Geometric attributes: dip, azimuth, and curvature

Dip: the slope of the local reflector

At every sample the local reflector has a slope, measured in two directions of the survey grid:

• $p_x = \partial t/\partial x$, how much the time of the reflector changes from one crossline to the next along an inline, in ms per trace;
• $p_y = \partial t/\partial y$, the same from one inline to the next along a crossline.

Together they describe a tilted plane. The dip magnitude $\sqrt{p_x^2 + p_y^2}$ is its steepness whatever the direction; the azimuth $\operatorname{atan2}(p_x, p_y)$ is the direction in which the time increases, the down-dip direction, a cyclic value from $0^\circ$ to $360^\circ$ that is best shown on a colour wheel. On a time-migrated volume both are time dips; a dip angle needs a velocity: $\sin\theta = V p/2$ with $p$ in seconds per metre, so 1 ms per 25 m trace at 2000 m/s is a dip of $2.3^\circ$.

Dip from the local gradients

Following a reflector means keeping the amplitude constant. Moving by $dy$ and $dt$ along it:

and likewise for $x$. This ratio is useless as it stands: $\partial a/\partial t$ passes through zero twice in every cycle of the wavelet. The usual cure is to fit one slope over a short window by least squares, which sums the numerator and the denominator separately:

The denominator is the energy of the time derivative over the window, positive wherever the window holds any reflection. The estimate is closely related to the gradient structure tensor used in interpretation software.

The gradients are taken by central differences, and that limits the estimate. For a plane wave of $\omega$ radians per sample dipping $p$ samples per trace, the estimator returns $\sin(\omega p)/\sin\omega$: exact at one sample per trace, a little high below that, low above it, largest at $p = \pi/2\omega$ and aliased beyond $\pi/\omega$. At 30 Hz and 4 ms ($\omega$ = 0.75) the estimate can never exceed about 5.8 ms per trace, the reading at a true dip near 8 ms per trace; a true dip of 12 ms per trace reads about 4.5. Steep dips need a longer stencil or a search over dips.

Curvature: how the reflector bends

Curvature measures how the dip itself changes across the survey. A flat reflector has none, and so has one that tilts the same way everywhere. Curvature appears where the geometry changes: anticlines, synclines, the flanks of channels and of salt, the bends beside a fault. The simplest single number is the mean curvature; for gentle dips, in time,

in ms per trace squared. This is the small-dip form of the mean curvature of Roberts (2001), written in time rather than depth. Read it geometrically, remembering that time increases downward:

• $k > 0$: the reflector bows upward, as on an anticline, a ridge or a dome;
• $k < 0$: it bows downward, as in a syncline, a valley or the trough of a channel;
• $k \approx 0$: flat, evenly tilted, or a saddle, where one principal curvature is positive and the other negative. Mean curvature cannot tell a saddle from a plane; the most-positive and most-negative curvatures can.

Curvature is a derivative of a derivative, so it amplifies noise twice. Estimated from dips measured trace by trace it is mostly noise; it is usually computed from dips averaged over many traces, and the averaging distance sets the wavelength at which the bending is read (Al-Dossary and Marfurt 2006). Figure 6.3 averages the curvature over $L \times L$ traces, which by linearity is the curvature of the averaged dip field.

What dip and curvature reveal

• Faults. Where a fault offsets the layers, the estimated dip jumps over a few traces, and the dip magnitude shows a thin bright line. The layers often bend into the fault, so curvature shows a band beside it. Coherence (Section 6.4) shows the break itself; between them the three attributes catch most faults.
• Folds. Anticlines and synclines are curvature anomalies, and coherence can miss them entirely when the layers bend without breaking.
• Channels. A channel cut into older rock is a trough, so it often shows as a negative ribbon on a curvature map, with its edges on coherence.
• Salt. Layers steepen toward the flanks of salt, and curvature is high where they wrap around it.
• Gentle closures. A four-way closure with no amplitude anomaly and little coherence response can still be outlined by curvature.

Common geometric-attribute pitfalls

• Acquisition footprint. Shot and receiver lines and binning leave regular patterns, and derivatives amplify them. Check whether a lineament runs along the acquisition directions before calling it geology.
• Migration artefacts. Residual diffraction smiles and over- or under-migration bend the reflectors and make curvature of their own.
• Window and wavelength. A very short time window (3 to 5 samples) follows single cycles of the wavelet; a long one averages several reflectors. Laterally, a short averaging distance gives noise, a long one smooths away small features. Both are choices to report with the map.
• Steep dips. The gradient estimate saturates and then aliases on steep dips, as above.
• Time is not depth. Time dips and time curvature change wherever the velocity above changes laterally; convert with a velocity model before using them for engineering or geomechanics.

Four questions about every sample

With this section you have the three families of Sections 6.1 to 6.3 and, in Section 6.4, coherence:

• amplitude: how strong is the reflectivity here?
• frequency: what is the spectral content of the wavelet here?
• geometry: which way is the reflector tilted, and how does it bend?
• coherence: how alike is this trace to its neighbours?

No single attribute answers every question. Section 6.5 combines them into blends and classifications, and Section 6.6 applies them to reservoir characterisation together with the rock physics of Part 5.