COMMUNICATIONS RANGE vs. PAYLOAD ALTITUDE and DISTANCE
Ralph Wallio, WØRPK

As a high altitude balloon/payload gradually ascends to higher and higher altitudes its horizon is gradually extended further and further away in all directions. This increasing distance to the horizon defines the payload's line-of-sight VHF/UHF radio communication footprint. A ground station within this footprint will perceive the payload to be above its horizon and radio communication is possible.

A first approximation of the relationship between payload altitude and maximum straight line-of-sight distance can be calculated with an easy formula that is a reduction of required trigonometry:

 Distance [miles] = 1.23 * SQRT(Altitude [feet]) 39 miles at 1,000ft 123 miles at 10,000ft 389 miles at 100,000ft

(See below for the trigonometric heritage of this formula.)

This formula can be applied to selected altitudes to compile a table of maximum distances:

 ALTITUDE [feet] 20,000 40,000 60,000 80,000 90,000 100,000 DISTANCE [miles] [km] 174mi             280km 246mi             396km 301mi             484km 348mi             560km 369mi             594km 389mi             626km

Distance values from this table can be transferred to a map by using a service available from the FCC at http://www.fcc.gov/mb/audio/bickel/circleplot.html . Enter a set of payload coordinates such as the launch site (NWS/Valley in this example, 41d19m11sN 96d21m58sW). Use DISTANCE [km] for the six circle radii. Turn off all Map Features Options. Submit Initial Data and then Go Get The Map. These inputs should result in this map.  Resulting distance circles on this map predict the maximum line-of-sight communications coverage area when the payload is at 20,000, 40,000, 60,000, 80,000, 90,000 and 100,000ft (red). (See High Altitude Balloon RADIO COMMUNICATIONS COVERAGE -
Second Approximation based on Topographic Terrain and Radio Mobile Deluxe
, for a more arduous prediction method.)

This process has sufficient accuracy for the vast majority of uses. However, it needs to be understood that the payload will drift with winds aloft and the center of the coverage circles needs to move accordingly. During the summer this drift is minimal, on the order of 20 miles or less, but during the winter drift can be 100 miles and more. One strategy for understanding the impact of drift is to make a map centered on launch coordinates, another map centered on predicted burst coordinates, a third map centered on predicted touchdown coordinates and then interpolate (gauge by eye) as necessary.

The formula constant 1.23 is for straight line-of-sight distances. However, VHF and UHF radio signals will bend slightly toward the earth's surface. This bending effectively increases the radio horizon and therefore slightly increases the formula constant. The ARRL Antenna Book (page 23-5) gives a constant of 1.415 for weak signals during normal tropo conditions. (This constant can increase significantly during rare VHF/UHF tropo and other openings.)

I have carefully observed signal strengths during distant missions that were transmitting on an otherwise quiet frequency. These observations were matched with payload coordinates from which distances were calculated. These distances were matched with payload altitudes so that actual formula constants could be calculated:

 Very weak signal via product detector Stronger signal breaking FM squelch Copy first short packet Copy first long packet Start of ~100% packet copy 1.38 1.34 1.28 1.27 1.25

Packet reception never reaches perfect reliability (see Packet Radio Bit Error Rate) but a formula constant of 1.25 represents roughly 24dB of quieting (Eb/No) required for almost 100% packet reception. Therefore, it is only slightly conservative for normal tropo conditions to use a value of 1.23 as in the table above.

Ground station look angle (elevation above the horizon) also changes with payload distance and altitude. Line-of-sight VHF and UHF paths yield very strong signals so antenna gain and resulting beamwidth are usually not issues. Beamwidth of low gain antennas is wide enough that most stations do not need to worry about using antenna elevation mounts. However, stations very close to the payload, e.g., chase vehicles, may have look angles approaching 90 degrees (straight up).

A Ground Station Look Angle table can be calculated with increasing distance and altitudes: A look angle of 0.0 degrees (the station's horizon) corresponds with a formula constant of 1.23, e.g., 123 miles at 10,000ft.

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Courtney Duncan, N5BF, offers the following trigonometric solution to derive this simple formula:

Range (miles) = 1.23 * sqrt (height in feet).

The whole problem is over a region of the earth and heights above it where the approximation
re >> ht and the approximation that the earth is spherical are reasonable.  (re is the radius of the earth, ht is height above the surface.)

For a spherical earth and no non-geometric propagation effects, the correct
way to calculate geometric range is

range = re * acos (re / (re + ht))

where 'acos' is "arc cosine" or "inverse cosine" in radians and re and ht are in the same units of length.

This represents a right triangle with the right angle at the maximum reception range (target at altitude "ht" is on the horizon), the hypotenuse connects the center of the earth with the target, the "opposite" side is the slant range from observer to target, and the "adjacent" side connects the observer with the center of the earth.  The angle at the center of the earth (in radians) times earth radius is the desired great-circle range.  In this problem, the angle is rather small, on the order of a tenth of a radian (5-6 degrees).

For small x, (i.e. x << 1) we can approximate

cos x ~ 1 - 1 / x^2

meaning

x ~ acos ( 1 - 1 / x^2)

substituting

u = 1 - 1 / x^2

and thus

x = sqrt(2) * sqrt(1 - u)

we have

acos u ~ sqrt(2) * sqrt(1 - u)

So now we have a good approximation for range:

range ~ re * sqrt(2) * sqrt( 1 - (re / (re + ht)))

Let's work on that last radical for a minute:

1 - (re / (re + ht)) = (re + ht - re) / (re + ht) = ht / (re + ht)

and further, since re >> ht

~ ht / re

That's better.  So now we have

range ~ re * sqrt(2) * sqrt( ht / re ) = sqrt ( 2 * re) * sqrt( ht )

Taking re as the WGS-84 earth equatorial radius, 6378.137 km, for all values in km, this gives us:

range (km)  ~ 112.94 * sqrt( ht )   (ht in km)

Now we just need to convert units.  You want this in statute miles with ht in feet.

The conversion from km to sm is sm = km * 0.62137.
The conversion from ft to km is km = ft / 3280.83.

So we have

range (sm) ~ 112.94 * 0.62137 * sqrt( ht (ft) / 3280.83)
= 70.178 * sqrt( 1 / 3280.83 ) * sqrt( ht )    (ht in feet)
= 1.225 * sqrt( ht ).

Pretty close to 1.23.

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