Bob,

Some information below on the effects of above-freezing and below-freezing temperatures on various VLF losses, including (a) ground-wave losses, (b) near-field ground effects losses and (c) skywave (waveguide) ground effects losses.

A short preface:

The discussion below is for a VLF system with a vertical transmit antenna.

In a VLF system with a vertical transmit antenna, high surface conductivity under the antenna and in the near and far fields is desirable.  (1)

In a VLF system with a horizontal antenna, low surface conductivity under the antenna is desired, and high surface conductivity is desired elsewhere in the near field, and in the far field.  (1)

This makes Antarctica a great place for VLF QRP: ice under the antenna; salt water elsewhere (whence the 42 km long dipole at Siple Station: https://s3.amazonaws.com/Antarctica/AJUS/AJUSvXVIIIn5/AJUSvXVIIIn5p270.pdf           http://nova.stanford.edu/~vlf/Antarctica/Siple/ )    

Temperature coefficients of conductivity:

As temperature drops, non-frozen soil conductivity (mhos, siemens) decreases by roughly 1.9%  per degree C. (2)

As temperature drops below freezing, soil VLF conductivity decreases rapidly; the rate of change is highly dependent on soil type, but a 2:1 change per 10 degrees C at 10 kHz (for below-freezing temperatures) is a reasonable example for some soils. (3)

Near field and ground wave losses:

As conductivity decreases (i.e. with decreasing temperature), VLF near-field and ground wave losses increase non-linearly (4)

At 30kHz, ground wave amplitude is low at 5000 km (4)

At 30kHz, ground-wave loss in non-frozen soil of poor conductivity (10^-4 mhos/m) might be 20dB greater (20dB more loss) than in non-frozen soil of fairly good conductivity (10^-2 mhos/m)   (4)

At 30kHz, ground-wave loss in frozen soil would be very high (4)

All of the above might suggest a very low amplitude ground wave at TA distances in mid-winter (ground wave including direct, ground-reflected, Norton surface and trapped surface waves)

In that case the long-path part of the problem reduces to skywave losses (skywave ionospheric losses and skywave ground-effects losses):

Skywave loss (waveguide loss in this case) due to ground effects is nominally:

d_alpha_gnd = [.046 * sqrt(f)]  *  1/{h * sqrt(s) * sqrt[1 – (fc/f)^2]}    (in units of dB per 1000 km)     (5)

where f = 29,500 (Hz), fc ~ 2100 (Hz) (day) or 1700 (Hz) (night), h ~ 70 (kilometers),  .0001 < s < .01  [s for sigma (conductivity), in mhos per meter; range of values from .01 for very good (highly conductive) soil to .0001 for frozen earth; values below .0001 occur in arctic regions, but such regions tend to have snow and ice above the soil. A different formula is used for snow and ice losses at frequencies below 1 MHz]

The formula above may be sufficient for analysis of cold-weather propagation losses over distances of 5 Mm or more. Besides propagation losses, there are near-field and antenna-system losses.

Near-field and antenna-system losses:

VLF near-field and antenna-system losses associated with ground effects, increase as temperature (and therefore conductivity) decline. The increase in near-field and antenna-system losses can be between sqrt (s) and s^1.5, but the nominal value of these losses is hard to determine analytically for an electrically-short antenna. The calculation for ice/snow cover for your antenna system may be easier to determine analytically; if you’re interested I’ll send a notional formula.

If your VLF signal reports do not change appreciably after a snowfall or an ice storm, that information can be used to help generate a temperature model of your ground-effects-related near-field and antenna-system losses, in which case you might have a good starting point for a whole-system model for all temperatures.  Noting what happens to signal reports after a freeze, after a snowfall and after an ice-storm can help to nail down the most difficult part of the system model for a VLF system with an electrically short antenna (antenna-system losses and near-field losses).

In summary:

A)     VLF near-field and antenna-system losses associated with ground effects, increase as temperature declines.  Noting what happens to signal reports after a freeze, after a snowfall and after an ice-storm can help to establish a good model for these. Such a model will be useful at all temperatures, and very helpful in the optimization of an electrically-short VLF antenna.

B)     The formula above for d_alpha_gnd (relative ground-effects losses incurred by the skywave in the waveguide) gives values from about 2 dB per megameter to 6 dB per megameter at 29.5 kHz for soil conditions ranging from fairly good to frozen. This formula is useful by itself for assessing the difference in far-field path loss at various temperatures. Total path loss includes other terms that add to and subtract from* d_alpha_gnd, but those terms are not necessary for assessing changes in ground-effects-related skywave path losses (in the waveguide) over temperature.

C)     Ideally, you would add (A) to (B) to obtain the sum of the predominant contributors to loss variation over temperature.

* terms such as the convergence factor, and reinforcements at discontinuities, subtract from path loss

(1)    Biggs, AGARD, 1970

(2)    a general approximation; Corwin 2005, Ma 2010

(3)    Moore 1992

(4)    Watt 3.2

(5)    Watt 3.4.33

73,  Jim AA5BW