earthSurfacePermittivity

Permittivity and conductivity of earth surface materials

Syntax

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('pure-water',fc,temp)

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('dry-ice',fc,temp)

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('sea-water',fc,temp,salinity)

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('wet-ice',fc,liqfrac)

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('soil',fc,temp,sandpercent,claypercent,specificgravity,vwc)

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('soil',___,bulkdensity)

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('vegetation',fc,temp,gwc)

Description

The earthSurfacePermittivity function computes electrical characteristics (relative permittivity, conductivity, and complex relative permittivity) of earth surface materials based on the methods and equations presented in ITU-R P.527 [1]. The earthSurfacePermittivity function provides various syntaxes to account for characteristics germane to the specified surface material.

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('pure-water',fc,temp) calculates the electrical characteristics for pure water at the specified frequency and temperature. For pure-water, the temperature setting must be greater than 0 ℃.

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('dry-ice',fc,temp) calculates the electrical characteristics for dry-ice at the specified frequency and temperature. For dry-ice, the temperature must be less than or equal to 0 ℃.

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('sea-water',fc,temp,salinity) calculates the electrical characteristics for sea water at the specified frequency, temperature, and salinity. For sea-water, the temperature must be greater than –2 ℃.

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('wet-ice',fc,liqfrac) calculates the electrical characteristics for wet ice at the specified frequency, and liquid water volume fraction. For wet-ice, the temperature is 0 ℃.

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('soil',fc,temp,sandpercent,claypercent,specificgravity,vwc) calculates the electrical characteristics for soil at the specified frequency, temperature, sand percentage, clay percentage, specific gravity, and volumetric water content.

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('soil',___,bulkdensity) sets the soil bulk density in addition to input arguments from the previous syntax.

[epsilon,sigma,complexepsilon] = earthSurfacePermittivity('vegetation',fc,temp,gwc) calculates the electrical characteristics for vegetation at the specified frequency, temperature, and gravimetric water content. For vegetation, the temperature must be greater than or equal to –20 ℃.

Examples

Compare Permittivity and Conductivity of Salt-free Sea Water to Pure Water

Compare the relative permittivity and conductivity for salt-free (zero-salinity) sea water to pure water.

Specify a carrier frequency of 9 GHz, temperature of 30℃, and salinity of zero.

fc = 9e9; % Carrier frequency in Hz.
temp = 30;
salinity = 0;

Compute the relative permittivity and conductivity.

[epsilon_pure_water,sigma_pure_water] = earthSurfacePermittivity('pure-water',fc,temp);
[epsilon_sea_water,sigma_sea_water] = earthSurfacePermittivity('sea-water',fc,temp,salinity);

Confirm that salt-free sea water and pure water have equal relative permittivity and conductivity.

isequal(epsilon_pure_water,epsilon_sea_water)

ans = logical
   1

isequal(sigma_pure_water,sigma_sea_water)

ans = logical
   1

Compare Permittivity and Conductivity of Wet Ice to Dry Ice

Compare the relative permittivity and conductivity for wet ice with no liquid water to dry ice at 0℃. Confirm the results differ by a negligible amount.

Specify a carrier frequency of 12 GHz.

fc = 12e9; % Carrier frequency in Hz.

Calculate the relative permittivity and conductivity for wet ice with zero liquid water by volume.

liqfrac = 0;
[epsilon_wet_ice_0,sigma_wet_ice_0] = earthSurfacePermittivity('wet-ice',fc,liqfrac); % Set liquid water volume fraction to 0.

Calculate the relative permittivity and conductivity for dry ice at 0 ℃.

temp = 0;
[epsilon_dry_ice_0,sigma_dry_ice_0] = earthSurfacePermittivity('dry-ice',fc,temp); % Set temperature to 0.

Compare the relative permittivity and conductivity for wet ice with no liquid to dry ice at 0℃. Confirm that wet ice with no liquid and dry ice at 0℃ have essentially equal relative permittivity and conductivity.

epsilon_wet_ice_0-epsilon_dry_ice_0

ans = 8.8818e-16

sigma_wet_ice_0-sigma_dry_ice_0

ans = -9.2179e-16

Plot permittivity and conductivity versus frequency for dry ice and for wet ice. For dry ice, vary the temperature. For wet ice, vary the liquid water volume fraction. Calculate the permittivity and conductivity values by using arrayfun to apply the earthSurfacePermittivity function to the elements of the arrayed inputs.

freq = repmat([0.1,10,20,40,60]*1e9,6,1);
temp = repmat((-100:20:0)',1,5);
liqfrac = repmat((0:0.2:1)',1,5);
[epsilon_dry_ice, sigma_dry_ice] = arrayfun(@(x,y)earthSurfacePermittivity('dry-ice',x,y),freq,temp);
[epsilon_wet_ice, sigma_wet_ice] = arrayfun(@(x,y)earthSurfacePermittivity('wet-ice',x,y),freq,liqfrac);

Display tiled surface plots across specified ranges.

figure
tiledlayout(2,2)
nexttile
surf(temp,freq,epsilon_dry_ice,'FaceColor','interp')
title('Permittivity of Dry Ice')
xlabel('Temperature (℃)')
ylabel('Frequency (Hz)')
nexttile
surf(temp,freq,sigma_dry_ice,'FaceColor','interp')
title('Conductivity of Dry Ice')
nexttile
surf(liqfrac,freq,epsilon_wet_ice,'FaceColor','interp')
title('Permittivity of Wet Ice')
xlabel('Liquid Fraction')
ylabel('Frequency (Hz)')
nexttile
surf(liqfrac,freq,sigma_wet_ice,'FaceColor','interp')
title('Conductivity of Wet Ice')

Calculate Permittivity and Conductivity of Various Soil Mixtures

Calculate relative permittivity and conductivity for various soil mixtures as defined by textual classifications in ITU-R P.527, Table 1.

Initialize computation variables for constant values and arrayed values.

fc = 28e9; % Frequency in Hz
temp = 23; % Temperature in °C
vwc = 0.5; % Volumetric water content
pSand = [51.52; 41.96; 30.63; 5.02]; % Sand percentage
pClay = [13.42; 8.53; 13.48; 47.38]; % Clay percentage
sg = [2.66; 2.70; 2.59; 2.56]; % Specific gravity
bd = [1.6006; 1.5781; 1.5750; 1.4758]; % Bulk density (g/cm^3)

Calculate the relative permittivity and conductivity for these textual classifications: sandy loam, loam, silty loam, and silty clay. Use arrayfun to apply the earthSurfacePermittivity function to the elements of the arrayed inputs. Tabulate the results.

[Permittivity,Conductivity] = arrayfun(@(w,x,y,z)earthSurfacePermittivity( ...
    'soil',fc,temp,w,x,y,vwc,z),pSand,pClay,sg,bd);

pSilt = 100 - (pSand + pClay); % Silt percentage
soilType = ["Sandy Loam";"Loam";"Silty Loam";"Silty Clay"];
varNames1 = ["Soil Textual Classification";"Sand";"Clay";"Silt";"Specific Gravity";"Bulk Density"];
varNames2 = ["Soil Textual Classification";"Permittivity";"Conductivity"];

ITU-R P.527, Table 1 specifies the sand percentage, clay percentage, specific gravity, and bulk density for soil mixtures with these soil textual classifications.

table(soilType,pSand,pClay,pSilt,sg,bd,'VariableNames',varNames1)

ans=4×6 table
    Soil Textual Classification    Sand     Clay     Silt     Specific Gravity    Bulk Density
    ___________________________    _____    _____    _____    ________________    ____________

           "Sandy Loam"            51.52    13.42    35.06          2.66             1.6006   
           "Loam"                  41.96     8.53    49.51           2.7             1.5781   
           "Silty Loam"            30.63    13.48    55.89          2.59              1.575   
           "Silty Clay"             5.02    47.38     47.6          2.56             1.4758   

The relative permittivity and conductivity for these soil textual classifications are included in this table.

table(soilType,Permittivity,Conductivity,'VariableNames',varNames2)

ans=4×3 table
    Soil Textual Classification    Permittivity    Conductivity
    ___________________________    ____________    ____________

           "Sandy Loam"               15.281            18.2   
           "Loam"                     14.563          16.998   
           "Silty Loam"               13.965          16.011   
           "Silty Clay"               12.861          14.647   

Calculate Permittivity and Conductivity of Vegetation

Calculate relative permittivity and conductivity versus frequency for vegetation, varying gravimetric water content and temperature.

Calculate relative permittivity and conductivity for vegetation at specified settings.

fc = 10e9; % Frequency in Hz
temp  = 23; % Temperature in °C
gwc = 0.68; % Gravimetric water content
[epsilon_veg,sigma_veg] = ...
    earthSurfacePermittivity('vegetation',fc,temp,gwc)

epsilon_veg = 20.5757

sigma_veg = 4.9320

Calculate values necessary to plot permittivity and conductivity by using arrayfun to apply the earthSurfacePermittivity function to the elements of the arrayed inputs.

For a range of temperatures, calculate values to plot permittivity and conductivity versus frequency for vegetation at a 0.68 gravimetric water content.

fc = repmat([0.1,10,20,40,60]*1e9,6,1);
gwc1 = 0.68;
temp1 = repmat((-20:20:80)',1,5);
[epsilon_veg_gwc,sigma_veg_gwc] = ...
    arrayfun(@(x,y)earthSurfacePermittivity('vegetation',x,y,gwc1),fc,temp1);

For a range of gravimetric water contents, calculate values to plot permittivity and conductivity versus frequency for vegetation at 10°C.

temp2 = 10;
gwc2 = repmat((0.2:0.1:0.7)',1,5);
[epsilon_veg_tmp, sigma_veg_tmp] = ...
    arrayfun(@(x,z)earthSurfacePermittivity('vegetation',x,temp2,z),fc,gwc2);

Display tiled surface plots across specified ranges.

figure
tiledlayout(2,2)
nexttile
surf(temp1,fc,epsilon_veg_gwc,'FaceColor','interp')
title('Permittivity of Vegetation at 0.68 gwc')
xlabel('Temperature (℃)')
ylabel('Frequency (Hz)')
nexttile
surf(temp1,fc,sigma_veg_gwc,'FaceColor','interp')
title('Conductivity of Vegetation at 0.68 gwc')
nexttile
surf(gwc2,fc,epsilon_veg_tmp,'FaceColor','interp')
title('Permittivity of Vegetation at 10°C')
xlabel('Gravimetric Water Content')
ylabel('Frequency (Hz)')
nexttile
surf(gwc2,fc,sigma_veg_tmp,'FaceColor','interp')
title('Conductivity of Vegetation at 10°C')

Input Arguments

fc — Carrier frequency

scalar in the range (0, 1e12]

Carrier frequency in Hz, specified as a scalar in the range (0, 1e12].

Data Types: double

temp — Temperature

numeric scalar

Temperature in °C, specified as a numeric scalar. Valid surfaces and associated temperature limits are indicated in this table.

Surface	Valid Temperature (℃)
pure-water	greater than 0
dry-ice	less than or equal to 0
sea-water	greater than or equal to –2
soil	any numeric
vegetation	≥ –20

Note

When the surface is wet-ice, the temperature is 0 ℃.

Data Types: double

salinity — Salinity of sea water

nonnegative scalar

Salinity of the sea water in g/Kg, specified as a nonnegative scalar.

Data Types: double

liqfrac — Liquid water volume fraction of wet ice

numeric scalar in the range [0, 1]

Liquid water volume fraction of the wet ice, specified as a numeric scalar in the range [0, 1].

Data Types: double

sandpercent — Sand percentage of soil

numeric scalar in the range [0, 100]

Sand percentage of the soil, specified as a numeric scalar in the range [0, 100]. The sum of sandpercent and claypercent must be less than or equal to 100.

Data Types: double

claypercent — Clay percentage of soil

numeric scalar in the range [0, 100]

Clay percentage of the soil, specified as a numeric scalar in the range [0, 100]. The sum of sandpercent and claypercent must be less than or equal to 100.

Data Types: double

specificgravity — Specific gravity of soil

nonnegative scalar

Specific gravity of the soil, specified as a nonnegative scalar. The specific gravity is the mass density of the soil sample divided by the mass density of the amount of water in the soil sample.

Data Types: double

vwc — Volumetric water content of soil

numeric scalar in the range [0, 1]

Volumetric water content of the soil, specified as a numeric scalar in the range [0, 1]. For more information, see Soil Water Content.

Data Types: double

bulkdensity — Bulk density of soil

nonnegative scalar

Bulk density, in g/cm³, of the soil, specified as a nonnegative scalar. For more information, see Soil Water Content.

Data Types: double

gwc — Gravimetric water content of vegetation

numeric scalar in the range [0, 0.7]

Gravimetric water content of the vegetation, specified as a numeric scalar in the range [0, 0.7]. For more information, see Soil Water Content.

Data Types: double

Output Arguments

epsilon — Relative permittivity

nonnegative scalar

Relative permittivity of the earth surface, returned as a nonnegative scalar.

sigma — Conductivity

nonnegative scalar

Conductivity of the earth surface in Siemens per meter (S/m), returned as a nonnegative scalar.

complexepsilon — Complex relative permittivity

complex scalar

Complex relative permittivity of the earth surface, returned as a complex scalar calculated as

complexepsilon = epsilon – 1i sigma / (2πfcε₀).

The computation of complexepsilon is based on Equations (59) and (9b) in ITU-R P.527 [1]. f is the frequency in GHz. c is the velocity of light in free space. ε₀ = 8.854187817e-12 Farads/m, where ε₀ is the electric constant for the permittivity of free space.

More About

ITU Terrain Materials

ITU-R P.527 [1] presents methods and equations to calculate complex relative permittivity at carrier frequencies up to 1,000 GHz for these common earth surface materials.

• Water

• Sea Water

• Dry or Wet Ice

• Dry or Wet Soil (combination of sand, clay, and silt)

• Vegetation (above and below freezing)

As described in ITU-R P.527, specific textural classification applies to these mixtures of sand, clay, and silt in soil with associated specific gravities and bulk densities.

Soil Designation Textural Class	Sandy Loam	Loam	Silty Loam	Silty Clay
% Sand	51.52	41.96	30.63	5.02
% Clay	13.42	8.53	13.48	47.38
% Silt	35.06	49.51	55.89	47.60
Specific gravity (ρs)	2.66	2.70	2.59	2.56
Bulk Density (ρb) in g/cm3	1.6006	1.5781	1.5750	1.4758

Soil Water Content

Soil water content is expressed on a gravimetric or volumetric basis. Gravimetric water content, gwc, is the mass of water per mass of dry soil. Volumetric water content, vwc, is the volume of liquid water per volume of soil. The bulk density, bulkdensity, is the ratio of the dry soil weight to the volume of the soil sample. The relationship between gwc and vwc is vwc = gwc ⨉ bulkdensity. When bulk density is not specified, the value of bulkdensity is computed by using ITU-R P.527, Equation 36:

bulkdensity = 1.07256 + 0.078886 ln(pSand) + 0.038753 ln(pClay) + 0.032732 ln(pSilt),

where

• pSand = sandpercent

• pClay = claypercent

• pSilt = 100 – (sandpercent + claypercent)

References

[1] ITU-R P.527-5. "Electrical characteristics of the surface of the Earth." International Telecommunications Union - Radiocommunications Sector (ITU-R). August 2019.

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