Module 06 · Geometric attributes
Module 05 left you with two numbers at every sample, k₁ and k₂, and two numbers are hard to look at. Recombining them into how strongly it bends and what shape it bends into turns out to be far easier to interpret — and the shape half of that pair asks a question no single curvature volume can answer: is this a dome, a ridge, a saddle, a valley or a bowl?
The shape you have built — a patch of reflector, seen obliquely
Set the two curvatures — everything follows from these
Work through the five steps in order, then try the exercises. Method and references are there when you want them.
Every point on a reflector carries two principal curvatures. Plot them as a point on a graph with k₁ across and k₂ up, and the whole vocabulary of shapes falls out of where the point sits rather than what its coordinates are.
Two things about that point turn out to matter, and they are independent of one another. How far it is from the origin says how strongly the surface is bent — that is curvedness. Which direction it lies in says what shape it is bent into — that is the shape index. A dome and a bowl sit on opposite sides; a saddle sits on the diagonal where k₁ and k₂ are equal and opposite; and the origin is a flat plane, where the shape question has no answer at all.
1 The k₁–k₂ plane your point, and the five named regions
2 The same point, in polar terms distance is curvedness, angle is shape
Shape index runs from −1 to +1 and every value along it has a name. Walk the preset buttons in the header from Dome to Bowl and watch the marker travel down the scale while the little surface in the header morphs from one shape into the next.
The useful thing about this scale is that it is continuous. Real geology is not restricted to five shapes; it sits somewhere between them, and the number tells you where. A value of +0.7 is a dome that has been stretched a bit, +0.3 is a ridge with a slight bowl to it, and so on. The five names are landmarks on a continuum, not boxes.
From one point to a whole map. Steps 1 and 2 worked on a single point whose two curvatures you set by hand. From here on the same arithmetic runs at every sample of a modeled surface, shown in the header: a bowl in the north-west and a dome in the south-east, a north–south ridge below the bowl and a syncline above the dome, and a broad gentle swell underneath the lot which makes saddles wherever it meets the sharper features. One of every shape, so every component in step 5 has something to find.
Curvedness is the other half of the pair, and it throws the shape away completely. A sharp dome and an equally sharp bowl have identical curvedness, because both are bent just as hard — one up, one down. It is always positive or zero.
On its own that makes it a good deformation intensity map. Bright means the reflector is strongly bent, whatever way it happens to be bending, so faults, flexures and fold hinges all come out bright and gently dipping monoclines come out dark.
1 Curvedness over the map deformation intensity, sign discarded
2 k₁ for comparison signed, so anticlines and synclines separate
Shape index is built from a ratio, and on a flat piece of reflector both curvatures are essentially zero, so the ratio is noise divided by noise. The attribute still returns a definite value of +0.5 or −1 for a patch of ground that has no shape at all.
Displayed raw, a shape index map is therefore fully saturated everywhere, and the strongest colors are often on the flattest ground. The standard fix is to modulate it by curvedness, so that shape is only allowed to show where there is enough bend for the question to mean anything. Compare the two panels — the second is the same attribute with its brightness governed by curvedness.
1 Shape index, raw saturated even where there is no shape
2 Shape index modulated by curvedness shape only where there is shape
Rather than ask you to read a shape index map by eye, the shape can be split into five separate volumes. Each one is simply curvedness, kept where the shape index falls in that shape's band and zeroed everywhere else. So the dome volume is bright where the surface is strongly bent and dome-shaped, and blank everywhere else.
These are easy to interpret and easy to co-render, and they are the form most people end up using. Pick a shape below and see where on this surface it occurs.
1 Shape index, modulated for reference, from step 4
2 The shape component you chose curvedness, gated by shape
Everything so far has been geometry. These two attributes are used in place of k₁ and k₂ for the following reasons.
Shape relates to process. Different geological processes make different shapes, so the shape index is a first, cheap indication of mechanism:
Curvedness carries intensity, and intensity relates to strain. Bending a brittle layer puts its outer arc into extension, so strongly curved zones are where open fractures are most likely to have formed. Curvedness is the attribute that answers "how hard has this rock been bent", independently of which way, which makes it a common input to fracture-density prediction and to picking where a horizontal well should be landed. Treat that as a correlation to be tested against well data, not a law — curvature is geometry, and fractures depend on lithology, burial history and the stress field as well.
And two numbers are easier to work with than two curvatures. Co-rendering k₁ and k₂ means reading two signed volumes at once. Curvedness modulated by shape index puts intensity in the brightness and mechanism in the color, which is one image instead of two, and the shape components turn it into a set of yes-or-no maps that can be handed to someone else without a tutorial.
The header sliders set k₁ and k₂ directly, so the first two steps are arithmetic on two numbers you control rather than anything extracted from data. The little surface patch is the quadratic z = ½(k₁x² + k₂y²), which is the surface those two principal curvatures describe near a point, drawn in oblique projection. The maps in steps 3 to 5 come from a modeled structure containing a dome, a ridge, a syncline and a bowl, with curvature computed the same way module 05 computes it.
Curvedness is C = √((k₁² + k₂²) / 2), always positive or zero. Shape index is s = (2/π)·tan⁻¹[(k₁ + k₂) / (k₁ − k₂)] with k₁ ≥ k₂, which gives +1 for a dome, +0.5 for a ridge, 0 for a saddle, −0.5 for a valley and −1 for a bowl. Those five values are checked numerically by the code rather than asserted. The shape components are curvedness kept inside a shape index band and zeroed outside it: bowl below −0.75, valley from −0.75 to −0.25, saddle from −0.25 to +0.25, ridge from +0.25 to +0.75, and dome above +0.75.
Conventions differ between packages, so check before you compare. Curvedness is written both as √(k₁² + k₂²) and as √((k₁² + k₂²)/2) in the literature. The second is Roberts (2001) and is the one used here; the two differ by a factor of √2, which matters if you are comparing absolute numbers and not at all if you are looking at a map. Shape index is also published with either sign. The convention here puts domes at +1, which is the more common one, but it is worth confirming which way round your own software has it before trusting the sign of a printed number.
Because shape index is a ratio, it is undefined where the surface is planar and both curvatures vanish. AASPI's approach, followed here, is to modulate the display by curvedness so that the shape is only visible where there is enough curvature to define it — and the shape components do the same thing by construction, since curvedness is what they are made of.
Simplifications. This is one surface rather than a volume. Everything here applies unchanged to amplitude curvature: the same curvedness and shape index are built from e_pos and e_neg in module 07, and AASPI outputs the same five shape components for them, prefixed e_.
Next up
The same arithmetic on a completely different surface: not the shape of the reflector, but how bright it is as you walk along it.
Open module 07 → All modules