## NOAA KLM User's Guide## Appendix I.4 |

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Figure I.3-1 represents a cross section of the earth with polar radius r_{p} and equatorial radius r_{e}. A feature is represented at
height h_{f }above Mean Sea Level. The height is measured along a line through the feature and perpendicular to the earth ellipsoid's surface. The distance
of the feature from the earth's center is r_{f}.

φ_{gcf} ≡ geocentric (or geometric) latitude of the feature.

-90^{0} ≤φ_{gcf} ≤ 90 ^{0}. North is positive.

φ_{gdf} ≡ geodetic latitude of the feature.

-90^{0} ≤φ_{gdf} ≤ 90 ^{0}. North is positive.

The boundary of the earth's cross section is assumed to be an ellipse formed by the equation

p>when the cross section in the xz plane.A. Conversion from Geodetic to Geocentric Latitude

Suppose that φ_{gdf} is known. Then the slope of the line tangent to the earth's cross section at φ_{gdf}
is

This slope is equal to the derivative dz_{s}/dx_{s}. From the equation for the earth's cross section,

So

or

where (x_{s}, z_{s}) is the point on the cross section boundary at geodetic latitude φ_{gdf}

Solving for x_{s} gives

Substitution in the equation for the earth's cross section gives

Substitution of this back in the equation for x_{s} gives

These equations for x_{s} and z_{s} can be put in the forms

where θ_{f} is the East longitude of the feature,

and

because x_{s} and y_{s} will always have the same signs as cos(θ_{f}) and sin(θ_{f}), respectively, and z_{s} will
always have the same sign as sin(φ_{gdf}). The coordinates (x_{f}, y_{f}, z_{f}) of the feature can now be calculated.

and

Finally, so

and the distance to the feature from the earth's center is

B. Conversion from Geocentric to Geodetic Latitude

Assume that φ_{gcf}, the geocentric latitude of a feature is known. If the features geodetic latitude is to be found, its height above Mean Sea Level must
also be known. The features height above Mean Sea Level can be calculated if its distance from the center of the earth is known.

Case 1 - h_{f} the height of the feature above Mean Sea Level, is known

Since φ_{gdf} will not be known initially for substitution into the right side of the above equation, substitute a guess, starting with φ
_{gdf} ≈ φ_{gcf}, to compute a better guess. Interate until the guesses converge.

Case 2 - r_{f}, the distance of the feature from the center of the earth, is known.

Since ,

Define

then
, and this equation could be solved for h_{f} if φ_{gdf} were known.

Since φ_{gdf} is not known initially, guess that φ_{gdf} ≈ φ_{gcf}, solve for the corresponding h_{f}, use this as a
guess height
together with the guess φ_{gdf} is the same as the most recent guess. Then use φ_{gdf} to find a new guess for h_{f}. Interate this procedure
until both φ_{gdf} and h_{f} converge to stable values.

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Last Updated Tuesday, 03-Mar-2009 10:29:33 EST

Please see the NCDC Contact Page if you have questions or comments.