NOAA KLM User's Guide

Appendix I.4

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Appendix I.4: Conversion Between Geodetic and Geocentric Latitude


Figure I.3-1 represents a cross section of the earth with polar radius rp and equatorial radius re. A feature is represented at height hf 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 rf.

φgcf ≡ geocentric (or geometric) latitude of the feature.

-900 ≤φgcf ≤ 90 0. North is positive.

φgdf ≡ geodetic latitude of the feature.

-900 ≤φgdf ≤ 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 φgdf is known. Then the slope of the line tangent to the earth's cross section at φgdf is

-tan(900 - φgdf) = 1 /tan(φgdf).

This slope is equal to the derivative dzs/dxs. From the equation for the earth's cross section,

two times x sub s divided by r sub e squared plus two times z sub s times the deriviative of z sub s by x sub s divided 
by r sub p squared equal 0
the derivative of z sub s by x sub s is equal to minus the quantity r sub p divided by r sub e squared times x sub s divided 
by z sub s

So

one divided by tan of phi sub gdf is equal to the quantity r sub p divided by r sub s squared times x sub s divided by z sub s

or

tan of phi sub gdf is equal to the quantity r sub e divided by r sub p squared times z sub s divided by x sub s

where (xs, zs) is the point on the cross section boundary at geodetic latitude φgdf

Solving for xs gives x sub s is equal to the quantity r sub e divided by r sub p squared times z sub s divided by tan of phi sub gdf

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

the quantity r sub e squared times z sub s squared divided by r sub p to the fourth power times the tan squared of phi sub gdf 
plus z sub s squared divided by r sub p squared is equal to one
or

z sub s squared is equal to one divided the quantity of r sub squared divided by r sub p to the fourth power times tan 
squared of phi sub gdf plus the quantity of one divided by r sub p squared and can be rewritten as the quantity r sub to the fourth power times tan squared of phi 
sub gdf divided by the quantity r sub e squared plus r sup p squared times tan squared of phisub gdf

z sub s is equal to plus or minus r sub p times tan of phi sub gdf divided by the square root of r sub e divided by r 
sub p squared plus tan squaredof phi sub gdf
Z sub s is equal to plus or minus r sub p times tan of phi sub gdf divided by the square root of r sub e divided by r 
sub p squared plus tan squared of phi sub gdf

Substitution of this back in the equation for xs gives

x sub s is equal to plus or minusnthe quanity r sub e divided by r sub p squared times tan of phi sub gdf divided by 
the quantity tan sub gdf times the square root of r sub e divided by r sub p squared plus tan squared of phi sub gdf
x sub s is equal to plus or minus r sub e squared divided by r sub p times the square root of r sub e divided by r sub 
p squared plus tan squared of phi sub gdf

These equations for xs and zs can be put in the forms

x sub s is equal to r sub e squared tims cos phi sub gdf times the cos of theta sub f divided by r sub p times the 
square root of r sub e divided by r sub p squared times cos squared of phi sub gdf plus sin squared of phi sub gdf (I-31)

where θf is the East longitude of the feature,

y sub s is equal to r sub e squared times cos of phi sub gdf times sin of theta sub f divided by r sub p times the 
square root of r sub e divided by r sub p squared times cos squared of phi sub gdf plus sin squared of phi sub gdf(I-32)

and

z sub s is equal to r sub p squared times sin phi sub gdf divided by r sub p times the square root of r sub e divided 
by r sub p squared times cos squared of phi sub gdf plus sin squared of phi sub gdf(I-33)

because xs and ys will always have the same signs as cos(θf) and sin(θf), respectively, and zs will always have the same sign as sin(φgdf). The coordinates (xf, yf, zf) of the feature can now be calculated.

x sub f is equal to the quantity r sub e squared divided by r sub p times the square root of r sub e divided by r sub p 
squared times cos squared of phi sub gdf plus sin squared of phi sub gdf plus h sub f.  All times cos phi sub gdf times cos theta sub f

y sub f s equal to the quantity r sub e squared divided by r sub p times the square root of r sub e divided by r sub p 
squared times cos squared of phi sub gdf plus sin squared of phi sub gdf plus h sub f.  All times cos phi sub gdf times sin theta sub f

and

 z sub f s equal to the quantity r sub p squared divided by r sub p times the square root of r sub e divided by r sub p 
squared times cos squared of phi sub gdf plus sin squared of phi sub gdf plus h sub f.  All times sin phi sub gdf

Finally, tan phi sub gcf equals z sub f divided by square root of x sub f squared and y sub f squared so

tan of phi sub gcf is equal to the quantity r sub p squared plus h sub f times r sub p times the square root of r sub e 
divided by r sub p squared time cos squared phi sub gdf plus sin phi sub gdf divided by the quantity r sub e squared plus h sub f time r sub p times the square root of r 
sub e divided by r sub p squared time cos squared phi sub gdf plus sin phi sub gdf divided by the times tan of phi sub gdf (I-34)

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

r sub f equals square root of x sub f squared plus y sub f squared plus z sub f squared

B. Conversion from Geocentric to Geodetic Latitude

Assume that φgcf, 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 - hf the height of the feature above Mean Sea Level, is known

tan of phi sub gdf is equal to the quantity r sub e squared plus h sub f times r sub p times the square root of r sub e 
divided by r sub p squared time cos squared phi sub gdf plus sin phi sub gdf divided by the quantity r sub p squared plus h sub f time r sub p times the square root of 
r sub e divided by r sub p squared time cos squared phi sub gdf plus sin phi sub gdf divided by the times tan of phi sub gcf

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

Case 2 - rf, the distance of the feature from the center of the earth, is known.

Since r sub f equals square root of x sub f squared plus y sub f squared plus z sub f squared ,



tan of phi sub gdf is equal to the quantity r sub e squared plus h sub f times r sub p times the square root of r sub e 
divided by r sub p squared time cos squared phi sub gdf plus sin phi sub gdf divided by the quantity r sub p squared plus h sub f time r sub p times the square root of 
r sub e divided by r sub p squared time cos squared phi sub gdf plus sin phi sub gdf divided by the times tan of phi sub gcf
r sub f squared is equal to the quantity r sub e squared divided by r sub p times the square root of the quantity r sub e 
divided by r sub p squared times cos squared of phi sub gdf plus sin squared of  phi gdf plus h sub f  quantity closed squared times cos squared of phi gdf plus

the quantity r sub p squared divided by r sub p times the square root of the quantity r sub e divided by r sub p squared
 times cos squared of phi gdf minus the sin squared of phi gdf plus h sub f quantity closed times sin squared of phi sub gdf

r sub f squared is equal to r sub e to the fourth power times cos squared of phi sub gdf divided by r sub p squared times the 
quantity of r sub e divided by r sub p squared times cos squared of phi sub gdf plus sin squared of phi sub gdf plus

two times r sub e squared time h sub f times cos squared of phi sub gdf divided by r sub p times the square root of r sub e 
divided by r sub p squared times cos squared of phi sub gdf plus sin squared of phi sug gdf plus h sub f squared times cos squared of phi sub gdf plus

r sub p to the fourth power times sin squared of phi sub gdf divided by r sub p squared times the quantity of r sub e divided 
by r sub p suared times cos squared of phi sub gdf plus sin squared of phi sub gdf plus

two times r sub p squared time h sub f times sin squared of phe sub gdf divieded by r sub p times the square root of r sub e 
divided by r sub p squared times cos squared of phi sub gdf plus sin squared of  phi sub gdf plus h sub f squared times sin of phi sub gdf

r sub  f squared is equal to h sub f squared plus two times h sub f times the quantity r sub e squared times cos squared of 
phi sub gdf plus r sub p squared tiems sin squared ofph sub gdf divided by r sub p times the square root of r sub e divided by r sub p squared times cos squared of phi sub 
gdf plus sin squared on phi sub gdf plus

r sub e to the fourth power times cos squared of phi sub gdf plus r sub p to the fourth power times sin squared of phi sub gdf 
all divided by r sub p squared times the quantity r sub e divided by r sub p squared times cos squared of phi sub gdf plus sin squared of phe sub gdf

r sub f squared is equal to h sub f squared plus two times h sub f tims r sub p times the square root of r sub e divided by r 
sub p squared times the cos squared of phi sub gdf plus sin squared of phi sub gdf plus

r sub p squared times the quantity r sub e divided by r sub p to the fourth power times cos squared of phi sub gdf plus sin 
squared of phi sub gdf divided by the quantity r sub e divided by r sub f squared times cos squared of phi sub gdf plus sin squared of phi sub gdf

Define

A is defined as 2 times r sub p times the square rood of r sub e divided by r sub p squared times cos2 of phi sub gdf plus sin 
squared of phi sub gdf

B is difined as r sub p squared times the quanitiy r sub e divided by r sub p to the fourth power times cos squared of phi sub 
gdf plus sin squared of phi sub gdf all divided by r sub e divided by r sub p squared time cos squared of phi sub gdf plus sin squared of phi sub gdf minus r sub f squared

then h sub f squared plus A times h sub f plus B is equal to zero , and this equation could be solved for hf if φgdf were known.

Since φgdf is not known initially, guess that φgdf ≈ φgcf, solve for the corresponding hf, use this as a guess height together with the guess φgdf is the same as the most recent guess. Then use φgdf to find a new guess for hf. Interate this procedure until both φgdf and hf converge to stable values.

Amended May 8, 2006


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