Module 02 · Geometric attributes
In module 01 we estimated dip along a single seismic line by finding the tilt that best aligned neighboring traces. A 3D volume allows that measurement in two directions, and one direction is not enough. In a volume the orientation of a reflector at a trace takes two numbers rather than one.
The reflector orientation — these two numbers drive every step
How much later the reflector arrives on the next trace along the inline direction.
How much later it arrives on the next crossline. The same reflector, measured in the other direction.
Work through the seven steps in order, then try the exercises. Method and references are there when you want them.
A volume of 25 inlines by 25 crosslines contains one planar reflector. Two vertical sections are cut through the middle of it, one along each acquisition direction. Both sections pass through the same trace at inline 13, crossline 13, marked in teal on each.
1 Inline section crossline held at 13
2 Crossline section inline held at 13
This is module 01's question, asked in the inline direction at the marked trace. Step one trace to the right and read how far the reflector moves down. The answer is written ∂t/∂x, and this section is all you need to measure it.
Only the inline component changes. The crossline component is untouched, and this section cannot see it.
The same question, turned ninety degrees, at the same trace. Step one line in the crossline direction and read how far the reflector moves. This is ∂t/∂y. It is measured independently of the inline answer, and neither section carries any information about the other.
Set the inline dip in step 2 to something non-zero and come back. This section does not change.
Steps 2 and 3 produced two numbers at one trace. That pair describes the reflector there. Move to the next trace and the measurement is repeated, and repeated again at every sample down each trace. A dip volume is the result of doing this everywhere, so it holds two numbers for every sample in the input.
With a planar reflector every trace returns the same pair, so the maps below come out a single flat color. Curve the reflector and the pair starts changing from trace to trace, which is what it does in real data and the reason the measurement has to be repeated rather than assumed.
1 Dip magnitude at every trace how steep, regardless of direction
2 Dip azimuth at every trace which way it dips
Steps 1 to 4 all cut the volume vertically. A time slice cuts it horizontally: pick one travel time and read the amplitude of every trace at that one time. The result is a map with a seismic value at every bin, and no time axis at all.
The diagram below shows why that map is useful. A horizontal cut through a dipping reflector meets it along a straight line. On one side of that line the slice sits above the reflector, on the other side below it, and along the line itself it is passing straight through the peak. So the reflector appears on the map as a band, and the band runs along strike.
Switch the dome on and the same argument gives a different shape. A horizontal cut through a domed reflector meets it along a closed ring rather than a straight line, and that is exactly the ring you see on the slice. The diagram redraws itself to match, so the picture on the left and the sketch above it can be checked against each other: the band and the ring are the same intersection, drawn on two different reflector shapes.
1 The time slice itself amplitude, not dip
2 The two components as one vector inline across, crossline up
Moving the slice sweeps the bands sideways. For a plane their spacing and direction do not change.
Make these different and watch the azimuth readout move while the pictures do not.
Needed only for the last readout. The volume itself contains no velocity information.
Inline and crossline are only two of the directions available. Cut a vertical section at any azimuth through the same reflector and it shows a dip of its own. Rotate the section and watch that dip change, reaching zero along one particular direction.
1 Where the section is cut looking down on the volume
2 The section at that azimuth sampled every 15 m along the line
3 Apparent dip at every azimuth a cosine, zero along strike
In module 01 the algorithm tilted one line, scored how well the traces lined up along it, and kept the best tilt. Here it has to tilt two lines at the same time: one along the inline, one along the crossline. Every combination of the two is tried.
Press run and watch the two lines below rock back and forth. Each pair of positions they take is one square on the grid, and that square is colored by how well the traces agreed.
1 Tilting along the inline the traces either side, inline direction
2 Tilting along the crossline the same trace, crossline direction
3 Score for each combination one square per pair of tilts
The same parameters as module 01, with one difference worth noticing: the search range and step apply in both directions at once, so halving the step multiplies the work by four rather than two.
The width and height of the grid. A reflector outside this range in either direction cannot be found.
How fine the grid is. The number of squares is the square of the number of steps along one side.
A square patch centered on the trace. The two panels show one row and one column of it; the score uses all of it at once.
The length of trace compared, as before.
Applied to the whole volume. Watch the bright spot on the grid spread out before the estimate itself moves.
Lower frequencies make a broader wavelet and a broader, less certain bright spot.
Module 01 found dip on a single line. Real surveys are volumes, and the moment you leave one line the question changes: a reflector does not have a dip, it has a direction it goes down and a steepness in that direction. Two numbers per sample, not one.
The second component is not bookkeeping. The following all depend on having both.
Dip azimuth is cyclic. An azimuth of 359° and one of 1° are two degrees apart, not three hundred and fifty-eight, and a color scale that does not wrap will draw a hard boundary across perfectly continuous structure — an artifact that looks exactly like a lineament. Step 4 uses a cyclic scale for this reason, and it is worth checking what your own software does before you interpret an azimuth map.
The same trap returns twice more in this course: curvature strike wraps at 180° because it is an orientation, and aberrancy azimuth wraps at 360° because it is a direction. Each of the three wraps at a different value, and a color scale that does not match the rule draws a false edge.
The volume is 25 inlines by 25 crosslines by 220 samples at 2 ms. The reflector is defined analytically as t(x, y) = t0 + px·x + py·y, plus a quadratic term when the dome is switched on, and the amplitude at each sample is a 30 Hz Ricker wavelet evaluated at the offset from that surface. The geometry is therefore exact rather than interpolated, and the true dip at any trace is the analytic derivative. The optional second and third beds are parallel copies 70 ms above and below.
The two maps in step 4 are produced by running the search at every second trace and interpolating for display, at a fixed 13 by 13 grid of candidate tilts regardless of the step-7 settings, because a full density scan on every slider movement would stutter. The decimation is a display choice, not part of the method. A production run estimates at every sample of every trace.
Azimuth is measured in degrees counter-clockwise from the inline axis and points downdip, so on the maps here, which put inline to the right and crossline upward, 0° is to the right and 90° is up. Real surveys are rotated relative to north, so a dip azimuth volume has to be corrected by the survey rotation before it means anything geographic, and packages differ over whether azimuth points updip or downdip and whether it is measured from north or from the inline. Check the convention before comparing two volumes.
The rotated section in step 6 is sampled at a fixed 15 m along the line, using bilinear interpolation between the four surrounding traces. That fixed spacing is a display choice. In a real volume the natural trace spacing along a section varies with azimuth — a section cut diagonally across square bins has traces √2 further apart than one cut along the inline — which is a further reason apparent dip is quoted per unit distance rather than per trace once you leave the acquisition directions.
The search is the discrete scan of Marfurt et al. (1998) extended to two components, one of the two algorithms in AASPI program dip3d. Refinement fits a parabola along each axis through the winning square. A true two-dimensional peak wants a paraboloid including a cross term; the separable form used here is what most implementations do and is accurate while the peak is not strongly skewed, which the elongated bright spot under noise shows it can be.
Simplifications carried over from module 01, plus new ones. Semblance is computed on the plain trace rather than the analytic trace. There is no Kuwahara search over windows offset around the analysis point. The reflector is smooth, so the estimate is never fighting a fault inside the window. And the alternative family of estimators, which builds a gradient structure tensor and takes its principal eigenvector rather than trying candidates, is not shown here at all.
Next up
Dip is now a direction at every sample. The next module is about what that direction is for: every attribute that compares neighboring traces has to decide where to look on each of them, and the difference between looking along the bed and looking straight across the screen decides what the attribute ends up imaging.
Open module 03 → All modules