundary at a wider angle than this triangle gives, by an amount that grows with offset. It also means the reflection point is not exactly at the midpoint once the layers dip. Module 04 handles both.
One boundary, no moveout. The geometry panels show a single reflector and say nothing about the extra travel time a longer offset costs, which is what has to be removed before the amplitudes at different offsets can be compared at all. That correction, and what it does to the data, is module 04.
Why the curves have the shape they do. Step 3 shows that they differ without explaining it. The explanation needs the shear response, which is modules 01 and 02, and the reflection equations themselves, which are module 05.
How well any of this survives real data. These curves are exact and noise-free. What happens to them once noise, a limited angle range and an unknown background are involved is modules 07 and 08.
Mode conversion, multiples, attenuation and anisotropy are all absent.

The porosity slider stops at 18%. The dry frame here is a soft-sand Hertz-Mindlin model, which describes a grain pack held together by contact stiffness and is not intended for well-cemented rock. Run below roughly 18% and it keeps returning numbers, but they describe a grain pack that no longer resembles the tight sand being asked about, and the difference between the two fluid cases at wide angle stops behaving monotonically for that reason rather than a physical one. A cemented-sand or constant-cement model would be the appropriate choice there.

The angle reaches 45° at the extremes of the sliders, at the shortest depth with the longest offset. On a marine gather at reservoir depth, 30° to 40° is a more usual practical limit: beyond that the reflection coefficient changes rapidly, the linear approximations most people use stop holding, and the processing that flattened the gather is less reliable. The curves here are exact Zoeppritz, so they remain correct across the whole range even where an approximation would not.

The shear velocities used in step 3 come out of the rock model rather than from a Vp-to-Vs trend. The shale takes its Vs from Castagna's mudrock line, which is a fit to shales. The sand takes its Vs from the shear modulus of the dry grain frame, which Gassmann's relation leaves unchanged when the pore fluid is swapped — the small rise in Vs from brine to gas is entirely the density drop, not a change in the frame.

References

Bedle, H., and Moreno-Ward, A., 2026, What Can You REALLY See in Seismic?: University of Oklahoma. — the prerequisite modules for this set.
Zoeppritz, K., 1919, Erdbebenwellen VIIIB: Göttinger Nachrichten, I, 66–84. — the exact reflection coefficients used in step 3.
Castagna, J. P., Batzle, M. L., and Eastwood, R. L., 1985, Relationships between compressional-wave and shear-wave velocities in clastic silicate rocks: Geophysics, 50, 571–581.
Mavko, G., Mukerji, T., and Dvorkin, J., 2020, The Rock Physics Handbook, 3rd ed.: Cambridge University Press.
Simm, R., and Bacon, M., 2014, Seismic Amplitude: An Interpreter's Handbook: Cambridge University Press.

The rock above

The rock below

Where the angle comes from

Step 3 only

1Four results, restated

2One assumption ran through all of it

3The reflection coefficient is not one number

WWhy it matters

Exercises

Key points

MMethod, simplifications and references

01 · Rocks, pores and stiffness