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EM1DFM  Help: Examples
 

Four examples are presented:

Here are links to return to the "EM1DFM" technical manual, or the GUI manual.

Example 1: DIGHEM-type sounding

For the first of three examples, consider synthetic DIGHEM-style data generated for the layered conductivity/susceptibility model shown by the dashed lines in Figure 1.
Figure 1. 
The layered conductivity/susceptibility model (dashed lines) for which the synthetic data-set used as the first example were generated, and the final conductivities and susceptibilities (solid lines) produced by the inversion of the synthetic data-set.

Forward-modelled data were computed for five frequencies, three of which (880, 7213 & 55840 Hz) were for the horizontal coplanar configuration, and two of which (1082 & 5848 Hz) were for the coaxial configuration. The transmitter-receiver pair were at a height of 40 m above the surface of the Earth, and their separation was 8.1 m for all frequencies (except 55840 Hz, for which the separation was 6.3 m). Gaussian noise (of standard deviation equal to 5 % of the absolute value of a datum, or 1 ppm, whichever was greater) was added to the forward modelled data to give the synthetic data-set that was inverted. The chi-squared measure of the total amount of noise that was added was equal to 7.48. The observations file ("test.obs") containing the data was: 

    1
    1.0   0.0    5
  880.    1
    1.0   -40.0  z    1
    1.0   8.1    0.0  -40.0  z  1  b  -7.867  30.8   v  1.0   1.61
 7213.    1
    1.0   -40.0  z    1
    1.0   8.1    0.0  -40.0  z  1  b  67.12   206.9  v  3.50  10.1
55840.    1
    1.0   -40.0  z    1
    1.0   6.3    0.0  -40.0  z  1  b  260.8   210.2  v  13.1  11.1
 5848.    1
    1.0   -40.0  x    1
   -1.0   8.1    0.0  -40.0  x  1  b  11.16   43.9   v  1.0   2.15
 1082.    1
    1.0   -40.0  x    1
   -1.0   8.1    0.0  -40.0  x  1  b  -0.523  11.1   v  1.0   1.0
The above data are values of the secondary magnetic field normalized by the free-space field, and expressed in parts-per-million. The standard deviations of the noise added to each datum were used as the measurement uncertainties. These are also given in ppm in the above file. Note the receiver dipole moment of -1Am2 for the coaxial configuration that is required to convert the normalization convention used for DIGHEM data to that used by program EM1DFM (see Section 2.1 of "Background for Program EM1DFM").

These data were inverted for both conductivity and susceptibility (with positivity) using the GCV criterion. The em1dfm.in  file was: 

test                   ! Root for output file names.
test.obs               ! Name of the observations file.
3                      ! mtype.
layers.dat             ! Starting conductivity model.
0.                     ! Starting susceptibility model.
0.0001                 ! Reference (smallest) conductivity model.
0.                     ! Reference (smallest) susceptibility model.
NONE                   ! Reference (flattest) conductivity model.
NONE                   ! Reference (flattest) susceptibility model.
NONE                   ! Additional weights.
1. 5. 0.001 1. 0.01 1. ! sigchibal, acs, acz, ass, asz.
3                      ! Type of inversion algorithm.
0.5                    ! Greatest allowed decrease in the trade-off parameter.
15                     ! Max number of iterations in the inversion.
DEFAULT                ! Stretch-factor for logarithmic barrier term.
0.02                   ! Small number for convergence tests.
DEFAULT                ! Number of explicit evaluations of Hankel kernels.
2                      ! Flag for amount of output.

The starting conductivity model file supplied to program EM1DFM was a thicknesses-only file (see Section 3.1.3 of the manual), and so the starting conductivity used in the inversion was the best-fitting halfspace. The reference conductivity model for the smallest component of the model norm was a homogeneous half-space of 0.001 S/m. The starting susceptibility model, and the reference susceptibility model for the smallest component of the model norm, were halfspaces of 0 SI units. There were no reference models for the flattest components of the model norm. The values of the various components of the objective function as the inversion progressed (as given in the status reports to the standard output and file em1dfm.out) were: 

Iter phid    beta    phim    gamma         phiLB    Phi     (Phi)
0    278.61          4.5326               -345.39
1    138.93  26.213  2.5452  1.4947        -316.71  679.02  913.66
2    90.235  13.107  2.0146  0.45704       -295.26  251.59  317.04
3    50.726  12.517  1.8084  0.30782       -274.82  157.96  206.34
4    10.314  28.962  1.6117  0.20854       -246.60  108.42  160.41
5    6.0239  35.200  1.4336  0.62364E-01   -227.31  70.661  82.423
6    6.0010  34.188  1.3481  0.46773E-02   -223.90  53.136  56.098
7    5.9749  35.346  1.2606  0.15723E-02   -234.35  50.900  54.002
8    5.9739  37.782  1.2264  0.15395E-03   -252.84  52.347  53.638
9    5.9651  37.192  1.1865  0.78292E-04   -271.02  50.113  51.604
10   5.9554  36.366  1.1692  0.23494E-04   -286.81  48.481  49.119

(see Section 3.2.1 for the meaning of each quantity). Convergence was reached at the tenth iteration. The final value of misfit (5.96) is slightly less than the amount of noise added to the synthetic data (= 7.48). The final conductivity/susceptibility model is shown in Figure 1. Both the conductivity and susceptibility parts of the constructed model are good representations of the structures in the model for which the data were generated, although they have the smeared-out character that is typical of models constructed by minimizing a sum-of-squares measure of structure. The forward-modelled data for the model constructed by the inversionare (file "test.prd"): 

    1
    1.0   0.0  5
  880.    1
    1.0   -40.0  z    1
    1.0   8.1    0.0  -40.0  z  1  b   -7.761  31.5
 7213.    1
    1.0   -40.0  z     1
    1.0   8.1    0.0  -40.0  z  1  b   67.00   199.5
55840.    1
    1.0   -40.0  z    1
    1.0   6.3    0.0  -40.0  z  1  b   266.3   219.2
 5848.    1
    1.0   -40.0  x    1 
   -1.0   8.1    0.0  -40.0  x  1  b   11.84   42.6
 1082.    1 
    1.0   -40.0  x    1
   -1.0   8.1    0.0  -40.0  x  1  b   -1.64    9.57

Example 2: EM31-type sounding

For a second example, consider the following observations file ("em31.obs") for a synthetic sounding generated for EM31-type survey parameters: 
   1
   0.0   0.0    1
9600.    4
   1.0   -1.0   z     1
  -1.0   3.66   0.0   -1.0   z  2  q  -0.333  v  0.020
   1.0   -1.0   y     1
  -1.0   3.66   0.0   -1.0   y  2  q  -0.211  v  0.020
   1.0   -0.05  z     1
  -1.0   3.66   0.0   -0.05  z  2  q  -0.447  v  0.021
   1.0   -0.05  y     1
  -1.0   3.66   0.0   -0.05  y  2  q  -0.285  v  0.020
The conductivity model for which these data were generated is shown by the dashed line in Figure 2. (The susceptibility was assumed to be zero).
 
Figure 2. 
The final model (solid line), and the model from which the synthetic data were generated (dashed line), for the EM-31 single sounding example.

These data are for two different heights of the instrument above the Earth's surface (waist height and on the ground), and for two different orientations (the normal instrument orientation, and the instrument on its side). The data in the above file are values of the quadrature part of the secondary magnetic field normalized by the free-space field and expressed as a percentage of the primary field. Gaussian noise of standard deviation equal to 5% of the magnitude of each datum, or 0.02%, whichever was larger, was added to the forward-modelled data to give the above data-set. The chi-squared measure of the total amount of noise added was 1.60. The measurement uncertainties in em31.obs above are the standard deviations of the noise added to each datum. Note also the receiver dipole moment of -1Am2 for all four orientations of the instrument to reconcile the normalization convention used with program EM1DFM (see Section 2.1 of "Background for Program EM1DFM") and that used for EM31 data. The data were inverted using the following em1dfm.in file: 

em31         ! Root for output file names.
em31.obs     ! Name of the observations file.
1            ! mtype.
layers.dat   ! Starting conductivity model.
0.01         ! Reference (smallest) conductivity model.
0.           ! Reference susceptibility model.
NONE         ! Reference (flattest) conductivity model.
NONE         ! Additional weights???
0.1 1.       ! alpha s & alpha z.
3            ! Type of inversion algorithm.
0.5          ! Greatest allowed decrease in the trade-off parameter.
15           ! Max number of iterations in the inversion.
0.001        ! Small number for convergence tests.
DEFAULT      ! Number of explicit evaluations of Hankel kernels.
3            ! Flag for amount of output.

The data were therefore inverted for conductivity only (mtype =1). The file "layers.dat" was a thicknesses-only file (see Section 3.1.3 of the manual) with 25 layers and a depth of 40 m to the basement halfspace. The program therefore used the best-fitting halfspace as the starting conductivity model. The reference model for the smallest component of the model norm was a homogeneous halfspace of 0:01 S/m, and there was no reference model for the flattest component of the model norm. The background susceptibility model was a non-susceptible halfspace. The GCV-based inversion algorithm was used (Algorithm 3).

The iteration-by-iteration status reports written to the standard output, and to the file em1dfm.out, during the inversion were: 

PROGRAM "EM1DFM" (v1.0).
Start at 16:52:02.670, 20/07/2000.

Sounding 1 (0.0,0.0).
Initial phid= 36.599, phim= 1.0948.
Iteration 1: phid= 18.875, beta= 5.5817, phim= 1.9090, Phi= 29.530 (42.710).
Iteration 2: phid= 3.2718, beta= 2.7908, phim= 2.3937, Phi= 9.9522 (24.203).
Iteration 3: phid= 1.4503, beta= 1.3954, phim= 2.8128, Phi= 5.3753 (6.6120).
Iteration 4: phid= 1.0292, beta= 0.69771, phim= 3.1536, Phi= 3.2295 (3.4128).
Iteration 5: phid= 0.98273, beta= 0.58493, phim= 3.2059, Phi= 2.8580 (2.8739).
Iteration 6: phid= 0.98277, beta= 0.58778, phim= 3.2058, Phi= 2.8671 (2.8671).
Convergence.

Final conductivity model written to "em31.con".
Predicted data written to "em31.prd".

The End! [16:52:21.195, 20/07/2000]

The model produced by the inversion is shown by the solid line in Figure 2. The constructed model exhibits the typical smooth character of a model created using a sum-of-squares measure for the model norm, which is the case for all models produced by program EM1DFM. However, even before taking this smoothing into account, there is good agreement between the constructed model and the model from which the data were generated. Also, the final value of the misfit (0.983) is slightly less than the total amount of noise that was introduced into this synthetic data-set (1.60). Note that the GCV-based approach has a tendency to sometimes over-fit the observations. The forward-modelled data for the final model are (file "em311.prd"; see Section 3.2.4): 

    1
    0.0   0.0 1
39200.    4
    1.0   -1.0   z 1
   -1.0   3.66   0.0   -1.0   z  2  q  -0.33986
    1.0   -1.0   y     1
   -1.0   3.66   0.0   -1.0   y  2  q  -0.19743
    1.0   -0.05  z     1
   -1.0   3.66   0.0   -0.05  z  2  q  -0.43953
    1.0   -0.05  y     1
   -1.0   3.66   0.0   -0.05  y  2  q  -0.29555

Example 3: A line of synthetic airborne data

For the third and final example, consider the three-dimensional model shown in Figure 3.
 

Figure 3. 
The model from which the synthetic data were computed for the third example. The conductivity and susceptibility of prism 1 were 0.1 S/m and 0.1 SI units respectively, and those of prism 2 were 0.5 S/m and 0.2 SI units. The conductivity and susceptibility of the background were 0.01 S/m and 0 SI units. Prism 1 extended from 15 to 30 m in depth, and prism 2 from 30 to 100 m. The dashed line indicates the line for which data were inverted.

Observations are synthetic horizontal coplanar data generated at 25 m intervals along the line x = 350 m (the dashed line in Figure 3) for the ten frequencies: 110, 220, 440, 880, 1760, 3520, 7070, 14080, 28160 & 56320 Hz (see Zhang & Oldenburg, Geophysics, 64, 33; 47, 1999). The entries in the observations file for the first sounding are: 

   29
  350.0   0.0    10
56320.    1
    1.0   -30.0  z     1
    1.0   0.0    10.0  -30.0  z  1  b  2638.0  2034.5    v  52.085  41.239
28160.    1
    1.0   -30.0  z     1
    1.0   0.0    10.0  -30.0  z  1  b  1630.8  1728.7    v  33.242  34.379
14080.    1
    1.0   -30.0  z     1   
    1.0   0.0    10.0  -30.0  z  1  b  965.91  1291.1    v  19.315  25.972
 7040.0   1
    1.0   -30.0  z     1
    1.0   0.0    10.0  -30.0  z  1  b  515.93   893.56   v  10.328  17.943
 3520.0   1
    1.0   -30.0  z     1
    1.0   0.0    10.0  -30.0  z  1  b  263.55   579.97   v  5.1279  11.476
 1760.0   1
    1.0   -30.0  z     1
    1.0   0.0    10.0  -30.0  z  1  b  121.54   335.85   v  2.4043  6.9002
  880.00  1
    1.0   -30.0  z     1
    1.0   0.0    10.0  -30.0  z  1  b  53.439   196.75   v  1.0790  3.9562
  440.00  1
    1.0   -30.0  z     1
    1.0   0.0    10.0  -30.0  z  1  b  23.986   106.72   v  1.0     2.1837
  220.00  1
    1.0   -30.0  z     1 
    1.0   0.0    10.0  -30.0  z  1  b  8.6005    58.106  v  1.0     1.1705
  110.00  1
    1.0   -30.0  z     1
    1.0   0.0    10.0  -30.0  z  1  b  5.4019    30.998  v  1.0     1.0
The data were contaminated with Gaussian noise of standard deviation equal to 2 % of the magnitude of a datum, or 1 ppm, whichever was greater. The transmitter & receiver coils were separated by 10 m in the y-direction, and were at a height of 30 m above the Earth's surface. The data are shown by the error bars in Figure 4a.

Figure 4a. The synthetic data inverted as the third example are shown by the error bars. The forward-modelled data for the models produced by the inversion are shown as dashed lines.

The data were inverted using the following em1dfm.in file: 
ubc4y350                ! Root for output file names.
ubc4y350.obs            ! Name of the observations file.
3                       ! mtype.
start.con               ! Starting conductivity model.
0.                      ! Starting susceptibility model.
0.01                    ! Reference (smallest) conductivity model.
0.                      ! Reference susceptibility model.
NONE                    ! Reference (flattest) conductivity model.
NONE                    ! Reference (flattest) susceptibility model.
NONE                    ! Additional weights???
1. 10. 0.001 1. 0.01 1. ! scbal, alphascon, alphazcon, alphassus, alphazsus.
3                       ! Type of inversion algorithm.
0.5                     ! Greatest allowed decrease in the trade-off parameter.
20                      ! Max number of iterations in the inversion.
DEFAULT                 ! Stretch-factor for logarithmic barrier term.
DEFAULT                 ! Small number for convergence tests.
DEFAULT                 ! Number of explicit evaluations of Hankel kernels.
1                       ! Flag for amount of output.

The starting model was a homogeneous halfspace of conductivity 0.003 S/m and susceptibility 0 SI units, and the reference model was a homogeneous halfspace of conductivity 0.001S/m and zero susceptibility. The status reports produced by the inversion were (see manual section 3.2.8): 

PROGRAM "EM1DFM" (v1.0).

Start at 21:32:34.709, 04/07/2000.

Sounding 1 (350.0, 0.0).
Convergence: n= 17, phid= 19.24, beta= 0.145E+06, phim= 0.314, phiLB= -322.5, Phi= 45642..

Sounding 2 (350.0,25.0).
Convergence: n= 19, phid= 23.97, beta= 0.309E+06, phim= 0.251, phiLB= -380.1, Phi= 77364..

Sounding 3 (350.0,50.0).
Convergence: n= 17, phid= 16.719, beta= 2155.2, phim= 0.3452, phiLB=-325.40, Phi= 760.8.

Sounding 4 (350.0,75.0).
Convergence: n= 18, phid= 14.875, beta= 927.01, phim= 0.3083, phiLB=-351.46, Phi= 300.7.
. . .
Sounding 17 (350.0,400.0).
Convergence: n= 15, phid= 26.531, beta= 1.1198, phim= 1.4454, phiLB=-138.98, Phi= 28.167.

Sounding 18 (350.0,425,0).
Convergence: n= 11, phid= 19.036, beta= 17.650, phim= 1.1546, phiLB=-112.92, Phi= 39.438.

Sounding 19 (350.0,450.0).
Convergence: n= 15, phid= 50.583, beta= 1.1310, phim= 2.2370, phiLB=-138.68, Phi= 53.277.
. . .
Final conductivity model written to "ubc4y350 con.mod".
Final susceptibility model written to "ubc4y350 sus.mod".
Predicted data written to "ubc4y350.prd".

The End! [22:48:55.498, 04/07/2000]
The final conductivity and susceptibility models are shown below in Figures 4c and 4d, and the corresponding forward-modelled data are included as dashed lines above in Figure 4a.


Figure 4b. The misfit and model norm for the final model at each sounding are indicated by filled and open circles respectively.


Figures 4c. Final 1D conductivity model at each location, concatonateded into a 2D section under the survey line.


Figures 4d. Final 1D susceptibility model at each location, concatonateded into a 2D section under the survey line.

Example 4:

Seven soundings were done with a MaxMin I8S system by students in applied geophysics at UBC. The location was a local peat bog. Measurements of the inphase and quad phase response were recorded at eight frequencies for every sounding, and the instrument was used in the horizontal loop (vertical dipole) mode with coil spacing of 30 metres. Results, data, etc. are provided via the following links: