Mediumwave Skywave Prediction #5 - Dissecting The Formulas

In the series:

Mediumwave Skywave Prediction #1 - A Measurement History

Mediumwave Skywave Prediction #2 - A Formula History

Mediumwave Skywave Prediction #3 - Introduction To Formulas

Mediumwave Skywave Prediction #4 - Slant Distance & Geomagnetic Latitude

Mediumwave Skywave Prediction #5 - Dissecting The Formulas

Mediumwave Skywave Prediction #6 - Wrapping Things Up

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:

https://www.movable-type.co.uk/scripts/latlong.html

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);

    //convert to radians

    lat1 = Math.toRadians(lat1);

    lat2 = Math.toRadians(lat2);

    lon1 = Math.toRadians(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);

    //answer in degrees

    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:

    ThetaM(radians) = 

        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.

AM Radio In South America

I've been working on compiling a list of world AM broadcast radio (530-1700 kHz) still on the air. South America is essentially complete. The goal is a worldwide list of stations and locations.

Here is what is left of AM broadcast radio in South America as of this date, August 2023.

Click image for the larger view.

AM Broadcast Radio, 2023

Mediumwave Skywave Prediction #4 - Slant Distance & Geomagnetic Latitude

In the series:

Mediumwave Skywave Prediction #1 - A Measurement History

Mediumwave Skywave Prediction #2 - A Formula History

Mediumwave Skywave Prediction #3 - Introduction To Formulas

Mediumwave Skywave Prediction #4 - Slant Distance & Geomagnetic Latitude

Mediumwave Skywave Prediction #5 - Dissecting The Formulas

Mediumwave Skywave Prediction #6 - Wrapping Things Up

We will define two important concepts in this article: Slant Distance and Geomagnetic Latitude, both critical to determining the base path loss factor. This is our first step in solving the mediumwave skywave prediction puzzle.

To review, here are our main formulas again.

The Wang Method:

The FCC Method:

The ITU Method:

SLANT DISTANCE

The skywave field strength calculation process must compute a path loss factor between transmitter and receiver. Several parameters come into play here. The obvious one is the distance between transmitter and receiver. Greater distance incurs greater loss, plainly evident to the early experimenters. For many years the great circle overland distance was used in all formulas. It was eventually found that the actual distance traveled by the signal, the slant distance, was a better fit and produced better figures, as the signal must travel from transmitter to the reflection point high up in the ionosphere, then back down to the receiver. This, the preferred distance, is referred to as the Dslant distance in the formulas.

Slant distance is easily calculated for any signal path. From the FCC document 47 CFR 73.190:

' D is the overland great circle distance from transmitter to receiver.
' hr is the ionospheric layer height in kilometers. For mediumwave, usually set to 100.

Let's do a few examples. We will see that the higher the reflective layer height, the greater the slant distance. For added interest I've calculated TA, shown below, which is the signal takeoff angle from the antenna.

At 275 km overland distance, slant distance can deviate greatly. Takeoff angle is also large:

' 275 km distant station and a 100 km layer height, Dslant = ~340 km, TA=35°
' 275 km distant station and a 120 km layer height, Dslant = ~365 km, TA=40°
' 275 km distant station and a 150 km layer height, Dslant = ~407 km, TA=46°

At 1000 km overland distance, slant distance is only just a little greater. Takeoff angle has come way down:

' 1000 km distant station and a 100 km layer height, Dslant = ~1019 km, TA=9°
' 1000 km distant station and a 120 km layer height, Dslant = ~1028 km, TA=11°
' 1000 km distant station and a 150 km layer height, Dslant = ~1044 km, TA=14°

And slant distance is basically negligible at 2000 km. Takeoff angle is right at the horizon:

' 2000 km distant station and a 100 km layer height, Dslant = ~2009 km, TA=1°
' 2000 km distant station and a 120 km layer height, Dslant = ~2014 km, TA=2°
' 2000 km distant station and a 150 km layer height, Dslant = ~2022 km, TA=4°

Out past about 900 km or so, the slant distance is very close to the actual overland distance. As we get closer in from 900 km, the difference starts to accelerate. The Dslant distance value is dependent on the E-layer height and Dslant (in km) is always higher than the exact overland distance value. Slant distance is now commonly used in all modern skywave formulas.

This slant distance is used in two places in the formulas. It becomes part of the basic path loss factor, and part of the ionospheric loss adjustment (the Kr term). In the basic calculation, the larger the slant distance, the greater the basic path loss factor. Secondly, and since the ionospheric losses are subtracted from the basic path loss, the larger the slant distance, the greater the effect it has on ionospheric losses, Kr.

Ionospheric losses, Kr, will be explained in further detail in the next article.

Each formula uses the inverse square law in the basic path loss calculation. This will be in dB. This simply says that for every doubling of distance, the strength is one-fourth of what it was. For example, the strength at 1000 km is one-fourth the strength found at 500 km. This is realized through the formula snippet 20 * Log10(Dslant). 20x gives us the value needed in dBµV/m to subtract from our start value since we are dealing with field strength in voltage units.

Here are some path loss examples for a layer height of 100 km:

Dslant = 250 km = 48 dB (190 km overland distance)
Dslant = 500 km = 54 dB (458 km overland distance)
Dslant = 1000 km = 60 dB (980 km overland distance)
Dslant = 2000 km = 66 dB (1990 km overland distance)

Each doubling of distance increases the loss by another 6 dB, also one S-unit. The Dslant contribution to the basic path loss is subtracted from our start value of 106.6 dB (ITU), (107 dB, Wang), (97.5 dB, FCC).

In the ITU formula (- 0.001 * Kr * Dslant), Dslant again modifies the ionospheric losses, Kr. So, as you can see, the greater the slant distance, the greater its contribution to the ionospheric losses too.

Wang handles the ionospheric losses, Kr, a little differently (- Kr * Sqrt(Dslant / 1000). Dslant again modifies Kr. We can see again the greater the slant distance, the greater its contribution to ionospheric losses.

Wang's Kr value modification by Dslant is used the same way in the FCC formula.

PERSONAL OBSERVATIONS GAINED FROM TESTING

Using the Dslant value has minimal effect on far stations, those out beyond 900 km or so, where Dslant is roughly equal to the exact distance value. An increasingly greater effect is evident on those stations as we narrow our distance to 250 km, and less.

A continuing problem still exists with accuracy for close in stations. Years ago, Wang suggested those stations less than 250 km distant should use a fixed E-layer height of 220 km, increasing the resultant Dslant value even more. That fixes the lowest slant distance at 506 km for any station closer than 250 km to the receiver. Consequently, 250 km becomes a hard "wall" which would make a station's calculated field strength at 251 km much stronger than one at 249 km. Nature undoubtedly has a proportional transition which must be accommodated.

It would be obvious that increasing the layer height also increases the transmitter signal's takeoff angle, generally resulting in a weaker facing signal to the receiver, resulting in the calculation further lowering the received field strength. By design, this was Wang's intent in raising the layer height to 220 km for stations closer than 250 km distance. It was not enough.

Experimenting.

The remaining paragraphs in this, the Slant Distance section, are ideas outside of the current formulas, and are food for thought. In my program which creates the mediumwave pattern map set, RDMW (Radio Data MW), I wrote a sandbox mode which allows me to experiment with different skywave propagation ideas. These include varying layer heights, varying attenuation factors, seasonal effects, and sunrise/sunset enhancements. Tweaks can be modified by frequency also. It has revealed some interesting facts.

Part of the problem with the current worldwide formula set is that, given a database of stations like the FCC mediumwave database, it will produce an acceptable list of varying field strengths, but the field strength order, channel by channel, isn't always what is heard during actual band scanning. I tested this on all three formulas and found this curious.

Material written is very explicit indicating that the E-layer is well-defined and exists between about 100 km and 115 km. Mediumwave skywave is considered (by formula) to be reflected or refracted off the E-layer at 100 km exclusively. I do not believe this to be the case, and it is borne out by the inaccuracies in the formulas for close in stations, those about 900 km and closer.

Modifying the layer height has been experimented with extensively, generally by raising it incrementally as we get closer and closer to the transmitter, starting at about 900 km and modifying by the inverse cosine of the distance. Results were better, bringing field strengths more in line. Still, the resultant skywave calculations using this method did not quite match signal strengths by band scanning. Actual signals are always less for close in stations, except at the sunrise/sunset enhancement periods where they exhibit a temporary strengthening.

A gentle transition of E-layer reflectivity height from 100 km to 280 km (acknowledged, 280 km is outside of the E-layer) is suggested, starting at 100 km with station distances about 900 km and raising it as we get closer to zero distance using an inverted cosine method. However, a maximum layer height of 280 km does not fully correct the field strength inaccuracy. We must add in an additional decay factor as the station distance is decreased. I would advise against increasing maximum layer height beyond 280 km as I think it presents an increasingly inaccurate picture of conditions.

An inverted waveguide?

The Earth's natural waveguide effect is well known for extremely low frequencies (ELF), those below 3 kHz. What if, instead, we treat the ionosphere from 100 to 140 km as a sort of mediumwave inverted waveguide? That is, make our reflecting layer heights dynamic - the lower frequencies (starting at 530 kHz) reflecting at the lowest layer height, and higher frequencies (ending at 1700 kHz) reflecting at highest layer height? We could set a layer height range of 100 to 140 km to fully contain all reflections within the banded E-layer. Or, we might even experiment with a range of 100 to 300 km to allow higher frequency reflections at the F-layer. The first scenario was experimented with and seems most promising. It delivers surprisingly good field strength results verified by what is actually heard by band scanning.

Skip distance.

Many of you, when studying radio propagation, will see charts or graphics showing a single hop track up to the ionosphere and reflected back to Earth. Sometimes beneath it are printed the words, "Skip Distance". They are referring to a zone of dead signal, that is, an area where the signal is "skipping overhead", and not receivable in the skip zone. Take care to note this applies almost exclusively to shortwave frequencies, that of 3 MHz and above, and hardly at all to mediumwave. Mediumwave tends to "fill in" in the skip zone, at varying levels. Nighttime skip reflections are detectable and receivable at very short distances, even under 60 km.

TAKEOFF ANGLE

For the curious, those wanting to calculate signal takeoff angle from a transmitter, this simple program will calculate it. Choose your layer height (hr) and your distance from receiver to transmitter (km).

Pi = 3.14159

hr = 100  'layer height, km

km = 900  'great circle distance, km

D = km / 40075 * 2 * Pi  '40075=circumference of earth, km

E = 6378 * Sin(D / 2)  '6378=radius of earth, km

F = E / Tan(Pi / 2 - D / 4)

G = Atn(F / E + hr / E) - D / 2

TA = G / Pi * 180  'TA in degrees

Takeoff angle is important. The ITU and Wang formulas include a basic gain/loss correction in dB referenced to 1 KW effective radiated power, ERP (the V cymomotive force parameter), but don't allude to any differences due to signal takeoff angle. The FCC formula accounts for the gain/loss correction in a different way, by normalizing its returned field strength value to 100 mV/m at 1 KW, still not alluding to any differences due to signal takeoff angle.

I'll show you an example of how ignoring takeoff angle can produce highly inaccurate results. We'll look at WBVP-1230 (1 KW) in Beaver Falls, PA. WBVP uses a single monopole tower at 0.64 wavelength tall. Their skywave signal takeoff angle from their antenna to Rochester, NY is 29.2 degrees, based on an E-layer height of 100 km, a substantial angle. If we ignore takeoff angle and assume to use their full 1 KW ERP (which we would only see at 0 degrees takeoff angle), we are calculating field strength at 1 KW "facing watts", that is, the ERP at the horizon, facing us. This isn't reality.

The reality is that our received signal is being delivered from the 29.2 degree angle, a very different effective radiated power than the angle at the horizon. At 29.2 degrees takeoff angle, with WBVP we only "see" 34 watts coming at us. WBVP will show up at very much less field strength on the dial than other stations because of this. Power differences because of elevated takeoff angle makes a huge difference in our calculation process and our resulting received field strength. It must be accounted for.

We move on to geomagnetic latitude and longitude.

GEOMEGNETIC LATITUDE & LONGITUDE

Normal latitude and longitude is referenced to as the north (or south) geographic pole, an actual latitude of 90°, and respectively, -90°. Longitude at the poles is irrelevant as they all converge at this point. Geomagnetic latitude and longitude uses the geomagnetic poles as our north-south reference instead. 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.

Imagine our Earth where the north pole was instead the geomagnetic north pole, currently (2023) in the extreme northwest corner of Greenland. The Earth's longitude lines would all emanate from that point, and it would be considered 90 degrees north latitude. The geomagnetic latitude of New York City, for example, would then be referenced to the geomagnetic north pole, not the actual north pole.

So, the result is this. Instead of New York City being at 40.75°N latitude actual, NYC is now at 49.95°N geomagnetic latitude. This makes a tremendous difference in our mediumwave skywave prediction. The geomagnetic north pole is where the auroral zone is centered in the northern hemisphere. The auroral zone greatly affects the mediumwave signal.

Geomagnetic location is sometimes called the geomagnetic dipole. Both the geomagnetic and magnetic poles have been wandering quite a bit over the last few years. In 1950 the geomagnetic pole was located at approximately 78.5°N and 68.8°W. Today, 2023, it has moved 4 degrees farther north and 2 degrees farther west. Here are the current and future predicted locations:

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

Geomagnetic dipole (Northern hemisphere):

2022   80.7°N  72.7°W

2023   80.8°N  72.7°W

2024   80.8°N  72.6°W

2025   80.9°N  72.6°W

So, in skywave analysis, first we must calculate the reflected signal's mid-path latitude and longitude and convert it to its geomagnetic reference. The mid-point latitude and longitude, before conversion, is generally half the distance between transmitter and receiver on a great circle line drawn between the two. We assume the geomagnetic north pole to be our new north pole at 90 degrees latitude. We first calculate the actual latitude and longitude of the path mid-point (the reflection point) between transmitter and receiver and reference its latitude (in degrees offset) to the geomagnetic north pole.

This mid-path latitude is then used in two places in the formulas. It becomes part of the basic path loss calculation, and part of the ionospheric losses (the Kr term). In the basic calculation, the higher the geomagnetic latitude, the greater the extra losses incurred. Secondly, and since the ionospheric losses are subtracted from the basic path loss, the higher the geomagnetic latitude, the greater the additional losses incurred.

In the ITU formula, the formula snippet 2 * Sin(ThetaM) establishes the basic geomagnetic loss relative to the path mid-point. At 40 degrees geomagnetic north latitude it is 1.28 dB, while at 60 degrees north it is 1.73 dB. So we see about a 0.5 dB difference (loss). Wang treats it differently, using Tan^2(ThetaM). In Wang's formula, and also the FCC formula, at 40 degrees geomagnetic north latitude the loss is 0.7 dB, where at 60 degrees north the loss is 3.0 dB. Wang is allowing greater compensation for North America as the mid-point approaches 60 degrees north.

In the next article we'll dive right into the formulas and put it all together.

Mediumwave Skywave Prediction #3 - Introduction To Formulas

In the series:

Mediumwave Skywave Prediction #1 - A Measurement History

Mediumwave Skywave Prediction #2 - A Formula History

Mediumwave Skywave Prediction #3 - Introduction To Formulas

Mediumwave Skywave Prediction #4 - Slant Distance & Geomagnetic Latitude

Mediumwave Skywave Prediction #5 - Dissecting The Formulas

Mediumwave Skywave Prediction #6 - Wrapping Things Up

Now that we've covered skywave prediction history in this series, let's look at a few actual formulas which are used to calculate skywave field strength. This will likely spill over into several articles as we describe the concepts and intricacies of skywave propagation.

THE SURVIVORS

By the turn of the millennium, three simplified formulas survived and are usable for worldwide mediumwave skywave field strength prediction. They each have viable options to consider.

They are:

The Wang Method:

The FCC Method:

The ITU Method:

Yes, they look cryptic at this point. Not to worry, we'll take these apart, item by item, and show you what they're attempting to do.

Where the ITU method attempts to provide a generalized worldwide formula, both the Wang and FCC methods are specialized for Region 2, the Americas, and specifically North America. It must be stressed that these are so-called "simplified formulas", though they do their job quite well. To wit, all have simplified the calculation process associated with hop loss, polarization coupling loss, and solar effects, boiling these down into a generalized expression, Kr. We will analyze Kr in due course.

DISSECTING THE FORMULAS

Each of the these formulas can be sub-divided into three parts.

They are:

1. Calculate a base path loss factor which is based on the path distance. Dslant in the formulas.

2. Calculate the extra loss due to the path's geomagnetic mid-point relative to the geomagnetic north pole. ThetaM in the formulas.

3. Finally, factor in any additional losses/gains like frequency, sea gain, and basic values for ionospheric absorption, polarization coupling losses, focusing and terminal losses, and losses between hops, sunspot number and solar activity. Kr in the formulas.

These additional losses or gains (items 2 and 3), in dB, are subtracted from or added to the base path loss factor to arrive at a final overall path loss value. The final result of the above calculations then give us a ballpark field strength for the midnight hour, or what is usually called SS+6, or sunset+6 hours. This is directly translated into dBµV/m, or dB relative to 1 microvolt per meter, the predicted field strength available at the receiver.

We may or not choose to continue on with even more losses or gains, not shown in the formulas above. If we do, these extras can be:

• Diurnal hourly losses/gains (skywave prediction for the hour of the day). 

• Sunrise and sunset enhancements (skywave prediction at these critical hours).

• Seasonally-driven losses/gains (skywave prediction for winter versus summer).

So, let's gather all the pieces we need to solve the prediction puzzle. We will ignore the extras for now.

The Basics:

1. Calculate Dslant, the "slant distance" and use it to derive a basic path loss factor (a new term - we use slant distance instead of the great circle distance from transmitter to receiver).

2. Calculate ThetaM, the mid-point geomagnetic latitude, also part of the basic path loss factor.

The Ionospheric Tweaks (all but Sea Gain calculated within the Kr term):

3a. Choose the ionospheric layer height (usually 100 km).

3b. Account for Hop losses.

3c. Account for Sea Gain (usually ignored).

3d. Account for Polarization Coupling losses.

3e. Account for Sunspots & solar activity.

Let's first describe the ionosphere at mediumwave and how our signal is reflected or refracted back to Earth. Later on we'll define two important concepts: Slant Distance and Geomagnetic Latitude, both critical to determining the base path loss factor.

IONOSPHERIC LAYERS

Nighttime mediumwave propagation has long been assumed to be reflected or refracted off the E-layer of the ionosphere. The ionosphere is layered as we go skyward, the layers being named the D, E, and F layers.

D-region 50-90 km (31-56 mi)

The D-region is a region of low electron density whose degree of ionization is determined primarily by solar photoionization. This region usually exists during the daytime, and it absorbs the energy of MF radio waves that pass through it. The MF sky wave is therefore highly attenuated as it enters the D-layer during the daytime. At night in the absence of the photo-ionization created by the sunlight, the ionization in the D-region is at a much lower level or is nonexistent, so the D-region no longer absorbs the energy from the MF sky wave passing through it.

Daytime skywave. Believe it or not, daytime skywave does exist and is present 24-7 in varying degrees depending on the season. In deep winter in the Northern Hemisphere (December, January), D-layer ionization during the day is strikingly less due to the lower solar position. Skywave signals, particularly at the upper end of the mediumwave band can pass right through it, and be reflected back to Earth off the E-layer at mid-day. Signals are weak, to be sure, but DX opportunities are abundant for those willing to dig for a signal. Deep winter D-layer absorption can be as much as 20-30 dB lower than at high summer (July, August).

The effect can be striking and unexpected in low-noise areas of the country where you are free from the extreme RF density of the east. I used to spend winters in southwestern Arizona. My custom was to do an annual Christmas trip to Denver, Colorado and I'd set my car radio on a frequency of one of the extremely distant powerhouse stations. I have received KFI-640, Los Angeles, in Trinidad, Colorado at the noon hour, a distance of 800 miles. At peak, the signal hovered right at or barely above the noise level, with long deep fades. Now, that to me is exciting DX.

Back at home in Arizona, I had a 25 ft. matched vertical, inductively-coupled to a variety of portable radios. Following is a sample of what was heard in deep winter during the middle part of the day.

Unusually good signals at noon:

KSL-1160 Salt Lake City, UT (506 miles) never went away at the noon hours. Week but very readable from 11:00-13:00 local, then back up to very nice strength again by 13:30.

KNBR-680 San Francisco, CA (524 miles)

KALL-700 N. Salt Lake City, UT (515 miles)

KCBS-740 San Francisco, CA (557 miles) with equal strength to two semi-locals KIDR-740 Phoenix and KBRT-740 Costa Mesa, CA.

KZNS-1280 Salt Lake City, UT (512 miles) was booming in with an outstanding signal at 12:30 local.

By 13:00 local:

KRVN-880 Lexington, NE appeared with decent strength. 944 miles.

KLTT-670, 50 KW Commerce City, Colorado (681 miles, suburban Denver) under stronger 198 mile groundwave 25 KW KMZQ-670, Las Vegas, NV

KNEU-1250 Roosevelt, UT at early afternoon. 515 miles but only a 5 KW station.

KGAK-1330 Gallup, NM 339 miles (another 5 KW).

E-region 90-140 km (56-87 mi)

During nighttime, the MF sky wave proceeds right on through the D-region to the E-region where it is refracted. The E-region ionization is from multiple sources that exist all of the time, so it is active during both the daytime and the nighttime. E-region ionization in the daytime is predominantly caused by solar ultraviolet and x-rays, while E-region ionization at night is caused predominantly by cosmic rays and meteors. The E-region is found at heights of 90 to 140 km, and it attains its maximum electron density near 100 km. This is the height within the E-region that is the predominant reflecting medium for MF propagation at night. The highly charged part of the E-region is a thin layer, roughly from 5 to 10 km (3 to 6 miles) thick.

Seasonal E-layer heights, as measured by ionosonde are:

Winter noon: 112 km, midnight: 118 km

Spring noon: 110 km, midnight: 108 km

Summer noon: 109 km, midnight: 104 km

Fall noon: 108 km, midnight: 111 km

These are actually measured sporadic-E heights, intense clouds of ionization within the E-layer itself, however evidence suggests that the reflective part of the E-layer may extend all the way to 140-150 km above the Earth. Though MF skywave calculations almost always fix the reflection layer at 100 km, it is evident that reflection or refraction of the MF signal surely does occur at varying altitudes, much dependent on time of day, frequency, and a host of other variables.

Critical frequency is a term used to describe the highest frequency above which radio waves penetrate the ionosphere and below which are reflected back. The critical frequency of the E-layer is mostly between 1.5 and 4 MHz, higher during a sunspot maximum than during a sunspot minimum.

This tells us two things. If our critical frequency has dropped to 1500 kHz or even lower (1.5 MHz, stated above), our MF signal may transit through the E-layer and be reflected back to us off the F-layer. Second, we may see this effect more during periods of lower solar activity. The F-layer, at night, settles in at about 250-300 km altitude. This can result in single hop distances upwards of 3000 km (1864 mi). Look to the upper range of the mediumwave band to sometimes provide unusual DX, particularly in the late night and early morning hours before sunrise.

The skywave/groundwave mixture. Skywaves and ground waves add vectorially. They can and do interfere with each other, the interference resulting in phase distortion in the audio you hear, and weakening (or strengthening) of the signal received at the receiver due to additive or subtractive combination. At night, at 500 kHz over average ground, the ground wave predominates over the skywave from the transmitter site out to distances of about 150 km, where the two signals are equal. The signals add as vectors, and destructive and constructive interference can occur. At 500 kHz at distances beyond 150 km, the sky wave is the predominant signal. At a signal frequency of 1500 kHz, the distance where the two signals are equal reduces to 45 km, because of the increased loss at the higher frequency.

F-region 250-400 km (155-250 mi)

The altitude of all the layers in the ionosphere vary considerably and the F-layer varies the most. During the daytime when radiation is being received from the sun, the F-region often splits into two: the lower and more insignificant one called the F1-region, and the higher and more significant one, the F2-region. Note also that the F1-region generally only exists in the summer. Typically the F1-layer is found at about an altitude of 300 km and the F2-layer at about 400 km.

At night, the two regions combine, and the combined F-layer then centers around 250 to 300 km. Like the D and E layers the level of ionization of the F-region varies over the course of the day, falling at night as the radiation from the sun disappears. However the level of ionization remains much higher than the lower regions.

The F-region is greatly affected by solar conditions. The maximum usable frequency, or MUF, is generally at least 15 MHz, but during the sunspot maximum period, the MUF may often exceed 50 MHz. The maximum usable frequency is the highest frequency that can be refracted off the ionosphere and returned to Earth (generally the F-region is implied).

Then we have what is called lowest usable frequency, or LUF. The sky would appear to be the limit here, but the problem we have is our signal must first transit through the D and E layers to get to the F-layer. This probably isn't going to happen during the day in the mediumwave frequency range due to the highly absorptive D-layer. So, during the daylight hours, the D-layer will limit the lowest frequency allowed to pass through. At night, it's a different story.

As we said in our description of the E-region, almost all MF signals will refract off the E-layer at night. But under certain conditions and at certain times of year, when the critical frequency of the E-layer drops to 1500 kHz or below, we have F-layer skywave in the AM broadcast band, a fascinating phenomena.

Let's summarize.

Practically, with all that said, our skywave prediction formula must choose a reflective layer height before we begin. The common choice is 100 km. Varied results will be found between 90 to 140 km, with the higher altitudes producing lower field strengths in general. The prediction experimenter might choose the higher altitudes for frequencies at the upper end of the mediumwave band, or they might even try forecasting for refraction off the F-layer at 250-300 km.

In the next articles, we'll discuss Slant Distance and Geomagnetic latitude. We'll also talk more about ionospheric layer heights, and how they affect the two.