## Monday, September 11, 2023

### Mediumwave Skywave Prediction #5 - Dissecting The Formulas

We'll get to the actual skywave prediction formulas shortly, but first let's talk about how to calculate the geomagnetic midpoint of our signal path. To get this, we'll need the latitude and longitude of both the transmitter and receiver sites. We'll also need the latitude and longitude of geomagnetic north, which moves by small increments each year. A nice chart can be found at:

https://wdc.kugi.kyoto-u.ac.jp/poles/polesexp.html

The following geomagnetic north pole coordinates are accurate for 2023:

dipoleN = 80.8° latitude (actual)
dipoleW = 72.7° longitude (actual) -use a positive number in the final mid-point formula

Note: dipoleN and dipoleW are the geomagnetic north pole, NOT the magnetic north pole. There is a difference. To reiterate from the previous post, 'geomagnetic poles (dipole poles) are the intersections of the Earth's surface and the axis of a bar magnet hypothetically placed at the center the Earth by which we approximate the geomagnetic field. They differ greatly from the magnetic poles, which are the points at which magnetic needles become vertical. The magnetic poles are what has been "wandering", a subject in the news lately, but they drag the geomagnetic poles with them too, albeit at a lesser rate.'

First we'll calculate the actual geographic mid-point latitude and longitude between transmitter and receiver. The Movable-Type scripts website has our formula to do that:

Many websites have geographic mid-point calculators as well. Those familiar with Javascript can use the formula below or convert it to a different language if you wish to do the calculation yourself.

lat1 = transmitter latitude in degrees
lon1 = transmitter longitude in degrees
lat2 = receiver latitude in degrees
lon2 = receiver longitude in degrees

double dLon = Math.toRadians(lon2 - lon1);

double Bx = Math.cos(lat2) * Math.cos(dLon);
double By = Math.cos(lat2) * Math.sin(dLon);
double lat3 = Math.atan2(Math.sin(lat1) + Math.sin(lat2), Math.sqrt((Math.cos(lat1) + Bx) * (Math.cos(lat1) + Bx) + By * By));
double lon3 = lon1 + Math.atan2(By, Math.cos(lat1) + Bx);

mid_lat = Math.toDegrees(lat3);
mid_lon = Math.toDegrees(lon3);

mid_lat and mid_lon is the actual geographic mid-point of our path.

Now, we'll use a separate formula to translate the actual mid-point latitude-longitude to geomagnetic latitude:

Asin(Sin(mid_lat) * Sin(dipoleN) + Cos(mid_lat) * Cos(dipoleN) * Cos(dipoleW + mid_lon))

ThetaM will be in radians and must be converted to degrees. To do so:

ThetaM(degrees) = (ThetaM(radians) * 180) / Pi

ThetaM is the geomagnetic latitude in degrees.

Let's dive right into the skywave prediction formulas. We know slant distance, geomagnetic latitude of the path mid-point, and we have everything we need to calculate Kr, the aggregated ionospheric losses. Take note that the skywave prediction normally is the prediction for the local midnight hour, equidistant between sunset and sunrise, commonly referred to as SS+6, or sunset + 6 hours.

WANG FORMULA DETAILS

The Wang method is the only method which offers good to excellent results for short and long paths alike at all frequencies in the LF/MF bands, at all latitudes, and in all regions. It has been demonstrated that the Wang method is the only method that can be considered a true worldwide method.

The Wang expression for field strength is:

Note: FS(dBu), is also known as dBµV/m.Where: FS(dBu) is the field strength in dBµV/m, V is the transmitter cymomotive force above the reference 300 mV in dB (better known as our effective radiated power (ERP) referenced to 1 KW in the direction of interest). ThetaM is the mid-point transmitter and receiver geomagnetic latitude, Dslant is the slant distance in km. Kr, or generalized ionospheric losses, are described below.

In the Wang method, the 107 (dB) factor is used for most of the world. New Zealand and Australia use 110 dB, giving that part of the world a 3 dB field strength improvement (half an S-unit).

To convert FS(dBu) back to millivolts per meter: mV/m = 10 ^ (FS(dBu) / 20) / 1000

The generalized ionospheric losses are found in Wang's Kr factor. Both Wang and the FCC method calculate Kr in this manner:

Kr is the loss factor in dB, to include ionospheric absorption, focusing and terminal losses, losses between hops, geomagnetic latitude influence, and basic polarization coupling loss.

Where: ThetaM is the geomagnetic latitude defined previously. Dslant, the slant distance, will modify Kr accordingly.

Wang recommends that the geomagnetic mid-point latitude, ThetaM, be between -60 (south) and +60 degrees (north). When compared to the ITU expression, Wang's expression is symmetrical about zero degrees latitude and is not dependent on frequency.

Let's do a Kr loss example for a 1500 km slant path and see what our ionospheric losses are.

Here are the results for a single hop, 1500 kilometer (932 miles) slant distance for various mid-point locations. Using this Wang formula, I've prepared a chart showing the additional losses, in dB, caused by geomagnetic latitude influence.

Basic Loss  Deviation  Geo-Lat Mid-Point (actual location of)
---------- ----------- ------- ------------------------------
7.854 dB      0          9.19   0°N, over the equator
8.347 dB     +0.493dB   18.15  10°N, over Venezuela
10.637 dB    +2.783dB   34.86  25°N, over south Florida
11.778 dB    +3.924dB   39.37  30°N, over north Florida
14.553 dB    +6.669dB   46.76  38°N, over Richmond VA
17.889 dB   +10.035dB   52.36  43°N, over Rochester NY
18.767 dB   +10.913dB   53.50  45°N, over Minneapolis MN
21.233 dB   +13.379dB   56.21  48°N, over Grand Forks ND

Basic Loss = the basic loss on this 1500 km path. The first entry has its mid-point (reflection point) over the equator.
Deviation = additional loss incurred as latitudes increase using the basic equator loss as the base.
Geo-Lat = the adjusted geomagnetic latitude of the reflection mid-point.
Mid-Point = the actual geographic location of the reflection midpoint.

As you can see, we have lost over 13 dB in field strength when the reflection point is at 48° actual latitude!

Here is an example of how geomagnetic positioning of the signal path affects the final field strength result. Reception of KFAB (1110 kHz), Omaha, Nebraska (41.23°N, 96.0°W) here in Rochester, NY, places the mid-point of our ionospheric reflection at a geomagnetic latitude of 51.355 degrees. The slant distance is 1548 km. An overall Kr loss of 17.45 dB gives an additional geomagnetic position penalty of some extra 9.596 dB over tropical paths!

Now, let's calculate an expected skywave field strength value for 50 KW KFAB-1110 here in Rochester, NY. From above, we already know our slant distance is 1548 km. Our Kr loss factor from the example above is 17.45 dB.

FS(dBu) = V(-18.739) + 107 - 20 * Log10(1548) - 17.45

FS(dBu) = 7.01

Converting to millivolts per meter:

mV/m = 10 ^ (FS(dBu) / 20) / 1000 ...convert dBu back to mV/m, or:

.00224 = 10 ^ (7.01 / 20) / 1000

.00224 mV/m is a weak signal indeed.

Why such a weak signal from a 50 KW powerhouse station at only ~1500 km? We are placed perfectly in KFAB's deep cardioid pattern notch at 76 degrees azimuth and a 4 degree takeoff angle. Facing us at those angles is a theoretical and nearly-microscopic 12.72 watts ERP. This is a primary lesson we learn from tower array pattern analysis, both skywave and groundwave. One will naturally think, "Well, it's a 50 KW station, and only a mere 960 miles distant. I should be getting a pretty good signal". Not necessarily so. If you are in a deep notch of a pattern, you may only be "seeing" a few watts facing you.

Take a look at the graphic below. You will see the deep cardioid notch of KFAB's nighttime pattern. Stations to the east suffer a great signal loss.

 KFAB Nighttime Pattern. Click for larger image.

Where did the V(-18.739) figure come from, you ask? That is KFAB's facing 12.72 watts aimed at us, referenced to 1 KW, in dBW. It is 18.739 dB down from 1 KW. How did we get this and the 12.72 watts figure? The FCC has some rather serious formulas which will calculate power and mV/m levels delivered at any azimuth and elevation angle for any tower array. The FCC website for the station also provides a basic chart for each compass degree around the tower array, listing mV/m levels. This is the easiest to use, although it is calculated for 0 degrees takeoff elevation.

FCC FORMULA DETAILS

The FCC method has close resemblance to the Wang method. The FCC expression for field strength is:

Note: FS(dBu), is also known as dBµV/m (normalized to 100 mV/m, in dBµV/m per 100 mV/m).

Where: FS(dBu) is the field strength in dBµV/m, ThetaM is the mid-point transmitter and receiver geomagnetic latitude, Dslant is the slant distance in km. Kr, or generalized ionospheric losses, are described below.

The FCC formula would appear to not include any system gain, referred previously as "transmitter cymomotive force above the reference 300 mV in dB". The field strength predicted is normalized to 100 mV/m (in dBµV/m per 100 mV/m). We must convert this back to the actual mV/m value by multiplying by the number of 100 mV/m "portions" we have in the total mV/m measurement at 1 km. The total mV/m measurement is calculated and published by the FCC for each compass degree. This figure also contains our tower array gain - our effective radiated power (ERP) referenced to 1 KW from a quarterwave monopole.

Converting to millivolts per meter, again:

mV100 = 10 ^ (FS(dBu) / 20) / 1000 ...convert dBu back to mV/m

mV/m = mV100 * (measured_mVm@1km / 100) ...corrected to actual mV/m

The FCC formula uses Wang's identical Kr factor. The generalized ionospheric losses are again found in it:

Refer to the previous discussion of Kr, above, in the Wang equation. Their usage is identical.

Wang again recommends that the geomagnetic mid-point latitude, ThetaM, be between -60 (south) and +60 degrees (north). It is not dependent on frequency.

ITU FORMULA DETAILS

The ITU expression for field strength is:

Note: FS(dBu), is also known as dBµV/m.

Where: FS(dBu) is the field strength in dBµV/m, V is the transmitter cymomotive force above (or below) the reference 300 mV in dB, Gs is the sea gain correction in dB, Lp is the excess polarization coupling loss in dB (defined graphically in ITU Recommendation 435-7), ThetaM is the mid-point transmitter and receiver geomagnetic latitude, Dslant is the slant distance in km. Kr, or generalized ionospheric losses, are described below.

Converting to millivolts per meter, again:

mV/m = 10 ^ (FS(dBu) / 20) / 1000 ...convert dBu back to mV/m

The ITU formula applies the basic path loss elements, the slant distance and the mid-point geomagnetic latitude influence. It also attempts to quantify some of the additional ionospheric losses I alluded to in an earlier post:

1. Sea gains (separately, as Gs)
2. Excess polarization coupling losses (separately, as Lp)
3. Sunspot influence (specified within Kr, as R)
4. Regional loss due to solar activity (calculated within Kr, as bsa * R)

The generalized ionospheric losses are found in the ITU's Kr factor:

Kr is the loss factor in dB, to include ionospheric absorption, focusing and terminal losses, and losses between hops, geomagnetic latitude factor, and basic polarization coupling loss. Unlike the Wang and FCC formulas, the ITU formula incorporates a sunspot factor and a frequency factor as well.

Where: f is the frequency in kHz, and ThetaM is the geomagnetic latitude defined previously. ThetaM must not exceed 60 degrees north or -60 degrees south. For paths shorter than 3000 km, the ITU suggests simply using the geographic mid-point between transmitter and receiver. Note: this, on average, skews results about 6 dB higher for North America.

Where: R is the twelve-month smoothed international relative sunspot number, bsa is the regional solar activity factor (bsa=0 for LF band; bsa=4 for MF band for North American paths, 1 for Europe and Australia, and 0 elsewhere). For paths where the terminals are in different regions we use the average value of bsa, for example: Europe to the USA, 2.5.

Note that we have a frequency correction, a geomagnetic latitude (ThetaM) correction, a regional correction in bsa (North America has the highest absorption), and a sunspot count correction.

The sharp analyst will notice that the ITU's frequency correction results in greater loss at higher frequencies, something perhaps theoretically sound, but not observed in North America (shown by measurements). The ITU suggests that for North America, a fixed frequency of 1000 kHz should be used.

Sea gain (Gs) is included in the ITU formula, but is usually set to zero and not accounted for since the transmitting or receiving station must be very close to a coastal point, generally within ten kilometers, and having a path length of thousands of kilometers. Lp, excess polarization coupling loss, is also included. This is an attempt to compensate for Lp differences in the generalized Kr part of the formula. We generally leave this at zero.

CAIRO CURVES FORMULA DETAILS

The modern day formula for the Cairo curve, adapted to Region 2, is presented for informational purposes. The resultant field strength should be further modified by subtracting ionospheric absorption losses (Kr), and adding any antenna gain.

The Cairo Curve, Revised for North America, Region 2

Where D is the overland great circle distance in kilometers between transmitter and receiver.

Again, we find our result in dBu per 100 mV/m (NTIA Report 99-368). It must be converted back to actual mV/m, as does the FCC formula.

mV100 = 10 ^ (FS(dBu) / 20) / 1000 ...convert dBu back to mV/m

mV/m = mV100 * (measured_mVm@1km / 100) ...corrected to actual mV/m

In the final part of this series on skywave prediction we will wrap up by discussing polarization coupling loss, sea gain, solar cycle losses, and diurnal and seasonal effects on mediumwave propagation.