GSJ Open-file Report, no.705
Modified GOTIC2: Software for estimation of ocean tidal loading effects at subsurface observation
KAMIGAICHI Osamu*1,*2,・MATSUMOTO Norio**2and HIROSE Fuyuki*3
*1The Japan Meteorological Business Support Center
*2Institute of Earthquake and Volcano Geology, Geological Survey of Japan, AIST
*3Meteorological Research Institute
We release the three files here as follows.
Explanation of the Modified GOTIC2
GOTIC2 (Matsumoto et al, 2001) is a software to calculate displacement and strain from solid Earth tide and ocean tidal loading effect at the earth’s surface by the convolution of ocean tidal loading distribution and Green’s function for the surface vertical point loading.
For the precise calculation of ocean tidal loading effect at deployment depth of borehole sensor such as borehole strainmeter, Green’s function evaluated at the deployment depth (not at the surface) is necessary (Kamigaichi, 1998; Kamigaichi et al., in review), and we modified GOTIC2 as follows:
1. Usage of Green’s function evaluated at deployment depth
The original GOTIC2 uses the Green’s functions evaluated at the surface when ocean tidal loading effect is calculated. In modified GOTIC2, we modified to use the Green’s functions evaluated at the deployment depth. In case of evaluating the Green’s functions, the spherically symmetric earth model in the original GOTIC2 was 1066A or Gutenberg-Bullen A, whereas the earth model in the modified GOTIC2 is changed to be PREM (Dziewonski and Anderson, 1981), whose uppermost oceanic layer is replaced by the crust (Tsuruoka et al., 1995).
The reasons why we use the Green’s functions evaluated at the deployment depth are as follows: Displacement and strain caused by the ocean tidal loading effect are calculated by the convolution of ocean tidal loading distribution and Green’s function. Behavior of the Green’s function at the surface and that at subsurface are substantially different in the vicinity of the loading point (Kamigaichi, 1998). Then, usage of Green’s function evaluated at the surface brings non-negligible error when we calculate ocean tidal loading effect at the deployment depth (Kamigaichi et al., in review）.
Green’s functions at subsurface change drastically in a vicinity of the loading point, especially the angular distance from the loading point θ < 10-2 degrees. Nevertheless, the Green’s function of the original GOTIC2 is discretized at only 1 x 10-4, 1 x 10-3 and 1 x 10-2 degrees when θ is ranged from 10-4 to 10-2 degrees. In order to accurately reproduce the drastic changes in the Green’s function, we modified number of discretization of the Green’s function of the modified GOTIC2 to 36 points when 10-4 ≤θ ≤ 10-2 degrees, that is, (1.0, 1.2, 1.6, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0)×10-4, (1.0, 1.2, 1.6, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0)×10-3，(1.0, 1.2, 1.6, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0)×10-2 degree. The Green’s function of the modified GOTIC2 is discretized at 81 points in total, whereas that of the original GOTIC2 is discretized at 50 points.
Theoretical values of volume strain in the subsurface must also be calculated because there are not only horizontal strainmeters but also volumetric strainmeters available as borehole strainmeters. At the earth's surface, the former can be calculated from the latter because a linear relationship between volume and area strains is established from free surface boundary conditions. However, the volume strain in the subsurface must be calculated independently of the area strain, as the relationship does not hold in the subsurface.
When calculating ocean tidal loading effects in GOTIC2, the Green's function in the mesh is approximated by a quadratic function and the convolutional integral is calculated in an analytical expression. Since there is no analytical expression for a term in the calculation of volume strain in the subsurface, the modified version of GOTIC2 uses numerical integration for this term only (Kamigaichi, 1998).
The program “loadgreenf4” is used for the Green’s function calculation at the deployment depth, and the eigenfunctions for the surface vertical load required for the Green’s function calculation are calculated by the program “static”. These programs are available from the Web site
https://mri-2.mri-jma.go.jp/owncloud/s/tjqx7HfK8bD3KQf.
2. The parameters for the calculation of solid earth tide are unified in PREM
Various parameters such as love number, which are used to calculate the displacement and strain caused by solid earth tides, were unified with the PREM as the spherically symmetric earth model, which was the basis for the Green's function used to calculate the ocean tidal loading effect. However, since solid earth tide is a very long-wavelength phenomenon, the effect of instrument installation depth is negligible, and therefore, the parameters are based on values at the surface. Specifically, the solid earth tides were calculated for the surface boundary condition of tidal forces n=2 in the program “static“.
3. Modification of the expression of three independent components of horizontal strain
The representation of the three independent components of horizontal strain is adapted to the horizontal strain representation commonly used in the calibration of the borehole strainmeter (e.g., Hart et al., 1996）and shown in εA=εxx+εyy, εD=εxx-εyy, and εS=2εxy, where x and y represent the east and north directions in the local coordinate system at the observatories.
Installation method of the program gotic2_mod
1. Download the packages of the original GOTIC2
The original GOTIC2 is property of the National Astronomical Observatory of Japan, and the terms of use of the original GOTIC2 is shown in “Terms of Use of the Website of NAOJ”
https://www.nao.ac.jp/en/policy.html.
Download the packages of original GOTIC from the following website.
https://www.miz.nao.ac.jp/staffs/nao99/index_En.html.
The required files are as follows:
2. Download the package of gotic2_mod
Click on the file below to download the package of gotic2_mod.
3. Installation of the original GOTIC2
Install the original GOTIC2 by following the steps below.
https://www.miz.nao.ac.jp/staffs/nao99/README_GOTIC2_En.html#install
However, you must skip only the following compilation process of the original GOTIC2:
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You will find source codes and Makefile in source directory.
Before compiling the program, modify the following lines in the
Makefile so that they are suitable to your computer platform;
FC = f77
FFLAGS =
(~/nao99b): cd gotic2/source
(~/nao99b/gotic2/source): make
Install and clean up after normal compilation.
(~/nao99b/gotic2/source): make clean
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4. Extraction of a diff file of the gotic2_mod (gotic2_mod_diff.taz)
Put gotic2_mod_diff.taz in "~/nao99b/gotic2/source", and then run the following procedure:
(~/nao99b/gotic2/source): tar xvzf gotic2_mod_diff.taz
(~/nao99b/gotic2/source): ./gotic2_mod_source.sh
5. Compilation of gotic2_mod
You can find the source code and Makefile in “~/nao99b/gotic2/source”. Before compiling, you must configure the following statements in the Makefile.
---
FC = gfortran
FFLAGS = -std=legacy
---
The compilation method is as follows.
(~/nao99b/gotic2/source): make gotic2_mod
Copy the executable file into one of the upper directories.
(~/nao99b/gotic2/source): cp gotic2_mod ..
After the compilation is successfully completed, do the following:
(~/nao99b/gotic2/source): make clean
6. Expansion of greens_functions.taz
Put ”greens_functions.taz” in "~/nao99b/gotic2/data" and run the following procedures:
(~/nao99b/gotic2/data): tar xvzf greens_functions.taz
~/nao99b/gotic2/data/greens_functions contains 149 files of the Green’s functions calculated at each depth as follows. The explanation of this file is described later.
---
loadgreenf4.out9.d0000_0.prem10k
~
loadgreenf4.out9.d1200_0.prem200k
---
7. Selection of the Green’s function at depth
Select the Green’s function whose depth is closest to the installation depth of the borehole strainmeter etc., and replace it with "~/nao99b/gotic2/data/grn1.data".
In this case, we select “loadgreenf4.out9.d0592_5.prem500k” (a Green’s function with a depth of 592.5 m) as a Green’s function to input “gotic2_mod”.
(~/nao99b/gotic2/data): mv grn1.data grn1.data.org
(~/nao99b/gotic2/data): ln -s greens_functions/loadgreenf4.out9.d0592_5.prem500k grn1.data
8. Operation check of gotic2_mod
Place “samples.taz” in “~/nao99b/gotic2” and expand it. “ftn05.gotic2_mod.icu” is shown after the expansion, which is a control file for calculating the M2 and O1 tidal strains at the ICU station (depth of the strain sensor is 588.5-589.9 m) in AIST. We use the Green's function at a depth of 592.5 m, which is the closest depth to the deployment depth.
(~/nao99b/gotic2):tar xvzf samples.taz
(~/nao99b/gotic2):./gotic2_mod < ftn05.gotic2_mod.icu > result.gotic2_mod.icu.log
To verify the result of the gotic2_mod, compare the output file result.gotic2_mod.icu from the above calculation with the result.gotic2_mod.icu.test, which is the result of gotic2_mod compiled with gfortran of CentOS 7.7
Example of an input file for the modified GOTIC2（gotic2_mod）
The contents of ftn05.gotic2_mod.icu stored in samples.taz are as follows:
*********************[ Mandatory Cards ]**********************
STAPOSD ICU1 , 136.1408 , 33.8967 , 0.0, 141.0
WAVE M2,O1
KIND ST
**********************[ Option Cards ]************************
GREENF 1
MESH4 ON
FULLMESH ON
UNIT6result.gotic2_mod.icu
END
The input file of “gotic2_mod” uses the same Mandatory and Option Cards as the original GOTIC2 (see （https://www.miz.nao.ac.jp/staffs/nao99/README_GOTIC2_En.html). However, the following point about the Option cards is different:
・GREENF CARD
Selects Green's function.
Format : GREENF Green's function number
(1 = PREM, 2 = 1066A model)
Example: GREENF 1
Default value : 1
---
In Modified GOTIC2, the Green's function for the Gutenberg-Bullen earth model (Farrell, 1972) cannot be used which you can select in the original GOTIC2.
Notes on the Mandatory and Option Cards of gotic2_mod are as follows:
The coastline data around Japan in the original and modified GOTIC2 is based on the Tokyo datum. When calculating oceanic tidal loading effects for stations around Japan, the coordinates must be given in the Tokyo datum.
In the Option Card, MESH4 and FULLMESH are always ON, and the 3rd meshes are automatically used for stations in Japan. Users must prepare 3rd mesh files and set MESH3 ON for stations outside Japan.
Reasons of MESH4 ON and FULLMESH ON in Modified GOTIC2
For physical quantities represented by the first-order derivative of displacements such as strains and tilts, it is necessary to ensure the accuracy of the calculation, especially in the vicinity of the observation point where the Green's function changes drastically, because the near-field contribution to the global convolutional integration is large (Kamigaichi et al., in review). In GOTIC2, the Green's function is approximated by a quadratic function in the mesh to evaluate the convolutional integration from each mesh. Therefore, it is necessary to set the mesh size so small that the behavior of the Green's function in the mesh can be represented by the quadratic function, especially in the very short range (if two peaks exist in the mesh, the quadratic approximation will fail). GOTIC2 has four different sizes of ocean load meshes depending on the distance from the observation point. In the case of the vicinity of Japan, the minimum mesh size is 2.25 seconds in the longitude and 1.5 seconds in the latitude (about 50 meters, or about 4 x 10-4 degrees angular distance). Therefore, it is necessary to select MESH4 ON to use 4th mesh
Furthermore, even if the area to which the 4th mesh is applied in GOTIC2, (by default, within 0.2°from the observation point), if the entire 3rd mesh size, consisting of 10 x 10 4th meshes, is the ocean, it is replaced by one 3rd mesh and the convolution calculation is performed. The use of large meshes in regions where the Green's function is highly variable causes the quadratic approximation of the internal Green's function to fail, so the FULLMESH ON option is needed to preserve the 4th mesh calculation in all ocean regions.
Description of the Green's function package calculated at different depths
When the file green_functions.taz is decompressed, the Green’s functions (149 files in total) are extracted with the depths of 0 m, 52.5 m, and 97.5 m and 7.5 m increments from 112.5 m to 1,200 m, which is a typical deployment depth for borehole strainmeters.
An example of file name is loadgreenf4.out9.d0592_5.prem500k, where "loadgreenf4.out9" is the output file of the program loadgreenf4, d???? _? is the Green’s function of depth ????.? m, and prem?k (or ?M) denotes that the earth model is PREM, and ninf (Kamigaichi, 1998; Kamigaichi et al., in review) is ??k or ?M.
Since the depth intervals of the pre-prepared Green's function are 7.5 m, the maximum difference from the actual installation depth of the strainmeter is 3.75 m. Considering that the length of the housing of the strainmeter is several meters, and that the Green's function does not make much difference when the difference in depth is several meters, we believe that the user's needs for calibration of the strainmeter are generally met by the Green's function prepared in this package. Users who need the Green’s functions for depths outside this range and for different earth models are encouraged to use “static”, a program for calculating the eigenfunction of the earth's static deformation, and “loadgreenf4”, a program for calculating the Green’s functions for surface vertical point loadings, which are released separately in https://mri-2.mri-jma.go.jp/owncloud/s/tjqx7HfK8bD3KQf.
References
Dziewonski, A. M. and Anderson, D. L. (1981) Preliminary reference Earth model Phys Earth Planet In, 25, 297–356.
Farrell, W. E. (1972) Deformation of the Earth by surface loads, Rev. Geophys., 10, 761–797.
Hart, R. H. G., Gladwin, M. T., Gwyther, R. L., Agnew, D. C. and Wyatt, F. K. (1996) Tidal calibration of borehole strain meters: Removing the effects of small-scale inhomogeneity, Journal of Geophysical Research, 101, 25553-25571.
Kamigaichi, O. (1998) Green Functions of the Earth at Borehole Sensor Installation Depth for Surface Point Load, Papers in Meteorology and Geophysics, 48, 89-100.
Kamigaichi, O., Matsumoto, N. and Hirose, F., Green’s function at depth of borehole observation required for precise estimation of the effect of ocean tidal loading near coasts, submitted to Geophysical Journal International.
Matsumoto, K., Sato, T., Takanezawa, T. and Ooe, M. (2001) GOTIC2: A Program for Computation of Oceanic Tidal Loading Effect, Journal of the Geodetic Society of Japan, 47, 243-248. Matsumoto, K. (2004), Readme of GOTIC2,
https://www.miz.nao.ac.jp/staffs/nao99/README_GOTIC2_En.html, accessed 13 October 2020.
Tsuruoka, H., Ohtake, M. and Sato, H. (1995) Statistical test of the tidal triggering of earthquakes: contribution of the ocean tide loading effect, Geophysical Journal International, 122, 183-194.