U S E R M A N U A L

                               "G R O W T H 3.0"                                                           

    A 3-D gravity inversion tool based on exploration of model possibilities   

            by Antonio G. Camacho with support from Jose Fernandez                                      
               Instituto de Geociencias (IGEO), CSIC-UCM
               Calle del Doctor Severo Ochoa, 7
               Ciudad Universitaria.
               28040 Madrid (Spain)            
               e-mails:  antonio_camacho@mat.ucm.es, jft@mat.ucm.es

1. INTRODUCTION

This computational tool carries out a 3-D inversion of gravity data.

The main characteristics of the method are the following:

(1) solution for 3-D model space, (2) acceptance of non-gridded non-planar low-precision imprecise data, (3) calculation can start for an empty body or optionally from an initial “seed” structure, (4) automatic determination of a linear regional gravity trend from the data, (5) automatic determination of density contrast values (6) simultaneous inversion for both positive and negative density contrasts, (7) inversion for irregularly-shaped structures composed of several individual bodies, (8) inversion accounting optionally for a downward density increase, (9) inversion accounting optionally for layered structures, (10) semi-automated data inversion routine.

The software package is composed of two parts:

(a) “GROWTH” to perform the inversion routine and to obtain a 3D anomalous density model, and (b) “VIEW” for visual representation of the input data, the inversion model and modelling residuals.

The current version 3.0 of the tool has been developed from an earlier code [1 and 2] and now incorporates several novelties:

(1) a more compact package with only two codes
(2) an optional determination of values for density contrast (3) compatibility with Windows 10 Pro (4) inversion accounting optionally for a downward density increase, (5) inversion accounting optionally for layered structures, (6) inversion accounting optionally for lateral smoothing of the source structures, (7) a new graphical presentation across the inversion process, (8) a simpler structure of the file with the resulting 3D model, (9) default values for all the parameters, and suggestions after running.

The main input is a file "GRA.DAT" containing the coordinates of the gravity stations and the observed anomaly (see #2.2). The method (see [1] and [2]) seeks to determine the geometry of an indefinite number of anomalous bodies with prescribed (fixed or variable, positive and/or negative) density contrasts. A 3-D grid of regular cells is taken to represent the subsurface volume. Then, the inversion method fills some of these cells, with the possible density contrast, to reproduce the anomalous structures.

This software is composed of two Fortran 77 codes: GROWTH.FOR and VIEW.FOR.

GROWTH.FOR is the main inversion program. VIEW.FOR provide graphics representation of the obtained models.

2. PROGRAM "GROWTH.FOR": 3-D gravity inversion.

2.1. AIM: to realize a 3-D gravity inversion, searching for a model (MOD.DAT) of the anomalous bodies by means of an exploratory process from a gravity data file (GRA.DAT). The program works step by step in a growth process of the anomalous bodies to fit the observed gravity anomaly.

2.2. INPUT.

2.2.1. Input file:

-> file "GRA.DAT": The basic input ASCII file, named "GRA.DAT" (as default), contains a line corresponding to each gravity benchmark with the following values: x-coordinate (m), y-coordinate (m), altitude (m) (throughout the software package, elevation and depth values are positive above sea level and negative below sea level, respectively), gravity anomaly (uGal) (1 uGal=10-8 m/s2). Two additional columns are optionally included: a gravity error estimate (uGal) for each anomaly value, and the value for terrain correction (uGal) corresponding to unit density (1 kg/m3). Values for error estimation will be employed and propagated only optionally for initial relative weighting of the data. Values for terrain correction will be employed only optionally for re-estimation of a value for terrain density. Both complementary values could be filled with arbitrary numbers and neglected in the process. The end of data will be the last line or can be indicated by a line of “zeros”.

example:

212876.1 3079402.0 560.30 9094 30 2917 212903.9 3079303.6 571.02 9041 30 3012 213149.2 3079854.5 533.59 5136 30 2824 213128.0 3080707.0 524.20 2059 30 2885 212667.0 3081849.6 403.01 1718 30 2370 .......................................... 201634.6 3073315.1 283.73 44390 30 4110 0 0 0 0 0 0

2.2.2. Values for several parameters in a dialog form through the Graphical User Interface (GUI):

First step (left hand pane):

• Names of the input and output files (default values are provided). Paths can be used. It is necessary to press “Run/OK” at the end of each step. The program’s response to successful data loading is displaying the number count of benchmarks.

Second step: geometrical parameters (default values are provided).

• Depth (m) of the model top. Negative values below sea level as usual.
• Maximum depth (m) of the model bottom. Negative values below sea level as usual.
• Mean length (m) for the side of the parallelepiped cells. Shallow cells will have smaller length and deep cells will have larger length.

Third step (bottom): repetition.

If the resulting number m of cells is too high (>95,000 in this version) the check box “Re-run” allows for a new calculation. Upon clicking “Run”, the resultant 3D cell partition model is saved by default in the file “GRI.DAT”.

Fourth step (centre and right hand pane): parameters for inverse modelling (default or example values are provided).

a) Regional trend.

• Mean value g0 (µGal). This is an assumed regional constant value or mean offset value for the Bouguer anomaly data. If the “Adj” button is not pressed, the inversion is based on the given input value, which is assumed as fixed. By pressing the corresponding “Adj” button (default option), this parameter is considered as unknown to the program and an optimal value is adjusted as part of the inversion. The offset parameter has a particular bearing on the deeper parts of the model domain. We suggest performing an initial inversion by automatically adjusting a value, and then re-running inversions by varying this value in order to probe changes in the deep parts of the resultant 3D density models.
• gx, gy (µGal/km). These are slopes of the linear regional trend. The values may either be obtained from a regional survey and then introduced here as fixed values or conversely, they can be regarded as unknowns (pressing “Adj” button, default option) to be adjusted simultaneously in the inversion process.

b) Inversion parameters.

• Coefficient r for random search. This is an integer value (>=1) that allows for an exploratory process with a more or less random character. For a value of 1, the program carries out a systematic exploration. For a value r>1, the program tests 100/r % cells, randomly selected, thus allowing for a faster (but less precise) solution.
• Balance factor. This positive value corresponds to the parameter for balance between “fitness” and “smoothness” of the model. The value is dependent on the quality of the data and on the complexity of the subsurface structure. For low values, the inversion results in the generation of an excessively complex model of the subsurface structure with very good fit to the data. For high values, the inversion produces a simple model but with poor fit to the data. The operator may try different values of it close to the default value and draw conclusions by means of the third program (“VIEW”, see below) and evaluation of the auto-correlation distribution.

c) Terrain density fit. This is an option to estimate a revised value (kg/m3) to the terrain density used in the gravity anomaly calculations. It should be applied when the topographic relief is highly variable over the survey area (large and different terrain coefficients for unit density) to ensure a reliable solution. The corrective term is determined simultaneously during the gravity inversion according to the same minimization constraints. In this case a new parameter is required: the terrain filtering radius (meters). This is a distance value used to carry out an initial filtering of the terrain coefficients.

d) Parameters for density distribution.

• Prescribed positive and negative limits for density contrast (kg/m3). Optionally, pressing button “Adj”, the program determines a density contrast values (positive and negative) satisfying the minimization conditions.
• Lateral smoothing "a": This positive value defines the pattern of density transition across the boundaries (geometrical smoothness) of the resultant anomalous bodies. For low values of a, the boundaries are abrupt and the bodies approximate homogeneous geometries (a=0) , such as cubes or spheres, with a single density value. For high values of a, the boundaries appear smoothed and the bodies are more diluted, with density values close to the extreme values towards the centre of the body and decreasing values towards their periphery.
• Flattening coefficient "f": It is connected with the downward continuous increase in density for the background medium. For a flattening coefficient f close to zero the inverse approach produces a 3D model with cup shape, corresponding to a homogeneous background medium. For a high value of this coefficient f, the inverse model becomes flattened, with small vertical size and bell shape, as corresponding to a background medium with high downward increase in density. For instance, for volcanic areas, with possible intrusive bodies vertically elongated, a f value close to 2 or 3 can be tried. But for sedimentary areas, dominated by sub-horizontal layers and increased compaction as a function of depth, a f value larger than 6 could be more suitable.
• Upward weighting u: It is an optional parameter to get shallower models, based on an unbalanced weighting in the fit. Top regions have higher weigth than deep regions. It is very useful for models characterized by a shallow main low density structure. This is the case of basin structures and other sedimentary structures. Value cero for u corresponds to neutral weighting. Higher value as 1,2, ... produce shallower models.
• Sub-horizontal discontinuity layers: This check box accounts for the particular case of structures composed by several discrete layers with known average depth and fixed density. This case requires a precise a priori information of the medium structure to get suitable results. In the case of select this option, a new dialog window appears to input the a priori values for the layers.

2.3. RUNNING

Once all the required parameters are introduced, the program initiates the inversion. The evolution of the numbers of filled cells, number of outliers, density contrast values, standard deviation, regional values, etc. are shown on-screen. A graphic presentation (plan and elevation views) shows also the model growth.

The process ends when it is not possible to aggregate a new cell within the model constrains. Then , a message of either correct execution or error appears. Moreover, a box with some characteristic parameters of the resulting model and another box with a graphical presentation of the mass vs. depth distribution are displayed.

In particular, resulting values for (a) autocorrelation, (b) mass inversion, and (c) mass/depth slope can help to decide modification of the input modelling parameters. For instance, a high autocorrelation value of the resulting residual values suggests that the assumed balance factor is too low and there is a remaining signal. A re-running with a lower balance factor would be interesting. A high resulting value of mass inversion suggest that there is some portion of the model with downward density decrease. A re-runnig of the inversion approach with a higher value of flattening would be interesting. Surely, the last conclusion could be also obtained by inspection of the mass vs. depth distribution picture.

2.4. OUTPUT

-> file "MOD.DAT": inversion model as 3-D partition of the subsurface volume into cells of anomalous density. The content and general format of this ASCII file is:

nc, side, step
x(1),y(1),z(1),tx(1),ty(1),tz(1),d(1)   
x(2),y(2),z(2),tx(2),ty(2),tz(2),d(2)          
..........................................
x(nc),y(nc),z(nc),tx(nc),ty(nc),tz(nc),d(nc),ds(nc),at(j),sc(j)

where:                                                         

nc:  total number of cells (parallelepiped) of the model                                  
side: adopted mean side (in metres) for the cells in the 3D partition                                    
step: distance (in metres) for step in the further correlation analysis. 

x(j),y(j),z(j): coordinates (in metres) UTM and depth for the center of the j-th cell of the model.
tx(j),ty(j),tz(j): horizontal sides along x,y,and z of the j-th cell of the model.
d(j): anomalous density contrast (kg/m3) of the j-th cell

ds(j): anomalous density contrast (kg/m3) of the j-th cell without including the medium effects.
at(j): mean quadratic atraction ugal for density=10 Tm/m3 in the j-th cell
sc(j): scale factor when filling of j-th cell
j=1,...,nc

At the end of this file MOD.DAT, several characteristic parameters of the inversion model are displayed: mean density contrast of the anomalous model, mean density contrast of the negative anomalies, mean density contrast of the positive anomalies, extreme density contrasts, depth of the full anomalous, negative and positive structures, etc.

  example:

97629 280 1497 199817 3070514 1161 224 144 224 0 798 66 0.0 199817 3070646 1161 224 120 224 0 798 64 0.0 199817 3070759 1161 224 106 224 0 798 64 0.0 199817 3070862 1161 224 100 224 0 798 64 0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225025 3082597 -9968 1203 557 1203 0 5586 -61 0.0 225025 3083167 -9968 1203 582 1203 0 5586 -62 0.0 -------------------------(-)-------(+)-----(anom)-----(min)-----(max)----- Mean Dens.(kg/m3) : -400. 502. 500. -20. 20. Depth of cells(m) : -1247. -4727. -1935. 540. -7472. Weighted depths : -1476. -5159. -5113. Number of cells : 610 9735 10345 Anomal. mass (kg) : 0.734E+13 0.340E+15 0.347E+15 (rel. 0.000E+00) Side of cells (m) : Min= 181 Max= 241 Average= 413 Ref= 280. Num.of stations= 179 Scale fact. = 0.990 Outliers(>3.5)= 0 ( 0.0%) Gravity anomaly dispersion (uGal) Obs.= 13581. Reg.= 7032. Loc.= 25532. Res.= 1827. Modelling parameters Lambda fac= 80.0 Up weighting= 0.0 Scale fact: in= 2543. fin= 20.00 A priori data weighting Y(1)/N(0):0 Random exploration coef.: 10 (1 sistem.) Trend = 6946.(1) uGal + ( -1085.(0) -92.(0) ) uGal/km Slope= 1089. uGal/km Azim.= N 85. E (1:adj,0:fix) Incremental coord. from (xm,ym,zm)= 203920 3072960 540 m Additional terrain density= 0. (0) kg/m3 Terrain filtering radius= 5986 m Residues (uGal): Mean= 6 Min.= -4844 Max.= 4125 Weight. StDv= 1827.3 Consist. Stdv= 1791.3 Fit.= 48737. uGal Mean Sensitivity for cells= 0.3 uGal for 100 kg/m3 Autocorrelation = 0.24 Correlat. step= 1497 m StDv Data = 25532. uGal SD Resid/SD Data = 0.07 Dens. pattern: Lateral smoothing (1-10)= 0.00 Flattening= 0.0 Downward Incr.: 0 kg/m3 each km Sub-horiz.discontin. (Yes(1)/No(0)): 1 Density contrast fit (Yes(1)/No(0)): 0 A priori max.densit.contrast= -20. 20. kg/m3

     Layers model:   Shape=2.00  Bottom=  -7380.
     Weight.model:  q1n=520 q2n=-12  q1p=  0 q2p= 40
     Num. of layers=14
     Depths(m)   1270   1270   1040    600    370    -70   -520  -1050  -2010
     Density      399    399    399    399    399    399    399    399    399
     Flattening    11.0   11.0   11.0   11.0   11.0   11.0   11.0   11.0   11.0
     Neg.weig:    NaN    NaN    NaN    NaN   1.05   0.82   0.86   0.89   0.98
     Pos.weig:    NaN    NaN    NaN    NaN    NaN   0.91   0.80   0.73   0.62
     Num.cells      0      0      0      0      1    120    294    568    955
     Aver.weights:   Neg= 0.88  Pos= 0.39  Tot= 0.41

-------------- Date:04-Mar-20-----------------

-> file "FIL.DAT": ASCII file containing, for each data point (in planar coordinates), the following values (uGal) resulting from the inversion process: regional gravity anomaly observed local anomaly (= observed - regional anomaly) modelled local anomaly residual values estimated weights (relative to median=1.0)

 example:

212876. 3079402. 12692. -3598. -3821. 223. 0.74
212877. 3079304. 12669. -3628. -3444. -184. 0.61
212878. 3079854. 12466. -7330. -7126. -205. 0.68
212879. 3080707. 12484. -10425. -10283. -143. 0.47
212880. 3081850. 12865. -11147. -11392. 244. 0.81
212881. 3082700. 13424. -15454. -14296. -1158. 3.84 *
212882. 3083784. 13356. -13964. -14031. 67. 0.22
                        . . . . . . . . . . .

2.4. DIMENSIONS

The memory requirements of the program execution depend on the following parameters:

ms:  max. number of gravity stations. 
mc:  max. number of whole prismatic cells for the 3-D grid.   

The values of these size parameters can be changed in the code according to the applications and the memory abailability.

2.5. INCLUDED SUBROUTINES

VAP: Vertical attraction (in microgal, positive upwards) of a parallelepiped cell of: center xc,yc (metres, from the point), horizontal sides dx,dy (metres), depth of the bottom and top faces zb,zt (metres, positive upwards), and density d (kg/m3) upon the coordinate origin. The main calculus is carried out in the subroutine PIC. STEP: Obtaining a default value for correlation step corresponding to the geometrical distribution of benchmarks. DMEDIAN: Obtaining the median of n different numbers. DIM: Output massage dimension limits are exceeded. COV2: Autocorrelation of two-dimensional data for a given correlation step. DIAL1, DIAL2, DIAL3, DIAL4, ABIL1, ABIL2, ABIL3, ABIL4, ABIL5, ABIL6, ABIL7, ABIL8, FIN: Dialog interfaces.

2.6. COMPILING SPECIFICATIONS.

Compiled with Intel Visual Fortran, by using a Quickwin Application Project. The complete compilation requires reveral additional files (that are available in the zip file of codes and):

Resource files: Growth.rc (resource script) Growth.ico (icono)

Fortran source files: Growth.fd Growth.for

C/C++Header: Growth.h (dialog window)

3. PROGRAM "VIEW.FOR": Graphic presentation of the inversion model.

3.1. AIM: to give a visual presentation of results coming from GROWTH gravity inversion. The program can offer several coloured pictures of the 3D inversion model by means of vertical profiles and horizontal cross sections. Maps of the gravity anomaly, altitudes, residuals, calculated anomaly, regional trend, etc are also availables. Optionally, output files can be obtained to export the figures to another more sophisticated graphical device.

3.2. INPUT.

-> file "GRA.DAT" (default name): ASCII file of coordinates, altitudes, observed gravity values and relative errors for the gravity stations. See paragraph #3.2 about content and format.

-> file "MOD.DAT" (default name): resulting 3-D model for the anomalous density contrasts, obtained as output of GROWTH.FOR. See paragragh #3.4 about content and format.

-> file "FIL.DAT" (default name): resulting regional, local, calculated and residual anomaly values from the inversion process, obtained as output from GROWTH.FOR. See paragraph #3.4 about content and format.

-> file "MAP.BLN" (default name): ASCII file containing a base map (coast, roads, etc.) to be included on the graphic presentation. The format of this file is:

   j1
 xx(1),yy(1)
 xx(2),yy(2)
 ..........
 xx(j1),yy(j1)
    j2
 xx(1),yy(1) 
 xx(2),yy(2)
 ...........
 xx(j2),yy(j2) 
   ...

where j1,j2,... are the number of points for each line and xx,yy are plane coordinates for the sucessive points of the lines.

example

 1528   * El Hierro

211600 3084000 0.4 211670 3084000 0.0 211720 3084000 0.1 211765 3083930 0.6 . . . . . . . . . .

3.3. OUTPUT

3.3.1. GRAPHICS Several kinds of pictures corresponding to program options can be obtained on the screen.

a. Plane maps (including the lines of MAP.BLN) for: station altitudes observed gravity anomaly relative errors of the observed anomaly regional trend values local anomaly (= observed anomaly - regional anomaly) calculated values for the local anomaly inversion residuals

b. Combined view of one horizontal section (across a desired depth) and two vertical profiles (with nearly W-E and S-N directions) of the 3-D model for:
density contrasts sensitivity values

3.4. DIMENSIONS

The memory requirements of the program execution depend on the following parameters:

ms:  max. number of gravity stations. 
mc:  max. number of prismatic cells for the 3-D grid.   
mb:  max. number of points for the base map file.    

The values of these size parameters can be changed in the code according to the applications and the memory abailability.

3.5. INCLUDED SUBROUTINES

ACOL: determinig a colour k corresponding to a variable dn. CMAP: displaying a coloured map of a plane data distribution x,y,g. CONT: displaying a base map of polygonal lines. PERTOP: determining a topographical line across a vertical profile of the 3-D model AM: interpolating and smoothing from a 3-D grid of cells. SST: mean sensitivity of the network for a variable point mass STRA: determining horizontal layers. COV2 and BUSCA: Autocorrelation of two-dimensional data for a given correlation step. TITULO: drawing a title. DIM: Output massage dimension limits are exceeded. EJES: Drawing axis and ticks. DIAL1, DIAL2, DIAL3, DIAL4, FIN: Dialog interfaces.

function IRVA : combining basic colours (red, green, blue)

3.6. COMPILING SPECIFICATIONS.

Compiled with Intel Visual Fortran, by using a Quickwin Application Project. The complete compilation requires reveral additional files (that are available in the zip file of codes and):

Resource files: View.rc (resource script) View.ico (icono)

Fortran source files: View.fd View.for

C/C++Header: View.h (dialog window)

REFERENCES:

[1] Camacho, A.G., Montesinos, F.G. and Vieira, R. (2001). A 3-D gravity inversion tool based on exploration of model possibilities. Computer & Geosciences 28, 191–204

[2] Camacho, A.G., Fernandez, J. and Gottsmann, J. (2011). The 3-D gravity inversion package GROWTH2.0 and its application to Tenerife Island, Spain. Computer & Geosciences 37, 621-633.

[3] Camacho, A.G. and Fernandez, J. (2019).Modeling 3D free-geometry volumetric sources associated to geological and anthropogenic hazards from space and terrestrial geodetic data. Remote Sens., 11(17), 2042; doi: 10.3390/rs11172042.