User:Cmarston1929/Sandbox

RESEARCH

A comprehensive numerical algorithm capable of evaluating complex permittivity of soil for arbitrary frequency, volumetric water content, and soil solids composition is constructed. The empirical polynomial modeling of soil dielectric properties developed by Hallikainen and Dobson [3-4](1400 MHz-18GHz) is extended to lower frequencies (.01 MHz -50 MHz) by numerical modeling of measured data and supplemented by other interpolative algorithms to uniformly span the intended frequency range of .01 MHz to 18 GHz. The log of the permittivity as a function of frequency is displayed in Fig.[1] which is labeled along the frequency axis to indicate the principal subdivisions within which the dominant data sources and methodologies determine the corresponding computational logic.The original work of Hallikainen et al [3] and of Dobson [4], in the frequency range 1400 MHz-18GHz, provided empirical methods consisting of polynomials quadratic is volumetric moisture with expansion coefficients explicitly dependent upon specific soil compositions expressed as gravimetric fractions of the three primary components silt, sand and clay.

An alternative semi-empirical formalism due to Hallikainen and Dobson [3-4](1400 MHz-18GHz) and Peplinksi [5-6] (300 MHz-1300 MHz) is also considered in this Report as a potential alternative to the empirical modeling. As indicated in the figure, the semi-empirical methods of Hallikainen, Dobson and Peplinksi [3-6] are utilized in the frequency range 300 MHz. In an intermediate frequency range 1-50 MHz, the measurements of Campbell [1-2] collected at                                               (1,2,5, 10,20 and 50)MHz are used to generate cubic polynomials in volumetric moisture at each frequency. The use of the polynomials to calculate complex permittivity in the range 1-50 MHz is then extended to the empirical regime with an interpolative transition between 50 and 1400 MHz ( or between 40 and 300 MHz if the semi-empirical option is utilized). Finally the algorithm is extended to frequencies as low as .01 MHz with the measurements of Sternberg [7] for sand and Montmorollinite clay which are each reported at single values of moisture from .001 MHz to the lower limit of the second frequency range reported here as 1.0 Mhz.

UNIT TESTING PLAN

I. PHENOMENOLOGICAL VALIDATION AND INITIAL NUMERICAL REPRODUCIBILITY The present algorithm was written to be consistent with measurements collected over the frequency range considered between .01 MHz and 18 GHz. This initial validation exercise is intended to demonstrate conformity with measurements and successful replication of numerical results from the developmental environment.

{1.1)Frequency Range 1400 MHz -18,000 MHz

The polynomials of Hallikainen and Dobson [3-4] which generate structure coefficients from provided composition fractions of sand and clay are incorporated directly into the model with slight modification of the lowest frequency (1400 MHz) to suppress non physicality of behavior and to compensate for the invalidity of the model for sand texture compositions exceeding 50%. In addition a logarithmic interpolator is used to compute complex permittivities for intermediate values of frequencies between the values at which measurements and polynomials coefficients are provided (1.4 ,4.,6.,8.,10.,12.,14.,16.,18.)GHz. The validation of the results is  obtained by comparing values returned by the model with those returned by an independent subroutine with explicit values of the polynomial texture-dependent expansion coefficients recommended by Dobson.

{1.2) Frequency range considered by Campbell [1-2], (1,2, 5) and(10,20,500) mHz.

The numerical measurements of Campbell, collected at the frequencies of 1,2,5 and (10,20 50) MHz over an extensive range of volumetric moisture (0< \zmv <.50) were digitized and modeled with linear regression to optimally adjust a third order polynomial foe each soil texture considered ( Hart sand, Fort Edwards Clay and Wilder Silt). Elementary linear algebraic methods were used to  compute another set of coefficients which generate the texture specific coefficients from know values of the soil solids texture ( gravimetric fraction of silt ,sand and clay). The numerical results are then accessible for arbitrary textures and volumetric moistures within the frequency range 1-50 Mhz. (As was implemented for the Dobson polynomials above, intermediate values of  frequency other than those  explicitly provided are generated in the algorithm by logarithm  interpolation in (logarithm of ) frequency to yield continuous values of permittivity at arbitrary   frequencies within the specified range) In the suggested  associated numerical exercise, the data of Campbell are superimposed on the values generated by the model for the three soil composition considered, and at the six frequencies and over the full range of moistures for which data was collected.

{1.3) Frequency range 0.01 MHz-0.90 Mhz

The measurements of Sternberg[7] in the frequency range .01 -0.9 Mhz (for Ca-Montmorollinte clay and Brookhaven sand), collected respectively at volumetric moistures of (aqueous volumetric fraction) zmv=0.706 and 0.0976 were used in the model as absolute values of the prospective permittivities in a scaling of the behaviour of soils of the same composition at 1 MHz (over a full range of  volumetric moistures) such that the experimental measurements are recovered when the general algorithm is evaluated at the frequencies for which measurements are available. {II) BRIEF DESCRIPTION OF UNIT TESTING PROCEDURE

{2.1) The unit test involves coordinated commands involving a header file,a source file for appropriate classes in the construction of EMPIRE and a unit test code which are located respectively in user versions of the following files which have been printed contiguously with their respective directory locations

/dfs/username/CREW_dielectrics/dev/include/EmpiricalSoilPermittivity.h /dfs/username/CREW_dielectrics/dev/src/empire/EmpiricalSoilPermittivity.cpp /dfs/username/CREW_dielectrics/test/uniti/EmpiricalSoilPermittivityTest.cpp

EMPIRE is constructed by the command \tt emake as issued in the second of these directories and the unit test is compiled and linked to EMPIRE with the command \tt emake as invoked from the above test directory

{2.2) commence execution of unit testing engine by issuing the command

/dfs/username/CREW_dielectrics/test/uniti/Linux/EmpiricalSoilPermittivityTest

{2.3) The results of the unit testing may then be examined in the file:

/dfs/username/CREW_dielectrics/test/DATA/dielectricModel/soil_regression_output_3.txt

which if successful will report zeros in the final column (cumulative deviations from expected error analysis results) as shown below.

N      zme           zLtd_r      rms_r       _i      std_i       rms_i        alt(rms_i)

1152 3.189264e-08 1.433703e-06 1.433436e-06 -1.372376e-06 2.229611e-05 2.232865e-05 2.232865e-05 1152 3.189264e-08 1.433703e-06 1.433436e-06 -1.372376e-06 2.229611e-05 2.232865e-05 2.232865e-05 1152 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.00e+00

1152 -3.744978e-09 1.692931e-06 1.692201e-06 -1.166014e-06 2.783002e-05 2.784236e-05 2.784236e-05 1152 -3.744978e-09 1.692931e-06 1.692201e-06 -1.166014e-06 2.783002e-05 2.784236e-05 2.784236e-05 1152 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.00e+00 .                .                 .                 . 2048 -2.599752e-06 7.350381e-06 7.794897e-06 -1.292476e-06 5.431557e-06 5.581926e-06 5.581926e-06 2048 -2.599752e-06 7.350381e-06 7.794897e-06 -1.292476e-06 5.431557e-06 5.581926e-06 5.581926e-06 2048 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.00e+00

The absolute values appearing along the first two rows of each set of three records indicate deviations form the standard regression files (validation) and the relative errors in the third row indicate reproducibility relative to saved results from an earlier execution of the same calculation

{ 2.4) Plot files may be generated by first commenting out the suppressed plotting commands which are of the form \tt include g5.h and by then linking the resulting test program to empire through the alternate commands:

{III. DESCRIPTION OF BATTERY OF UNIT TESTING EXERCISES

In each of the following phases of the unit testing, elements of specified sets of input regression files confirmed by validation studies will be scanned for various computational results including input parameters and essential output results relative permittivity and effective conductivity. The input parameters are then passed to the tested module for a testing replication of the calculation and the results are written to a file (unit 8 below)  for final compilation. The contents of file unit 8 for the current unit test are then compared with the results of a  previous unit test (file 9 below ) and both sets of results are written as the contents of a relative error file directed to unit  10 described below.

Directory containing files described below: /dfs/username/CREW_dielectrics/test/DATA/dielectricModel/

file name                 unit #      purpose

soil_regression_output_2.txt  8      current unit testing results soil_regression_input(0).txt  9      previous confirmed unit testing results soil_regression_output_3.txt 10      comparison of above two files (relative errors)

{3.1) The first unit test involves an initial general data replication file designated

/dfs/username/EMPIRE/test/DATA/dielectricModel/soil_regression_input(1).txt

The first several and final records of the file have been printed below.

1    2.5000000     2.5000000  1400.0000000     0.0100000     0.4605203    -0.4129327  2     2.5000000     2.5000000  1400.0000000     0.0156693     0.4665905    -0.3758497  3     2.5000000     2.5000000  1400.0000000     0.0213386     0.4736978    -0.3405790  4     2.5000000     2.5000000  1400.0000000     0.0270079     0.4817908    -0.3069351  5     2.5000000     2.5000000  1400.0000000     0.0326772     0.4908130    -0.2747605         .         .         .         . 126     2.5000000     2.5000000  1400.0000000     0.7186614     1.8287903     1.1365077 127     2.5000000     2.5000000  1400.0000000     0.7243307     1.8351817     1.1421677 128     2.5000000     2.5000000  1400.0000000     0.7300000     1.8415284     1.1477908

{3.2) Replication of Results from Previously Validated Soil Dielectrics Calculations

The portion of the general algorithm specialized for frequencies exceeding 1400 MHz and which therefore invokes the empirical polynomials due to Hallikainned and DoBson [3-4], is next tested by reading the eight files listed below each of which is specialized to examine either one of the three primary soil compositions (silt, sand and clay ) or one of each of the five loamy soils examined in Reference [3-4]. Whereas the Dobson frequencies are examined only in the files appearing in the first column belos, all files in the list are processed in a identical fashion in which their contents are scanned, input parameters are passed to the tested algorithm to recover results which are stored in a transient file (unit right) described in the previous section

/dfs/username/CREW_dielectrics/test/DATA/dielectricModel

UT(1)_0.txt    UT_CMP(1)_0.txt             UT_LE(ZLFMHZ)_(6)_0.txt UT(2)_0.txt    UT_CMP(2)_0.txt             UT_LE(ZLFMHZ)_(7)_0.txt UT(3)_0.txt    UT_CMP(3)_0.txt             UT_LE(ZLFMHZ)_(8)_0.txt UT(4)_0.txt    UT_LE(ZLFMHZ)_(1)_0.txt UT(5)_0.txt    UT_LE(ZLFMHZ)_(2)_0.txt     UT_STRNBRG(9)_0.txt UT(6)_0.txt    UT_LE(ZLFMHZ)_(3)_0.txt     UT_STRNBRG(10)_0.txt UT(7)_0.txt    UT_LE(ZLFMHZ)_(4)_0.txt UT(8)_0.txt    UT_LE(ZLFMHZ)_(5)_0.txt

A representative record from the first of the above files is printed below: index of soil variety | | excursion entry (1-128) | |   frequency (MHz) | |     |           volumetric moisture content | |     |          |       gravimetric sand percentage | |     |          |        |       gravimetric clay percentage | |     |          |        |         |    Log_10( relative permittivity =e'/e0) | |     |          |        |         |       |   Log_10( Im{e_c/e0) | |     |          |        |         |       |   (e' -je")/e0 = e_r -j sg/(w e0)                  | |      |          |        |         |       |        |<---Log_10(e"/e0)

sa,cL,zLer,zLei= 1 1 1400.000000 0.001000 2.500000 2.500000 0.453123 -0.476084 sa,cL,zLer,zLei= 1 2 1400.000000 0.006740 2.500000 2.500000 0.457518 -0.435161 sa,cL,zLer,zLei= 1 3 1400.000000 0.012480 2.500000 2.500000 0.463045 -0.396473 sa,cL,zLer,zLei= 1 4 1400.000000 0.018220 2.500000 2.500000 0.469663 -0.359765 sa,cL,zLer,zLei= 1 5 1400.000000 0.023961 2.500000 2.500000 0.477322 -0.324827 sa,cL,zLer,zLei= 1 6 1400.000000 0.029701 2.500000 2.500000 0.485964 -0.291478 sa,cL,zLer,zLei= 1 7 1400.000000 0.035441 2.500000 2.500000 0.495530 -0.259567

The above files contain the same information as in the input files described above with the exception of the last two columns which are the results for the \log_{10 of the complex relative permittivity e= (e\'-je\")/e0 which are compared with same values in the standard regression files

IV. EXHAUSTIVE SOIL COMPOSITION VARIATIONAL EXERCISE

The conclusion of the unit testing described here exhaustively scans a uniform sampling of the soil solids phase diagram with uniform spacing (5\% per increments) along the silt axis and 5\% along the axis perpendicular to the the silt axis, axis Clay^{\prime. At each locus of the phase diagram so selected, contour plot data is generated spanning the full range of the arguments frequency from .01 MHz to 8Ghz (sampled uniformly in \log_10 from -2 to 4.2552). Approximately 231 compositions are considered since the triangle with base 200/sqrt(3)) and a height of 100 has an area of 5773.5027 which contains 231 subdivisions each of area 25 square units.

V. REFERENCES

[1]Campbell, J.E., Dielectric Properties And Influence Of Conductivity In Soils At One to Fifty Megahertz, Soil Science Society American Journal 54(3),332(1990).

[2]Campbell, J.E., Dielectric Properties of Moist Soils at RF and Microwave Frequencies, Doctoral Dissertation, Dartmouth College, 1988), University Microfilms International, Ann Arbor, Michigan (1988)

[3]Hallikainen, M.T., Ulaby, F.T., Dobson, M.C., El-Rayes, M.A., and Wu, Lin-Kun, Microwave Behavior Of Wet Soil, Part I, IEEE Transactions on Geoscience and Remote Sensing 23(1), 25 (I985)

[4]Dobson, M.C., Ulaby, F.T., Hallikainen, M.T., and El-Rayes, M.A., Microwave Behavior Of Wet Soil, Part II, IEEE Transactions on Geoscience and Remote Sensing 23(1), 35 (I985)

[5] Peplinksi, N.R., Ulaby, F.T., and Dobson, M.C., Dielectric Properties of Soils in the 0.3 -1.3 GHz Range, IEEE Transactions of Geoscience and Remote Sensing 33(3), 803 (1995)

[6] Peplinksi, N.R., Ulaby, F.T., and Dobson, M.C., Corrections to Dielectric Properties of Soils in the 0.3 -1.3 GHz Range, IEEE Transactions of Geoscience and Remote Sensing 33(3), 803 (1995)

[7]Sternberg, B.K., Levitskaya, T.M.,  Electrical Parameters Of Soils In Frequency Range 1 kHz To 1 GHz Using Lumped Circuit Methods, Radio Science 36(4),709 (2001)