Structural curvature | How Geometric Attributes Actually Work

1From dip to curvature

One profile, differentiated twice

The seismic line above the tabs is the whole of the input. It runs west to east through a gentle fold and one small fault, and it stays on screen for every step so you can always see what the numbers are describing. The three curves below line up with it trace for trace. Two of them have already been built, and they are listed here in the order they were introduced.

Dip came first, in modules 01 to 03: which way the reflector is heading at every sample. On this line it is largest on the flanks of the fold and passes through zero at the crest and the trough.
Coherence came next, in module 04. It compares neighboring traces along that dip and asks whether they still look alike. It goes dark at the fault, where the reflector is broken. Across the fold it sees nothing whatever, because a fold is perfectly continuous: nothing is broken, the beds simply bend.
Curvature is the next step in the same chain — the rate at which the dip is changing. It lights up across the fold that coherence ignored, and it also reacts at the fault.

The three attributes form a sequence. Dip tells you the orientation. Coherence asks whether the reflector is broken. Curvature asks whether it is bent. Coherence finds breaks; curvature finds bends, and a fault too small to break the reflector will still bend it. That is what curvature adds to coherence.

Curvature has a direct geometric definition. At any point on the reflector, fit the circle that best follows the bed. Curvature is k = 1/r, one over that circle's radius: a tight fold is a small circle and a big number, a gentle warp is a huge circle and a small one, and a straight bed is a circle of infinite radius, so k = 0. The sign follows the shape — anticlines positive, synclines negative, and zero on any plane, however steeply that plane dips.

Now try the fault-throw slider in the header. Bring it down to 5 m and watch the two curves part company: coherence returns to a flat 0.99, exactly as if there were no fault at all, while curvature still marks the spot. Below the seismic resolution the offset stops being a break and becomes a flexure — and a flexure is precisely what curvature measures. That is why a fault interpretation is normally run on coherence and curvature together, and why very small faults appear on one and not the other.

Finally, look at where the curvature peaks sit relative to the seismic. They are not on the steep flanks — the flanks are where curvature crosses zero. The peaks are at the crest and the trough, exactly where the reflector is momentarily flat. A bright curvature anomaly means a place where the dip is changing quickly, not a place that is steep.

Curvature at the fold crest—

Curvature at the fold trough—

Steepest dip on the profile—

Lowest coherence on the line—

The sign convention—

Curvature is quoted here in reciprocal kilometers, which is what you get when you fit a circle to the reflector and take one over its radius. A curvature of 1 km⁻¹ means a fold you could fit inside a circle of 1 km radius; 0.1 km⁻¹ is a gentle 10 km-radius warp. Anticlines are positive and synclines negative, which is the usual convention but not a universal one.

2Curvature depends on which way you cut

The same problem apparent dip had, one derivative later

Module 02 showed that apparent dip belongs to the section, not the reflector. Curvature has exactly the same problem, and it is worse, because a surface can curve one way in one direction and the other way at ninety degrees to it — a saddle. Cut a profile through the marked point at any azimuth and it has a curvature of its own.

Rotate the profile below and watch the number swing. It reaches a maximum in one direction and a minimum ninety degrees away, and those two extremes are what get called the principal curvatures, k₁ and k₂.

Click anywhere on the map to move the analysis point. The plunging nose at the end of a ridge is a useful place to look: try the northern end of the western anticline, near the top of the map.

Profile azimuth 0°

1 Where the profile is cut through the marked point

2 Curvature at every azimuth the maximum and minimum are k₁ and k₂

This profile's curvature—

k₁, the most positive—

k₂, the most negative—

Strike of k₂—

What this point is doing—

The two extremes are always ninety degrees apart, which is not obvious but is a genuine result about smooth surfaces. Euler's theorem then gives the curvature at any other azimuth from those two alone: k(φ) = k₁cos²(φ − ψ) + k₂sin²(φ − ψ), where ψ is the direction k₁ points. So the whole curve on the right is determined by two numbers and an angle.

3k₁ and k₂ across the whole map

Two volumes, and the strike that goes with them

Now compute those two numbers at every point rather than one. k₁ picks out anticlines, ridges and the upthrown edges of faults; k₂ picks out synclines, valleys and the downthrown edges. Run them together and a fault shows as a paired bright and dark lineament, which is the signature interpreters look for.

Software menus often label these two most-positive and most-negative curvature. Those names belong to a related pair of attributes that is not quite the same thing, and step 4 is about the difference. Everything on this page is k₁ and k₂.

1 k₁, the more positive crests and ridges

2 k₂, the more negative troughs and valleys

3 Strike of k₂ which way the lineaments run

The strike map uses a cyclic color scale, for the same reason dip azimuth did in module 02: an orientation of +89° and one of −89° are almost the same direction, and a scale with different colors at its ends would draw a boundary that is not in the ground. Strike is meaningless where the surface is a dome, a bowl or a flat — there is no preferred direction — so in practice it is displayed modulated by curvature so those areas stay pale.

Largest k₁ on the map—

Largest negative k₂—

At the marked point—

Reading the pair—

4Most-positive curvature is not k₁

Two attributes with confusingly similar names

Software offers both k₁ and something called k_pos, and they are not the same thing. The principal curvatures fit a circle to the reflector, which means correcting for how steeply the reflector is already dipping. The most-positive and most-negative curvatures skip that correction and are, in the words of the AASPI documentation, better read as a second-derivative map of structure than as a curvature.

On flat-lying structure the two agree almost exactly. Add regional dip with the slider in the header and watch them separate — k_pos hardly moves while k₁ falls. Tilting the whole surface does not change its second derivatives, so k_pos, which is built from those alone, barely changes. k₁ does, because fitting a circle to a dipping bed measures the bend along the bed rather than against the vertical, and a dipping bed is longer than its map projection. That foreshortening is the dip correction. The practical consequence is that on dipping structure k_pos reads systematically high.

The fold is identical on both sides. All that has changed is that the right-hand one sits on a tilt.

1 k₁, dip corrected a true principal curvature

2 k_pos, not corrected the second derivative of structure

3 k_pos − k₁ blank until the structure dips

4 How far apart they drift k_pos divided by k₁ at the strongest anticline, against regional dip

Each point is one setting of the regional dip slider. At 1.0 the two attributes agree exactly. The vertical line marks where the slider is now, so you can see how much of the disagreement you have dialed in.

Regional dip now—

k₁ at the crest—

k_pos at the crest—

k_pos as a multiple of k₁—

Verdict—

5Long and short wavelength

The one parameter that decides what you see

Differentiating twice multiplies the high wavenumbers twice over, so any short-wavelength wobble in the dip field comes out amplified. The usual sources are acquisition footprint, a regular pattern left by the shot and receiver geometry; statics, small trace-to-trace time shifts introduced by the near surface; and migration noise. That is why curvature is computed with an operator length rather than a bare difference, and why the same volume computed two ways looks like two different pieces of geology.

Turn on the short-wavelength texture and the noise in the header, then move the operator length. A short operator picks out the fine fabric; a long one shows only the broad fold. Neither is the right answer — they are answers to different questions, and choosing between them is an interpretation.

1 k₁ at the operator length you chose the header slider drives this

2 k₁ with a long operator fixed at 15 traces, for comparison

Operator length—

Typical |k₁| at this length—

Typical |k₁| at 15 traces—

Amplification—

What you are looking at—

?Why bother with a third attribute

What curvature finds that dip and coherence cannot

Step 1 made the case in one picture: coherence needs a break and curvature needs only a bend. Below the seismic resolution a fault stops breaking the reflector and starts flexing it, and that is where curvature takes over. Walk the fault throw down in step 1 and coherence returns to its no-fault value at 5 m while curvature is still marking the spot.

What it is used for

Subtle faults and flexures. The ones too small for coherence, which are often the ones that matter for compartmentalization — a 5 m offset can still juxtapose sand against shale.
Fracture prediction. Bending a brittle layer puts its outer arc into extension, so strongly curved zones are where open fractures are most likely. This is a correlation to be tested against image logs and production, not a law, but it is the standard first-pass map for a fractured reservoir.
Differential compaction. A channel, a reef or a fan compacts differently from the mud around it, and drapes the overlying reflectors into a gentle ridge or sag. Curvature maps that drape cleanly, so you can see a body you cannot see directly.
Karst and collapse. Sinkholes and collapsed chimneys show as tight closed lows, which is exactly the shape module 06's bowl component isolates.
Structural framework. k₁ and k₂ run together give the paired bright and dark lineament that reads as a fault, with the upthrown and downthrown sides labeled for you.

The three ways it will mislead you

Bright does not mean steep. Curvature peaks where the bed is momentarily flat, at a crest or a trough, and passes through zero where the bed is steepest — because a steeply dipping bed at its inflection is momentarily straight, and straight means not bending. On the default structure the steepest point reads a hundredth of a reciprocal kilometer against nearly one at the crest, a factor of some seventy the other way from most people's intuition.

Operator length is an interpretation, not a setting. Long-wavelength curvature maps folds and short-wavelength curvature maps fabric, and the same surface honestly gives both. Which one you chose should travel with the volume, because nothing in the data says which is right.

It inherits the errors in the dip field, amplified. Curvature is a second derivative, so short-wavelength error in dip is amplified twice over. Put the noise slider to 30% at an operator length of 3 and the typical curvature reads several times what a long operator gives on the same surface, almost all of it noise. That is the practical reason modules 01 to 03 spent so long on getting dip right.

k₁ and k_pos are different attributes, not two names for one. On dipping structure the uncorrected version reads systematically high, and it reads highest where the structure dips most steeply, so the bias does not average out.

Exercises

Shrink the fault until only curvature can see it. In step 1, walk the fault throw down from 60 m to 5 m, reading the coherence value and the fault curvature at each stop.
Hint
At 60 m throw coherence sits around 0.36 and the fault is unmistakable. By 10 m it has climbed to about 0.97, and at 5 m it reaches exactly the value it reads with no fault at all. Coherence no longer registers the fault at all. Curvature meanwhile falls roughly in step with the throw, by about a factor of eight over the same range: reduced, but still clearly there at every stop. Once the offset drops below what the wavelet can resolve, the reflector is no longer broken, it is bent, and only an attribute that measures bending will find it. This is the classic progression — large offsets to coherence, small offsets to curvature, and offsets below even that to aberrancy, which has a module of its own. Notice too that the fold sits at the same curvature throughout, near 0.9 km⁻¹, and coherence never sees it at any setting.
Find where curvature is zero on a steeply dipping bed. In step 1, set the fold relief to about 100 m and look at where the middle and bottom panels cross zero.
Hint
The dip curve and the curvature curve cross zero in completely different places, a quarter of a wavelength apart. On the default structure, curvature at the steepest point on the fold is a hundredth of a reciprocal kilometer against nearly one at the crest, a factor of some seventy. Curvature is zero exactly where the bed is steepest, and largest where the bed is level at a crest or a trough. The reason is worth saying carefully: at the steepest point the bed is momentarily straight, not level. It is dipping hard, but it is not bending, and bending is the only thing curvature measures. This is the single most common misreading of a curvature volume: a bright curvature anomaly is not a steep place, it is a place where the dip is changing quickly.
Find a saddle. In step 2, click near the plunging nose at the north end of one of the ridges, then rotate the profile through 180°.
Hint
The curve crosses zero twice, and k₁ comes out positive while k₂ comes out negative at the same point. How large each one is depends on exactly where you clicked; what matters is the pair of signs, and on the strongest saddles here the two are comparable in size. The reflector arches up along the ridge and falls away along its plunge, so it curves one way in one direction and the other way at right angles. No single number describes that, which is the whole reason two principal curvatures are reported instead of one average. A mean curvature here would be close to zero and would tell you nothing at all. Saddles are common on real surfaces, and on this modeled one they cover more than half the map.
Separate k_pos from k₁. In step 4, start with the regional dip at 0° and raise it to 25°, watching the ratio readout.
Hint
At 0° the two agree to within a percent. By 25° k_pos reads about half again what k₁ does. Notice which one moves: k_pos does not budge at all while k₁ falls away steadily. Adding a tilt does not change a second derivative, and k_pos is nothing but second derivatives. k₁ falls because it measures the bend along the bed, and a bed dipping at 25° is longer than the map distance it covers, so the same rise is spread over more reflector. Dividing that foreshortening out is what the dip correction does, and it is why k_pos reads high wherever the structure dips. On a structurally dipping survey that difference is systematic rather than random, so it does not average out — it biases the whole map in the direction of steepest regional dip. If you are comparing curvature between two areas of a survey that dip differently, this matters.
Watch the second derivative amplify noise. Set the operator length to 1 in the header, then put the dip-field noise up to about 15%.
Hint
The fold almost disappears under speckle. Now take the operator length up to 15 and the fold returns while the speckle goes. At an operator length of 1 the typical curvature reads an order of magnitude more than the same surface gives at 15 traces, and almost all of that is noise rather than geology. Noise in a dip field is not a cosmetic problem for curvature — differentiating twice multiplies short-wavelength content twice over, so a dip volume that looked acceptable in module 01 can be unusable once it has been differentiated. This is the practical reason modules 01 to 03 spent so long on getting dip right.
Choose a wavelength on purpose. Turn the short-wavelength texture up to about 20 and compare the two panels in step 5.
Hint
The short operator maps the fine fabric and loses the fold; the long operator maps the fold and loses the fabric. Both are correct computations of curvature on the same surface. If the fine fabric is fractures you want the short operator; if it is footprint you want the long one, and nothing in the data tells you which it is. That decision is yours and should be stated when the volume is handed on.

The main points

Curvature is the rate of change of dip. Flat beds and uniformly dipping beds both give zero. Only changes in dip produce an anomaly.
Bright curvature does not mean steep. Curvature peaks where the bed is flat at a crest or a trough, and passes through zero where the bed is steepest.
One number is not enough. A surface can curve up in one direction and down at right angles to it, so two principal curvatures are reported, together with the direction they point in.
k₁ and k_pos are different attributes. The principal curvatures correct for the dip the reflector already has; the most-positive and most-negative ones do not, and on dipping structure they read systematically high.
Operator length is an interpretation. Long-wavelength curvature maps folds, short-wavelength curvature maps fabric, and the same surface gives both.
Errors in the dip field arrive here amplified. Curvature is a second derivative, so short-wavelength error in dip is amplified twice over.

Method, sources, and what has been simplified

The structure is a grid of depths to a single reflector, built from a small number of analytic features — two anticlines, a syncline, and a dome — on an optional regional tilt, with optional short-wavelength texture and optional noise. Depth is used rather than time so that curvature has its proper units of one over length. Working from a time-migrated volume instead requires converting the dip components with a velocity, which AASPI does with a single constant and which its documentation describes as crude; the resulting error carries into curvature magnitude but has little effect on strike or azimuth.

Curvature is computed the way a volumetric attribute computes it, not analytically. The inline and crossline dip components are found by differencing the depth grid, smoothed along the operator length, and then differenced again to give the three second derivatives. From those, with p and q the dip components:

k_mean = [(1+q²)r − 2pqs + (1+p²)t] / 2(1+p²+q²)^{3/2}, k_gauss = (rt − s²) / (1+p²+q²)², and the principal curvatures are k₁ = k_mean + √(k_mean² − k_gauss) and k₂ = k_mean − √(k_mean² − k_gauss), with k₁ ≥ k₂ by construction. The uncorrected most-positive and most-negative curvatures are the eigenvalues of the same second derivative matrix without the terms in p and q, which is exactly the dip correction the AASPI documentation says they omit. Strike comes from the eigenvector direction, ψ = ½·atan2(2s, r − t).

Apparent curvature at an arbitrary azimuth uses Euler's theorem, k(φ) = k₁cos²(φ − ψ) + k₂sin²(φ − ψ). That the two principal directions are always perpendicular is a property of smooth surfaces, not an assumption made here.

Simplifications. This is a single surface rather than a volume, so there is no vertical derivative and none of the attributes that need one: reflector convergence, rotation about the normal, and the vertical smoothing that curvature3d applies. The operator here is a simple smoothing length rather than the fractional-derivative operator AASPI uses, which its documentation describes as a first derivative followed by a low-pass filter; the effect demonstrated is the same one. Curvedness, shape index and the shape components have a module of their own, as do amplitude curvature and aberrancy.

Roberts, A., 2001, Curvature attributes and their application to 3D interpreted horizons: First Break, 19, 85–100.
al-Dossary, S., and K. J. Marfurt, 2006, 3D volumetric multispectral estimates of reflector curvature and rotation: Geophysics, 71, P41–P51.
Bergbauer, S., T. Mukerji, and P. Hennings, 2003, Improving curvature analyses of deformed horizons using scale-dependent filtering techniques: AAPG Bulletin, 87, 1255–1272.
Mai, H. T., K. J. Marfurt, and S. Chávez-Pérez, 2009, Coherence and volumetric curvatures and their spatial relationship to faults and folds, an example from Chicontepec basin, Mexico: SEG Technical Program Expanded Abstracts, 1063–1067.
Chopra, S., and K. J. Marfurt, 2007, Seismic Attributes for Prospect Identification and Reservoir Characterization: SEG Geophysical Development Series, 11.
AASPI, Geometric Attributes: Program curvature3d, University of Oklahoma.

Curvature: how fast the dip is changing

The folds

The data

1From dip to curvature

2Curvature depends on which way you cut

3k₁ and k₂ across the whole map

4Most-positive curvature is not k₁

5Long and short wavelength

?Why bother with a third attribute

What it is used for

The three ways it will mislead you

Exercises

The main points

06 · Curvedness and shape index