Posts Tagged True Altitude

ILS Glideslope and the Earth’s Curvature

Have you ever wondered what the errors are in following the glideslope?  Obviously, the ILS system is capable of being approved for category III-c approaches so we can assume the glideslope is fairly accurate.  I would agree but we aren’t really talking about those issues.  Have you ever looked at an approach plate to see a 3° glideslope and did the math to find that your calculation is actually a little bit higher than 3°.  I always assumed the approach plate was incorrect; when actually I was wrong all along.

First off, the glide slope transmitter is station about 600 feet off to one side of the runway in the touchdown zone.  This transmitter is installed so that a beam travels at an average of 3° upward.  If we were to do the math, we would realize that for every 1 NM from the GS transmitter the beam raises 318 feet AGL as compared to the elevation of the of the transmitter.   Drawing out a triangle we get a formula which you can paste into excel of =tan(radians(3))*6076 = 318 feet  (we need to convert degrees to radians for excel to use).  We can use a triangle because the transmitter sends out a straight beam.

Since the earth is a sphere we need to account for the curvature of the earth.  Imagine you have a ball and you place a ruler exactly on the top of the ball.  When you go out  to the edge of the ruler, The distance between the ruler and the ball is greater than it is at the top of the ball.  The same concept applies here.

The farther we are from the glideslope the more the earth curves away below us.  So we were viewing the glideslope beam from the side while we were standing on the ground.  It would appear to us that the glideslope actually curves up.  At 1NM the curvature adds around 1 foot.  At 2NM it adds around 4 ft.  At 15NM, the earth curvature adds just under 200 feet.  Let me show you graphically what I mean.

The graph on the right plots the true altitude of the glideslope with and without the earth curvature correction.  The purple line on the bottom shows the difference between the two altitudes.  As you can see, the farther we get from the transmitter the more the curvature of the earth plays a role in the true altitude.

The FAA has long since known this issue and have published a formula in the TERPS manual.  I am going to use FAA Order 8260.54A.  On page 69 of the order has the heading “Determining the Glidepath Altitude at a Fix for both an ILS/LPV and LNAV/VNAV.  For my example, we are going to use the ILS 11L approach into Tucson, AZ (KTUS).

If you can get your hands on a textual description of the instrument approach procedure you can get exact coordinates for all the fixes as well as the Landing Threshold Point (which includes elevation) and threshold crossing height.  Finding these textual procedures are not easy but you can view textual procedures for approaches the FAA is implementing here.   I plugged these numbers into the formula below.

The glide slope intercept altitude (GSIA) is at POCIB and the MSL altitude is 4600’.  If I were to calculate the standard with the curvature corrected I would get an altitude of 4566’.

The textual description of this approach shows the LTP elevation of 2578’ feet (the touchdown zone elevation is 2599’) and the threshold crossing height is 55’.  The distance between POCIB and the LTP is 6.07 NM or 36,881’.

The landing threshold point is pretty cool.  It is defined as the exact longitude and latitude and elevation of the first brick on the runway.  It is used as an alignment point for the procedure.  There is also something called a FTP or fictitious threshold point for procedures not aligned with the runway.  Here is the LTP for runway 11L in Tucson.  Notice the red line showing the final approach course from POCIB to RW11L.