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

*. 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.*

**20 * Log10(Dslant)**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

*, 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.*

**(- 0.001 * Kr * Dslant)**Wang handles the ionospheric losses, Kr, a little differently

*. Dslant again modifies Kr. We can see again the greater the slant distance, the greater its contribution to ionospheric losses.*

**(- Kr * Sqrt(Dslant / 1000)**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.__**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 = 3.14159hr = 100 'layer height, kmkm = 900 'great circle distance, kmD = km / 40075 * 2 * Pi '40075=circumference of earth, kmE = 6378 * Sin(D / 2) '6378=radius of earth, kmF = E / Tan(Pi / 2 - D / 4)G = Atn(F / E + hr / E) - D / 2TA = 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°W2023 80.8°N 72.7°W2024 80.8°N 72.6°W2025 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

*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*

**2 * Sin(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.*

**Tan^2(ThetaM)**In the next article we'll dive right into the formulas and put it all together.