Flattening

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.

First and second flattening

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

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

• The flattening of the Earth in WGS-84 is 1:298.257223563 (which corresponds to a radius difference of 21.385 km of the Earth radius 6378.137 - 6356.752 km) and would not be realized visually from space, since the difference represents only 0.335 %.
• 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 and that of the Moon barely 1:900.

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 squared half-angle tangent of $o\backslash !\backslash varepsilon\backslash ,\backslash !$:

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

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.

See also

• Angular eccentricity
• Astronomy
• Earth rotation
• Gravitational field

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