Flattening

From Wikipedia, the free encyclopedia

Ellipticity redirects here. For the mathematical topic of ellipticity, see elliptic operator.

The flattening, ellipticity, or oblateness of an oblate spheroid is the "squashing" of the spheroid's pole, down towards its equator.

[edit] First and second flattening

The first, primary flattening, f, is the versine of the spheroid's angular eccentricity (" $o\!\varepsilon\,\!$ "), equaling the relative difference between its equatorial radius, $a\,\!$ , and its polar radius, $b\,\!$ :

$f=\operatorname{ver}(o\!\varepsilon)=2\sin\left(\frac{o\!\varepsilon}{2}\right)^2=1-\cos(o\!\varepsilon)=\frac{a-b}{a};\,\!$

The flattening of the Earth is 1:298.25275 (which corresponds to a radius difference of 21.385 km of the Earth radius 6378.135–6356.75 km) and would not be realized visually from space;
The flattening of Jupiter (1:16) and Saturn (nearly 1:10), in contrast, can be seen even in a small telescope;
Conversely, that of the Sun is less than 1:1000.

The amount of flattening depends on

the relation between gravity and centrifugal force;

and in detail on

size and density of the celestial body (see Figure of the Earth, last chapter);
the rotation of the planet or star;
and the elasticity of the body.

There is also a second flattening, f' (sometimes denoted as "n"), that is the half-angle tangent² of $o\!\varepsilon\,\!$ :

$f'=\tan\left(\frac{o\!\varepsilon}{2}\right)^2=\frac{1-\cos(o\!\varepsilon)}{1+\cos(o\!\varepsilon)}=\frac{a-b}{a+b};\,\!$

[edit] Flattening without picking

Flattening without picking is an efficient full-volume automatic dense-picking method for flattening seismic data. First, local dips (step-outs) are calculated over the entire seismic volume. The dips are then resolved into time shifts (or depth shifts) relative to reference trace using a non-linear Gauss-Newton iterative approach that exploits Discrete Cosine Transforms (DCT's) to minimize computation time. At each point in the image two dips are estimated; one dip in the x direction and one dip in the y direction. Because each point in the image has two dips, each horizon is estimated from an over-determined system of dips in a least-squares sense.