computer programs for calculating and optimization linear 1-d
TRANSCRIPT
APPROACHES TO UPGRADE THE ALGORITHM FOR STOCHASTIC SIMULATION OF GROUND MOTION
PROJECT 1
Computer Programs for Calculating and Optimization
Linear 1-D Multi-Layers Deposits System & Response Spectra
December, 2004 Report No. 555/070/04
By
Dr. Y Zaslavsky, N. Perelman, M. Mikenberg, M. Gorstein and
V. Avirav
Prepared for
The Steering Committee for
National Earthquake Preparedness and Mitigation
Seismology Division Tel: 972-8-9785853 Fax: 972-8-9255211 אגף סייסמולוגיהWWW.GII.CO.IL
2
TABLE OF CONTENTS Page No.
LIST OF FIGURES
LIST OF TABLE
1. ABSTRACT 5
2. INTRODUCTION 6
3. DESCRIPTION OF THE PROGRAM STRUCTURE 9
4. COMPUTATION OF ANALITICAL TRANSFER FUNCTION 11
4.1. Input data 11
4.2. Computation parameters 11
4.3. Computation algorithm steps 11
5. GRAPHIC USER INTERFACE OF THE PROGRAM 13
5.1. Display organization 13
5.2. Menus and commands 15
5.2.1. FILE Menu 15
5.2.2. MODEL Menu 15
5.2.3. EDIT Menu 19
5.2.4. TRANSFER FUNCTION (TF) Menu 21
5.2.5. RESPONSE SPECTRA (RSP) Menu 23
5.2.6. HELP/INFO Menu 27
5.2.7. DATA Menu 31
5.2.8. SCALE Menu 33
5.2.9. ZOOM Menu 34
5.2.10. Edit plot Menu 35
6. COMPUTATION OF S-WAVE VELOCITY PROFILE AND SEDIMENT
THICKNESS BY AMBIENT VIBRATION MEASUREMENTS 39
6.1. Algorithm of the optimization 40
6.2. Examples 41
7. DISCUSSION AND CONCLUSIONS 44
8. ACKNOWLEDGMENT 46
REFERENCES 47
APPENDIX A. The source text of the MATLAB procedure cmp_shake3.m
for the computation of Analytical Transfer Function. 48
APPENDIX B. The table of old and new sets of oscillators 52
APPENDIX C. 53
3
1. The site model Test1.mod for the computation of Analytical
Transfer Function by LERA program 53
2. The site model Test1.modJ for the computation of Analytical
Transfer Function by Joyner's program 53
APPENDIX D. Program of inversion of local S-wave velocity structures and sediment
thickness from average H/V ratios 54
List of Figures
Figure 1. The screen of the program LERA. The structure of display organization. ........ 14
Figure 2. The system pull-down menu "File". ................................................................... 15
Figure 3. The "MODEL" pull-down menu ........................................................................ 15
Figure 4. An example of implementation command "Open and load model (read only)". 16
Figure 5. An example of site model parameters frame. ..................................................... 16
Figure 6. The command "Converting model". ................................................................... 17
Figure 7. An example of implementation the "Save as and reloading" command ............ 18
Figure 8. The "Compute" command .................................................................................. 18
Figure 9. An example of implementation "View any model (without reloading)"
command. ........................................................................................................................... 19
Figure 10. The "EDIT" pull-down menu ........................................................................... 19
Figure 11. Example of implementation the "Edit" command ............................................ 20
Figure 12. Example of the implementation the "Delete" command . ................................ 20
Figure 13. The pull-down menu "TRANSFER FUNCTION (TF)". ................................. 21
Figure 14. Example of the implementation "Drawing one TF for this model"
subcommand. ..................................................................................................................... 21
Figure 15. Selection several transfer functions from the list for drawing. ........................ 22
Figure 16. Example of the implementation "Drawing few Transfer Functions"
subcommand ...................................................................................................................... 22
Figure 17. The command "View ASCII file". ................................................................... 23
Figure 18. Example of the implementation subcommand "View TF of this model"......... 23
Figure 19. The pull-down menu "RESPONSE SPECTRA (RSP)". .................................. 24
Figure 20. An example of the implementation of "Drawing one RSP for this model"
subcommand. ..................................................................................................................... 24
Figure 21. Example of the implementation of subcommand "Drawing normalize TF and
RSP for this model" ........................................................................................................... 25
Figure 22. Example of the selecting from list for drawing several Response Spectra. ..... 25
Figure 23. Example of implementation of the "Drawing few Response Spectra"
subcommand. ..................................................................................................................... 26
Figure 24. The "View ASCII file" command .................................................................... 26
Figure 25. Example of implementation of "View RSP of this model" subcommand ........ 27
Figure 26. The pull-down menu "Help/Info". .................................................................... 27
Figure 27. Example of implementation of the "About this program" command. .............. 28
Figure 28. Example of implementation "Converter coordinates of Israel" command ....... 28
Figure 29. The HELP-information on the application The "Converter coordinates of
Israel". ................................................................................................................................ 29
Figure 30. Example of implementation of the "Computation IS413 for PGA of site"
Israel" command (input data). ............................................................................................ 30
4
Figure 31. Sample output file for the "Computation IS413 for PGA of site" command
(ASCII file) ........................................................................................................................ 30
Figure 32. Sample of result "Computation IS413 for PGA of site" (graph ). ................... 31
Figure 33. The pull-down menu "DATA". ........................................................................ 31
Figure 34. Example of implementation the "AXES SETTING" subcommand ................. 32
Figure 35. Example of subcommand "Window SETTING" implementation ................... 33
Figure 36. Example of implementation of "SAVE GRAPH for Graph Editor" command 33
Figure 37. The pull-down menu "Scale". ........................................................................... 34
Figure 38. The pull-down menu "Zoom". .......................................................................... 34
Figure 39. The pull-down menu "Edit plot". ..................................................................... 35
Figure 40. Example of the implementation "LABEL - TITLE " command. ..................... 37
Figure 41. Example of the implementation " SET LIMITS X Y " command. .................. 37
Figure 42. Example of the implementation " ADD TEXT " command. ............................ 38
Figure 43. Example of the implementation "POSITION PLOT " command. ................... 38
Figure 44. Example of H/V spectral ratio .......................................................................... 39
Figure 45. Comparison of the experimental and optimal analytical transfer function for
Model 1. ............................................................................................................................. 41
Figure 46. Comparison of experimental and optimal transfer function for Model 2. ........ 42
Figure 47. Comparison of experimental and optimal transfer functions for Model 3. ...... 43
Figure 48. Comparison experimental and optimal transfer function for Model 4 ............. 44
Figure 49. Comparison of the transfer functions computed using the SHAKE and Joyner's
programs for the Test1 site model ..................................................................................... 45
Figure 50. Comparison of the site-specific uniform hazard acceleration spectra calculated
for the Test1 site model. Response functions for this model are computed using the
SHAKE program (black line) and Joyner's program (blue line). The sets of oscillators for
calculation of the response spectra are 64 and 31 for SHAKE and Joyner's program
correspondingly. ................................................................................................................. 45
LIST OF TABLES
Table 1. The list of executable (external) modules of the program LERA
5
1. ABSTRACT
The Stochastic Evaluation of Earthquake Hazard (SEEH) procedure of Shapira
and van Eck (1993) requires the analytical response function to assess seismic hazard in
terms of Uniform Hazard Site-Specific Acceleration Spectra. In order to improve the
procedure, the two programs are created:
1. The Linear Earthquake site Response Analysis (LERA) based on the SHAKE
algorithm for estimation of the analytical transfer function of the 1-D multi-
layer soil-rock system taking into account density, S velocities, shear modulus
and attenuation parameters for each layer; and
2. a program for inversion of local S-wave velocity structures and sediment
thickness from average H/V ratios
6
2. INTRODUCTION
Development of resonant phenomena in sedimentary sequence, usually referred to
as "site response", is one of the important effects controlling spatial variations in seismic
intensity. Appearance of the resonant effect is in an amplification of ground motions,
which can be as large as a factor ten relative to the rock sites at different frequencies
between 0.3 to 15 Hz. This is particularly important since most of urban settlements are
built over soft surface deposits. Most of studies comparing various techniques for
estimating site response have been based on real data (from earthquakes, explosions, and
ambient vibrations). It should be mentioned that site response function determination is an
important stage in the overall process of seismic hazard assessment despite the fact that
the function itself has no direct engineering application. In order to estimate the ability of
buildings at a certain site to withstand seismic activity, one needs to obtain the site
specific acceleration spectrum. This design acceleration spectrum is essentially a
representation of the maximum acceleration amplitudes for a prescribed probability of
occurrence, developed on a family of one degree of freedom oscillators with a given
damping ratio.
A simple and powerful method for simulating ground motion is based on the
assumption that the amplitude of ground motion at a site is regarded as deterministic,
while the phase spectrum is random and modified so that the motion is distributed over a
duration related to the earthquake magnitude and the source distance. This method of
simulating ground motion often goes by the name "the stochastic method" (Boor, 1983;
Boor, 1984; Boor, 2000; Boor et al., 1997; Atkinson and Silva, 2000). A stochastic
procedure to synthesize the uniform-hazard site specific response spectrum, termed here
as SEEH (Stochastic Estimation of the Seismic Hazard), has been developed by Shapira
and van Eck (1993). This method uses the Monte Carlo process for simulating seismicity
in seismic areas neighboring the investigated site and quantifies the source parameters of
the earthquakes in the simulated categories. Stochastic simulations are then applied to
each of the listed earthquakes to generate the synthetic free surface S-wave ground
accelerations on hardrock at distances at which the near field effects can be reasonably
neglected.
The time history of ground accelerations is simulated for different earthquakes
which are characterized by their magnitude, stress drop, distances and etc. Where the site
is a sedimentary basin, the synthetic S-wave acceleration is convolved with an analytical
7
model of site. The synthetic free surface S-wave accelerations are used to compute the
response spectra for a given damping.
The methodology of preparing an earthquake damage scenario for Israel developed
by Shapira et al., (2001) is based on the concept that vibratory motions, especially those
leading to resonance motion of building, are the main cause of extensive damage and
destruction. Consequently, the method is based on comparison of the design acceleration
response spectrum of a building with a predicted site specific acceleration response
spectrum from prescribed earthquake.
In an earthquake damage scenario for Israel, the linear analytical response functions
at different sites are computed using the FORTRAN program for calculating nonlinear
seismic ground response (Joyner, 1977). The program was designed for calculating the
nonlinear response of a system of horizontal soil layers underlain by a semi-infinite elastic
medium representing rock (half space). Excitation is a vertically incident shear wave in
the underlying medium. The non-linear hysteretic behavior of the soil is represented by
the model proposed by Iwan (1967) and provides the attributed attenuation of seismic
motion with time and distance. With small vibrations the non-linear effect is negligible,
thus no attenuation is obtained using this model.
During computation the Joyner's program generates a numerical high- frequency
noise. This noise is generally not noticeable in the surface particle velocity time history
and has little effect on response spectral values, but it is conspicuous in the surface
acceleration time history. In general it is preferable to filter out this noise and the program
provides an option for digital filtering output time histories. This filter has two parameters
– F1 and F2. It is recommended that F1 be assigned equal to the desired frequency
resolution fR and that F2 be assigned the value 2fR.
We use different values of F1 and F2 in order to provide quasi-equivalent linear
damping in the horizontally layered soil-bedrock system. The choice of F1 and F2 is
based on the fact that during an earthquake amplifications at resonance frequencies of
higher modes (first 3-4 modes) are possible, but such a choice it is not physically justified
since we are not taking into account available specifications of damping for different
lithological units.
The severity of vibratory structural response to seismic motion largely depends on
the seismic ground motion characteristics and the structure's dynamic characteristics.
Some of the important ground motion characteristics are peak motion parameters, such as
acceleration, and the frequency content of the ground motion. The ground motion
8
frequency content can be generally described as a measure of relative predominance of
different frequencies present in the ground motion. Spectrum shapes are one the measures
of ground motion frequency content and are important in estimating seismic structural
response.
A response spectrum is defined as a plot of the maximum values of a response
parameter of a family of linearly elastic single-degree-of-freedom system with different
frequency characteristics and with a given ratio of system damping when subjected to a
ground motion time history versus the frequency characteristics of the system.
It should be emphasized that the first version of the SEEH procedure was relatively
high in computer time consumption. The response spectrum was, therefore, computed for
a family only of thirty one single-degree-of-freedom system (oscillators), which is
insufficient for correct presentation of spectrum shapes.
As previously mentioned, the influence of geological conditions can be calculated
using different analytical models of multi-layered media. The approach involves
knowledge of the depth to the reflector and its shear-wave velocity (Vs), spatial
distribution of softer materials above the reflector with corresponding a Vs for each layer.
Many geophysical tools may be used to obtain these parameters, but their employment in
urban areas is difficult and expensive.
Nakamura (1989, 2000) proposed and revised a method by which the effect of
source function might be minimized by normalizing the horizontal spectral amplitude in
terms of the vertical one. Assuming that the S-waves dominate in microtremors, he
indicated that the horizontal-to-vertical spectral ratio of microtremors at a site equals the
S-wave transfer function between ground surface and bedrock at the site. We analyzed the
relationship between the main resonance frequency of the soil, its thickness and its shear
velocity, based on the transfer function obtained from the H/V spectral ratios of
microtremors (Zaslavsky et al., 2004). The S-wave structures for different sediments were
deduced by trial-and-error fitting of theoretical transfer functions to those observed at
drilling sites. In turn, fixing the S-wave velocities, it became possible to adjust the
thickness of sedimentary layers to match analytical response functions to experimental
spectral ratios for sites where borehole data is not available. The manual technique in
order to receive the best-fitting S-wave velocities or depth to reflector is sufficiently
laborious process.
The aims of the present study are:
9
to upgrade and improve the site response analysis procedure designed in a PC-
Windows environment;
to create an user-friendly graphic interface for calculating and drawing the
analytical transfer function (ATF) using the MATLAB package;
to apply the SHAKE algorithm to compute the ATF for a multi-layers system
and convolve with accelerograms synthesized in the SEEH program for
calculation of the Site-specific Uniform Hazard Acceleration Spectra;
to extend the available set of single-degree-of-freedom oscillators for more
accurate calculation of the Site-specific Uniform Hazard Acceleration Spectra;
to include the calculation of Israel Building Code (IS 413) for a given site
to find automatically those parameters of the 1-D analytical model which provide
the best approximation of the function obtained experimentally.
3. DESCRIPTION OF THE PROGRAM STRUCTURE
The LERA program is a complex multi-modules software package that includes one
main executable module (MATLAB stand-alone application) for creating Graphic User
Interface (GUI) and calls the others external modules. These modules were written on
MATLAB (Ver. 5.3) and FORTRAN 90 programming languages. The list of executable
(external) modules of the program LERA is given in Table 1.
10
Table 1 The list of executable (external) modules of the program LERA
No
Name
Type of
Module
Function
Language
1
CSEEHTF3
Main
Creating GUI, computation ATF using
SHAKE program and call the others
modules
MATLAB
2
TSTSITE3
Slave
Computation ATF using Joyner's
program FORTRAN
3
DSSRS6E
Slave
Computation RSP by Joyner
FORTRAN
4
FSEEHTF1
Slave Computation RSP by SHAKE
FORTRAN
5
CNVCRD2A
Slave
Converter coordinates of Israel
MATLAB
6
CSIS413A
Slave
Computation of spectrum according to
the Israel Building Code (IS413) for
different sites
MATLAB
Note
GUI - Graphic User Interface; ATF - Analytical Transfer Functions;RSP – Response
Spectra
For running MATLAB stand-alone application is required also set of MATLAB dynamic
link libraries (DLL), whose list is given below:
sgl.dll
hg_sgl.dll
uiw_sgl.dll
gui_sgl.dll
hardcopy_sgl.dll
mpath.dll
libmmfile.dll
libmat.dll
libmcc.dll
libmatlb.dll
libmx.dll
libut.dll
It is also necessary to create a subdirectory, named \bin, that contains the MATLAB menu
bar and toolbar figure files used by the MATLAB stand-alone application.
The program LERA may operate under Microsoft Windows (from 95 to XP versions).
11
4. COMPUTATION OF THE ANALITICAL TRANSFER FUNCTION
4.1. Input data
The input data of this program contain soil data of the investigated site.
These soil data are following five parameters for every layer of the investigated site:
1. Thickness (h), [m];
2. Shear velocity (Vs), [m/s];
3. Density (ρ), [gr/cm^3];
4. Attenuation value (β), (< 1.0).
5. "Number of layers " (Nlayer), including half-space (bedrock).
Input parameters are contained in the file, whose name is specified interactively.
4.2. Computation Parameters
The analytical transfer function will be calculate for defined interval of frequency
([F1 F2], [Hz]), which is divided on Nfreq points.
(the parameters assigned for the computation are: F1 = 0 , F2 = 50, Nfreq = 2048).
4.3. Computation Algorithm Steps
1. Calculation of Real shear module (G) for all layers including half-space:
where ρj and Vsj are accordingly density and shear velocity from Input data,
respectively and j is the number of layers (from 1 Nlayer ).
Note that all these values are independent of frequency.
2. Calculation of the following complex parameters for the first of the (Nlayer - 1)
layers, (a half-space is not included):
a) Complex Shear module ( G* ):
b) Complex Impedance Ratio at the interface between j and j+1 layers ( ά*
j):
where G is Real shear module (see Eq. (1)),
β is Attenuation value ( from Input data ) , and i is imaginary unity.
2
sjjj VG (1)
iGG jjj 21* (2)
*
11
*
*
jj
jj
jG
G
(3)
12
12 lfl
c) Temporary Complex variable (γ) for following calculates:
where ρj is Density,
Gj* is Complex Shear module;
j is number of layer (from 1 to Nlayers).
3. Calculation of transfer function A1Nlayers with respect to the displacement (and also
velocity and acceleration) at the top of layers 1 and Nlayers. This transfer function
depends on frequency and will be calculated with frequency step Δf:
where F1 and F2 are lower and upper limits of defined interval of frequency
respectively, and Nfreq is number of points (steps of computation) of this interval.
3.1. Begin external loop on points (steps of computation):
l = 1: Nfreq.
Calculate round frequency (ω)
Δf is step of frequency (see (5)),
l is index of point of interval frequency
3.2. Begin internal loop on layers (without halfspace):
j = 1: ( Nlayers - 1)
Calculate Complex Wave Number (K):
where ωl is round frequency,
γ is temporary Complex variable (see (4)) .
Calculate is temporary Complex variable (δ):
where Kl,j is Complex Wave Number,
hj is thickness of layer j from input data.
*/ jjj G
freqN
FFf 12
(5)
jljlK ,
(6)
(7)
jjl hiK , (8)
(4)
13
Calculate amplitudes direct UP and DOWN (accordingly E and F) by recursive
algorithm:
This recursive algorithm is started at the top of the free surface, for which there is
no shear stress, which implies
E1 = F1
Equations (9) and (10) are the applied successfully to layers 2 to j.
End of internal loop (on layers).
Calculate the transfer function (A1N):
End of external loop (n frequency).
The source text of MATLAB procedure cmp_shake3.m for computing the
analytical transfer function as described in this algorithm is given in Appendix B.
5. GRAPHIC USER INTERFACE OF THE PROGRAM
5.1. Display organization
The screen of the LERA program consists of the following elements:
* Title Bar
* Menu Bar
* System Tool Bar
* Graph Display Area
For the structure of display organization see Figure 1.
eFeEE jjljjljl 115.0 ,,1,
eFeEF jjljjljl 115.0 ,,1,
(9)
(10)
NkNkNk
kk
NkEFE
FEA
,,,
1,1,
1,
5.0
1)1(
14
Figure 1. The screen of the program LERA. The structure of display organization.
Title Bar
The Title Bar at the top left hand corner indicates the name of the site model
currently being processed. The standard Windows control buttons for minimizing or
maximizing the display and exiting the program are located at the top right hand corner.
Menu Bar
The Menu Bar is located directly below the Title Bar. All pull-down menus are
located inside a Menu Bar. Each menu contains a list of commands that one can select
with the mouse or keyboard. The arrangement of the menus, designed with ergonomic
criteria, follows the logical order of the operations, inhibiting access to further operations
until all necessary data have been entered.
System Tool Bar
The System Tool Bar is located directly below the Menu Bar. The toolbar includes
several buttons which perform the exact same function as functions in the system pull-
down menu File. Owing to possible system problems, it it is preferable to use only the
Print button.
15
Graph Display Area
This is the large center section of the display where graphs are displayed and
manipulated: see below "Edit Plot" Menu section for more details.
5.2. Menus and Commands
5.2.1. FILE Menu
The system pull-down menu, above in this manual menu, "FILE" contains several
commands, but user can use only the Print command to produce a hardcopy for default
printing. The system pull-down menu "FILE" is shown in Figure 2.
Figure 2. The system pull-down menu "File".
5.2.2. MODEL Menu
The pull-down menu "MODEL" contains commands for opening and saving
models, converting, drawing a plot of a site model, computation and exit from the
program. Until a model has been opened, the program will not permit the use of other
menus, except the "Help/Info" menu. The pull-down menu "MODEL" is shown in Figure
3.
Figure 3. The "MODEL" pull-down menu
Command "Open and load model (read only)"
The command " is used to open the file of a site model. To be able to work with
the program and create a new model, one should, first of all, open a pre-existing file using
16
" Open and load model (read only)" and save it with a new filename (command "Save as
and reloading" from this menu). Then user can then edit the contents of the file (the model
parameters) using the pull-down menu "Edit". To perform this command, the user must
call up the Windows "Open" Dialog Box. The name of the file being used is shown inside
the title bar of the main window. An example of the execution the command "Open and
load model (read only)" is shown in Figure 4.
Figure 4. An example of implementation command "Open and load model (read only)".
The example of appearance of the selected site model after opening is shown in Figure 5.
Figure 5. An example of site model parameters frame.
17
Command "Converting model"
The "Converting model" command is used to convert a SHAKE site model to a
Joyner's site model and back. The command "Converting model" is shown on Figure 6.
Figure 6. The command "Converting model".
Command "Draw"
The "Draw" command is used to draw the current site model. The appearance of
the graph may be edited using the "EDIT PLOT" menu(see below). The graph may be
printed using the "FILE" menu, "Print" command (see above) and saved as a graphical file
by the "DATA" menu, command "Save graph for graph. editors" (see below). An example
of implementation command "Draw" is shown in Figure GUI1.
Command "Save as and reloading"
The "Save as and reloading" command is used to save the current site model with
a new name, as shown in Figure 7. The name of the file should not contain spaces and/or
punctuation marks and its length must be less than or equal to 8 characters. It is not
necessary to indicate extensions, since the extension “.mod” is added automatically: for
example, if one enters the name “TEST1” the file name will be “TEST1.mod". An
example of implementation command "Save as and reloading" is shown in Figure 7.
18
Figure 7. An example of implementation the "Save as and reloading" command
"Compute" Command
The "Compute" command is used to compute the Transfer function (TF) and
Response Spectra (RSP) for the site model opened or for several models. The "Compute"
command is shown in Figure 8.
Figure 8. The "Compute" command
"View any model (without reloading)" command
The "View any model (without reloading)" command is used only for viewing any
existing site model without its reloading into the program. First, the site model already
opened does not change. An example of the implementation command "View any model
(without reloading)" is shown in Figure 9. the User has to select only one site model from
appearance list of all existent site models for viewing.
19
Figure 9. An example of implementation "View any model (without reloading)"
command.
"Exit" Command
The command "Exit" is used to quit the program, when user has finished his work.
"Help" Command
The command "Help" is used to review help information about this menu.
5.2.3. EDIT Menu
The pull-down menu "EDIT" contains the commands for editing and deleting site
models, was opened before in the command "Open and load model (read only)" of the
MODEL Menu. The pull-down "EDIT" menu is shown in Figure 10.
Figure 10. The "EDIT" pull-down menu
20
"Edit" Command
With this command it is possible to edit the file of the opened site model by
system text editor Notepad. An example of the implementation command "Edit" is shown
in Figure 11. After finishing of work with editor user has to close it, otherwise the other
commands of the main menu will be disabled.
Figure 11. Example of implementation the "Edit" command
"Delete" Command
The "Delete" command is used to delete the opened site model file. After selecting
this command, the user will be asked to confirm his decision by clicking the "OK" button
or rejecting it by clicking "Cancel" button. An example of the implementation "Delete"
command is shown in Figure 12.
Figure 12. Example of the implementation the "Delete" command .
21
5.2.4. TRANSFER FUNCTION (TF) Menu
The pull-down menu "TRANSFER FUNCTION (TF)" contains the commands for
drawing and viewing transfer functions of the site models. To compute transfer function
see the "Compute" command on the "MODEL" menu. The pull-down "TRANSFER
FUNCTION (TF)" menu is shown in Figure 13.
Figure 13. The pull-down menu "TRANSFER FUNCTION (TF)".
"Drawing" Command
The "Drawing" command is used to draw existing transfer functions. The first
subcommand "Drawing one TF for this model" is used to drawing transfer tunction for the
opened site model. An example of implementation of this subcommand is shown in
Figure 14.
The subcommand "Drawing few TF" is used to draw several transfer functions on
one plot. The transfer function to be drawn must be selected from the full list of existing
transfer functions. An example of selection from a list of two transfer functions is shown
in Figure 15. An example of the implementation of this subcommand is shown in Figure
16.
Figure 14. Example of the implementation "Drawing one TF for this model"
subcommand.
22
Figure 15. Selection several transfer functions from the list for drawing.
Figure 16. Example of the implementation "Drawing few Transfer Functions"
subcommand
23
"View ASCII file" Command
The "View ASCII file" command is used to view ASCII files of the current
transfer functions. The "View ASCII file" command is shown in Figure 17. This
command contains two subcommands. In this manual we demonstrate only the first
subcommand "View TF of this model", since the second subcommand ("View any TF") is
very similar. An example of the implementation this subcommand is shown in Figure 18.
Figure 17. The command "View ASCII file".
Figure 18. Example of the implementation subcommand "View TF of this model".
5.2.5. RESPONSE SPECTRA (RSP) Menu
The pull-down "RESPONSE SPECTRA (RSP)" menu contains the commands for
drawing and viewing existing Response Spectra of site models. To compute Response
Spectra see above menu "MODEL", command "Compute". The pull-down menu
"RESPONSE SPECTRA (RSP)" is shown in Figure 19.
24
Figure 19. The pull-down menu "RESPONSE SPECTRA (RSP)".
"Drawing" Command
The "Drawing" command is used to plot available Response Spectra . The
subcommand "Drawing one RSP for this model" is used to draw Response Spectra for the
opened site model. An example of implementation of this subcommand is shown in
Figure 20.
The subcommand "Drawing normalized TF and RSP for this model" is used to
draw normalized transfer functions and response spectra for the site model in a one plot.
An example of implementation of this subcommand is shown in Figure 21. The
subcommand "Drawing few RSP" is used to drawing several response spectra in one plot.
The response spectra for drawing must be selected from the full list of available response
spectra. An example of selection from the list for drawing several response spectra is
shown in Figure 22. An example of the implementation of this subcommand is shown in
Figure 23.
Figure 20. An example of the implementation of "Drawing one RSP for this model"
subcommand.
25
Figure 21. Example of the implementation of subcommand "Drawing normalize TF and
RSP for this model"
Figure 22. Example of the selecting from list for drawing several Response Spectra.
26
Figure 23. Example of implementation of the "Drawing few Response Spectra"
subcommand.
"View ASCII file" Command
The "View ASCII file" command is used to view ASCII files of the existign
Response Spectra. The command "View ASCII file" is shown in Figure 24. This
command contains two subcommands:. "View RSP of this model" and "View RSP of any
model". The use of these subcommands is the same as the equivalent command for TF.
An example of the implementation this subcommand is shown in Figure 25.
Figure 24. The "View ASCII file" command
27
Figure 25. Example of implementation of "View RSP of this model" subcommand
5.2.6. HELP/INFO Menu
The pull-down menu "Help/Info" contains the commands for help and some
additional computations. The pull-down menu "Help/Info" is shown in Figure 26.
Figure 26. The pull-down menu "Help/Info".
"About this program" Command
With the command "About this program" one can see information on this
program. An example of the implementation this command is shown in Figure 27.
28
Figure 27. Example of implementation of the "About this program" command.
The "Converter coordinates of Israel" command.
The "Converter coordinates of Israel" command is used to converted (X, Y)
coordinates into (Latitude, Longitude) coordinates and vice versa. The geographical data
were taken from the Survey of Israel (Israel 1928, Cassini-Soldner Grid conversion). The
"Converter coordinates of Israel" is an external WINDOWS application, which is called
from program LERA by command named also "Converter coordinates of Israel". An
example of the implementation command "Converter coordinates of Israel" is shown in
Figure 28.
Figure 28. Example of implementation "Converter coordinates of Israel" command
29
The HELP-information on the application The "Converter coordinates of Israel" is shown
in Figure 29.
Figure 29. The HELP-information on the application The "Converter coordinates of
Israel".
"Computation IS413 for PGA of site" Command
The command "Computation IS413 for PGA of site" is used to compute the Israeli
Standard (IS-413) for PGA corresponding to a given site in Israel. The Israeli Building
Code 413 or Israeli Standard refers to the horizontal Peak Ground Acceleration (PGA) as
a representing the seismic hazard parameter and uses the mapped PGA to rescale the
standard response spectrum. The input data for this computation are two parameters the
site name and the PGA value for this site. An example of the implementation
"Computation IS413 for PGA of site" Israel" command (input data) is shown in Figure
30. The result of this computation is four curves ( S1 , S2 , S3 and S4) for different kinds
of soil ( four classes : from rock to very soft soil) as an ASCII file and graph. An
example of results from "Computation IS413 for PGA of site" is shown in Figure 31
(ASCII file ) and Figure 32 (graph).
30
Figure 30. Example of implementation of the "Computation IS413 for PGA of site"
Israel" command (input data).
Figure 31. Sample output file for the "Computation IS413 for PGA of site" command
(ASCII file)
31
Figure 32. Sample of result "Computation IS413 for PGA of site" (graph ).
5.2.7. DATA Menu
The "Help/Info" pull-down menu contains the commands to control data graphical
image screen for review and save. The pull-down menu "DATA" is shown in Figure 33.
Figure 33. The pull-down menu "DATA".
"Settings" Command
The command "Settings" is used to review and editing initial parameters (settings)
of graph. These parameters are saved in the special profile-file and relevant until next
editing. This command includes two following subcommands:
Axes settings
The user must define all parameters of axes position (origin, width and height) in
normalized units (0 -1). An example of the implementation subcommand "AXES
SETTING" is shown in Figure 34.
Window Setting
This subcommand is used to define location and size of window (graph) for
hardcopy. The user has to define all parameters of window position (origin, width and
32
height) in pixels units. An example of the implementation subcommand "Window
SETTING" is shown in Figure 35.
"SAVE GRAPH for Graph Editor" Command
This command saves an image of the current plot as BITMAP (*.bmp) and
MATLAB (*.fig) files on hard disk in special folder PLOT. The user selects only the
filename, extension .bmp and .fig will be added automatically. After saving, these
graphical files the data may be processed by any graphic editors for WINDOWS and
MATLAB. An example of the implementation command "SAVE GRAPH for Graph
Editor" is shown in Fig. 36.
Figure 34. Example of implementation the "AXES SETTING" subcommand
33
Figure 35. Example of subcommand "Window SETTING" implementation
Figure 36. Example of implementation of "SAVE GRAPH for Graph Editor" command
5.2.8. SCALE Menu
The "Scale" pull-down menu is used to toggle the scale of the current plot between
logarithms and linear. The pull-down menu "Scale" is shown in Figure 37.
34
Figure 37. The pull-down menu "Scale".
This pull-down menu contains four very simple options (commands):
LINEAR X; LINEAR Y
This option is used to toggle the scale of current plot to linear mode for all
respective- X (abscissa) and Y (ordinate).
LOG X; LINEAR Y.
This option is used to toggle the scale of current plot to logarithmic mode for
respective axis - X (abscissa) and to linear mode for respective axis Y (ordinate).
LINERAR X; LOG Y.
This option is used to toggle the scale of current plot to linear mode for respective
axis - X (abscissa) and to logarithmic mode for respective axis Y (ordinate).
LOG X; LOG Y.
This option is used to toggle the scale of current plot to logarithmic mode for all
respective axes - X (abscissa) and Y (ordinate).
5.2.9. ZOOM Menu
The pull-down menu "Zoom" is used to display and zoom the graph (plot). The
pull-down menu "Zoom" is shown in Figure 38.
Figure 38. The pull-down menu "Zoom".
This pull-down menu contains three simple options (commands) and is organized
as follows:
ZOOM ON
This option is used to activate Zoom mode for Mouse. To zoom, use the left
mouse button to draw a rectangle around the area of interest and release. Zooming
35
may be repeated as necessary, allowing more and more detail. To stop zooming,
press the right mouse button.
ZOOM OFF
This option deactivates Zoom mode for Mouse, and it is needed for using of the
context menu (press right button of Mouse) on object of plot - line and text.
ZOOM OUT
This option returns the plot to its initial size. This is also needed to use the context
menu.
5.2.10. EDIT PLOT Menu
The pull-down EDIT PLOT menu allows editing the current plot usually before
printing. The pull-down menu "Edit plot" is shown in Figure 39.
Figure 39. The pull-down menu "Edit plot".
This pull-down menu is organized as follows options (commands):
LABEL - TITLE
This option is used for editing X, Y axes labels and the title of the current plot in
the interactive mode. An example of the implementation this command is shown
in Figure EP2.
LEGEND of PLOT
This option puts a legend on the current plot
HIDE LEGEND
This option removes a legend from the current plot.
GRID ON
This option adds grid lines to the current axes of the plot. Hot Key : <Ctrl> + <G>.
GRID OFF
This option takes them off.
SET LIMITS X Y
36
This option edits limits for axes X and Y of the current plot. An example of the
implementation this command is shown in Figure EP3.
ADD TEXT
This option allows adding text to the current plot. User determines coordinates of
the text-box origin. An example of the implementation this command is shown in
Figure EP4.
POSITION
This option defines the location and size of axes of the current plot within the
figure. User defines all parameters of the plot position within. Example of the
implementation this command is shown in Figure EP5.
FREQUENCY DOMAIN
This option is used for setting Transfer Function or Response Spectra on the
current plot into Frequency Domain [Hz].
PERIOD DOMAIN
This option is used for setting Transfer Function or Response Spectra on the
current plot into Period Domain [s].
(where <Period > = 1/<Frequency >)
AXIS SQUARE
This option is used to makes the current axis box square in size.
AXIS NORMAL
This option is used to restores the current axis box to itsoriginal size.
This undoes the effects of AXIS SQUARE.
SEEKING POINTS OF EXTREMUM
This option is used to find peaks (MIN & MAX) for the current plot.
The coordinates of extremum found are written into ASCII file <extrmpnt.txt> and
shown on the current plot by following symbols :
for MIN - triangle (down) ;
for MAX - triangle (up).
These symbols may be edited using context menu. To show context menu press
right mouse button.
37
Figure 40. Example of the implementation "LABEL - TITLE " command.
Figure 41. Example of the implementation " SET LIMITS X Y " command.
38
Figure 42. Example of the implementation " ADD TEXT " command.
1.
Figure 43. Example of the implementation "POSITION PLOT " command.
39
6. COMPUTATION OF LOCAL S-WAVE VELOCITY STRUCTURES
AND SEDIMENT THICKNESS USING AVERAGE H/V SPECTRAL
RATIOS
The H/V spectral ratio of the ambient vibration technique provides estimates of the
amplification level at the dominant frequency and has no resolution power to reveal the
higher modes of resonance. We therefore use H/V measurements together with available
geological and geophysical information to construct a subsurface model for the
investigated region. This model may, in turn, be used for estimating the expected site
effects during earthquakes. In order to construct an analytical mode,l detailed knowledge
of the site conditions in terms of S-wave velocity, thickness, density and damping for
every layer is essential. Within the range of plausible S velocities and layer thicknesses
we select those that yield a good fit of the dominant (resonance) frequency and its
amplitude level between the observed H/V spectral ratios and the calculated response
functions. Similarity of shape of analytical and experimental transfer functions is also
well.
We use the SHAKE (Schnabel, 1972) program to calculate the transfer function.
The algorithm for the optimization of model parameters, program and some examples to
illustrate the results we present below. An example of H/V spectral ratio curve obtained
experimentally is shown on the Fig.44.
Figure 44. Example of H/V spectral ratio
40
6.1. Algorithm of optimization
In the SHAKE program a responses is considered to be associated with vertical
propagation of shear waves through the linear viscoelastic system. The system consists of
n horizontal layers which extend to infinity in the horizontal direction and has a halfspace
as the bottom layer (SHAKE [1972]). Each layer is homogeneous and isotropic and is
characterized by the thickness, density, velocity and damping factor. In SHAKE model is
calculated transfer function
)(AA , (1)
where is a frequency and A is an amplification.
The transfer function, as is generally known, is characterized by two parameters:
the fundamental frequency and amplification. Since a problem of approximation of the
experimental transfer function by the analytical one should have a single solution, we can
optimize only one or two parameters (in our case these parameters are velocity or
thickness of layer). The other parameters such as density and damping must be fixed.
Densities of layers are computed according to the relation between density and velocity.
Damping values for different layers are taken from different literature sources.
Stages of the optimization algorithm
Stage 1
We find a frequency and amplification A of the experimental spectral ratio curve and
choose small numbers and , which are an error estimation of the fundamental
frequency and amplification respectively. Values of and depend on convergence of
the optimization process. For the Initial value we usually assign 0.05 Hz (two samples);
and is 20-25% of amplification observed at corresponding fundamental frequency.
Stage 2
We choose intervals of variation of an unknown parameters and grid of points on them. It
is important to maintain two conditions in selecting these intervals: true values of
parameters in question should belong to these intervals and length of intervals should be
sufficiently small.
Stage 3
For every point of the grid we calculate a transfer function using the SHAKE analytical
model.
41
Stage 4
From transfer functions calculated we select those, fundamental frequency and
amplification pertaining to the interval ),( and ).,( AA
correspondingly. If such transfer functions are not found, we have to widen and
values, or increase the grid density at the Stage 2.
Stage 5
We choose an interval I including fundamental frequency (usually its length is about
0.5 Hz). We calculate in this interval I even norm of transfer functions selected at the
Stage 4 and then choose a transfer function, for which even norm is minimal.
Programs in FORTRAN and MATLAB were written for use with this algorithm.
6.2 Examples
Model 1. In the first example we will consider Model 1 consisting of two layers overlying
the bedrock (halfspace). The following parameters are known:
densities of layers are 4.23;8.12;5.11 , g/cm3
damping factors of layers are 03;02.02;03.01 ;
S-wave velocities of layers are 20003;3502;1701 vvv , m/sec
Index 3 corresponds to the halfspace.
The parameters to be optimized are the thicknesses of the first and second layers (h1 and
h2). We look for the thickness of the first layer in the interval [0, 10 m] and of the second
in the interval [0, 55m]. Optimal thicknesses found using the optimization procedure are
6m and 20m for first and second layer correspondingly (Fig. 45)
Figure 45. Comparison of the experimental and optimal analytical transfer function for
Model 1.
42
Model 2
This is another example of two layers model for which we know following parameters:
4.23,1.22,5.11 ;
03,02.02,03.01 ;
smvsmvsmv /23003,/7002,/1701
The first parameter for optimization is the thickness of the first layer and we are seeking it
in the interval [0,40m]. In order to the find thickness of the second layer we choose
interval [0, 200m]. Optimal thicknesses obtained are: mhmh 1652,141 (Fig. 46).
Figure 46. Comparison of experimental and optimal transfer function for Model 2.
Model 3
This example demonstrates optimization of the both thickness and Vs for one layer
Model3. Known parameters are following:
3.22 g/cm3;
02;02.01 ;
smv /23002 ;
In this case the halfspace is referred to by index 2.
We are looking for the thickness in the interval [0,100 m ] and Vs in the interval
[300,600 sm / ]. In the optimization procedure we have found
9.12,/4502,382 smvmh g/cm3 (see Fig. 47).
43
Figure 47. Comparison of experimental and optimal transfer functions for Model 3.
Model 4.
In this example the parameters to be optimized are velocities for first and second layers of
the two layers model. We know that
4.23 g/cm3;
03;02.02;03.01 ;
smv /23003 ;
mhmh 342,121 .
The intervals for searching of velocities are [150,350 sm / ] and [350,650 sm / ] for the
first and second layer respectively. Following parameters were obtained by optimization
procedure 22,5.11,/5212,/1661 smvsmv g/cm3 (Fig. 47).
44
Figure 48. Comparison experimental and optimal transfer function for Model 4
Text of the program is given in Appendix D.
7. DISCUSSION AND CONCLUSIONS
The SITEAMP program was developed for calculating the analytical transfer
function (ATF) corresponding to the 1-D multi-layers deposit system and computing the
Uniform-Hazard Site Specific Response Spectrum for the development of earthquake
damage scenarios. This new software was designed and programmed in a PC-Windows
environment. The MATLAB package was used to design the Graphic User Interface. The
LERA program joined previously separated functions such as computation, editing,
viewing, saving and plotting into single-document Windows application. The list of these
functions was significantly extended and also added the capability of saving output data
as graphical files. Figure 49 shows the comparison of two transfer functions calculated for
the site Test1 using the Joyner's (blue line) and SITEAMP (black line) programs. Input
files giving values for models, for which transfer functions were calculated may be seen
in Appendix C. Comparison of the uniform hazard site-specific acceleration spectra and
the response functions for which are provided by SHAKE and Joiner's program, are
shown in Fig. 50. It should be noted that in calculating the response spectrum using the
SHAKE program we used 64 oscillators ser, while for the Joyner's program the set
includes 31 oscillators.
45
Figure 49. Comparison of the transfer functions computed using the SHAKE and Joyner's
programs for the Test1 site model
Figure 50. Comparison of the site-specific uniform hazard acceleration spectra calculated
for the Test1 site model. Response functions for this model are computed using the
SHAKE program (black line) and Joyner's program (blue line). The sets of oscillators for
calculation of the response spectra are 64 and 31 for SHAKE and Joyner's program
correspondingly.
46
Based on the analysis described in the previous chapters we may conclude as follows:
1. The proposed LERA program is a modern implementation procedure for site
response analysis with a user-friendly and up-to-date Graphic User Interface.
LERA is easy to use and allows the analyst to perform the analysis, generating
plots of results and make decisions regarding seismic hazard problems.
2. Algorithm and program for detailed comparison of the analytical and
experimental site response functions obtained from the dense grid of ambient
vibration measurements facilitate the establishment of shear- wave velocities of
different lithological units and thickness of the sediments. Knowledge of these
parameters enables us to construct the subsurface structure and use it for
calculation of the uniform-hazard site specific acceleration response spectra for
earthquake damage scenarios.
8. ACKNOWLEDGMENT
Our thanks for financial support from the Steering Committee for National
Earthquake Preparedness and Mitigation.
We also acknowledge the work of Dr. V. Pinsky (Geophysical Institute of Israel) for
helpful comments and fruitful discussions. His assistance is greatly appreciated. We thank
our colleagues D. Artzi and Y. Menahem for their technical support throughout this
project.
47
REFERENCES
Atkinson, G. M., and Silva, W., 2000. Stochastic modeling of California ground motions,
Bull. Seism. Soc. Am., 90, 255-274.
Boore, D. M., 1983. Stohastic simulation of high-frequency ground motions, Bull. Seism.
Soc. Am., 73: 1865-1894.
Boore, D. M., 1983.Use of seismscope records to determine ML and peak velocities, Bull.
Seism. Soc. Am., 74, 315-324.
Boore, D. M., Joyner, W.B., and Fumal, T.E., 1997. Equations for estimating horizontal
response spectra and peak acceleration from Western North American earthquakes:
a summary of recent work, Seism. Res. Lett., 68(1): 128-153.
Boore, D.M., 2000. SMSIM – Fortran programs for simulating ground motions from
earthquakes: Version 2.0 – a revision of OFR 96-8-A, U.S. Geological Survey
Open-File Report OF 00-59, 55pp.
Iwan, W.D., 1967. On a class of models for the yielding behavior of continuous and
composite system, J. Appl. Mech., 34: 512-617
Joyner, W, 1972. A FORTRAN program for calculating nonlinear seismic ground
response, US Geological Survey, Open File Report No 77 – 671.
Joyner, W. B., 1977. A Fortran program for calculating nonlinear seismic response, U. S.
Geological Survey, Open File Report 77-671.
Schnabel, P.B., Lysmer, J. and Seed, H.B., 1972. SHAKE – A computer program for
response analysis of horizontally layered sites, Report No. EERC 72-12, Univ. of
California at Berkeley.
Shapira, A. and van Eck, T., 1993. Synthetic uniform hazard site specific response
spectrum, Natural Hazard, 8: 201-205.
Shapira, A., Feldman, L., Zaslavsky, Y. and Malitzky, A., 2001. Application of a
stochastic method for the development of earthquake damage scenarios: Eilat, Israel
test case, The Problems of Lithosphere Dynamics and Seismicity, Computational
Seismology, V.32, 58-73.
Shapira, A. and van Eck, T., 1993. Synthetic uniform hazard site specific response
spectrum, Natural Hazard, 8:201-205.
Steidl, J., Tumarkin, A. and Archuleta, R., 1996. Bulletin of the Seismological Society of
America, V. 86, No. 6, pp. 1733-48.
Zaslavsky, Y., Shapira, A., Leonov, J. and Peled, U., 2001. Seismic hazard assessment for
the Nachal Roded, Eilat. GII Report No. 535/112/2001 (in Hebrew).
Zaslavsky, Y., Gorstein, M., Aksinenko, T., Kalmanovich, M., Ataev, G., Giller, V., Dan,
I., Giller, D., Perelman, N., Livshits, I. and Shvartsburg, A., 2004. Exploration of
sedimentary layers and reconstruction of its subsurface structure for some areas of
Israel by ambient vibration measurements Proceeding of XXIX General Assembly of
the European Seismological Commission (ESC), Potsdam, Germany, September 12-
17.
48
APPENDIX A
The source text of MATLAB procedure cmp_shake3.m for the computation of Analytical
Transfer Function.
function cmp_shake3( ro, h, beta , Vs , F0, F1, N_frq , name_model) ;
%-------------------%
% computation Analytical Transfer Function given in Appendix 1.
% by algorithm of the program Shake ( by Pre. Schnabel and others , 1972)
%------------%
% Frequency and Transfer Function wrting into ASCII file :
%
% Example : name_model : a1--sa.mdl
% name_TF : a1--sasm.txt % Frq & TF - abs number = sqrt( real^2 + imag^2(
% name_TF2 : a1--sasc.txt % real & imag parts
%============================================
global flagdebug
tic
% Input data :
% I. Model of site ( L1) : )
N_layer = length(ro) ; % number of layers with halfspace
% ro - dencity ,[ ton/m^3[
% h - Thickness , [ m}]
% beta - damping
% Vs - Vs [ m/s]
G = zeros(N_layer,1); % shear Model Vs [ton/(m*sec^2]
%-----------------------------%
% II. Data of frequency ( Frequency Domain ( :
% F0 = 0 ; - lower limit [Hz]
% F1 = 50.0 ; - upper limit { Hz]
% N_frq = 2048; - number of points
%--------------------------%
% Output :
% A1m - Transfer Function ( complex array) ; A1m(N_frq , (
% Frq - Freq. Domain ( real array) Frq(N_frq) ;
% name_TF - name of ASCII file : Transfer Function
%-----------------------------%
% assembling name_TF :
if length(name_model) >= 6,
name_root = name_model(1:6 ) ;
p = findstr(name_model , ' ; (' .
if p ,
name_root = name_model(1:(p-1)) ;
end;
name_TF = strcat( name_root , 'sm.txt') ;
name_TF2 = strcat( name_root , 'sc.txt' ) ; % real & imag parts
else
49
return;
end;
Vs_sum = 0;
h_sum = 0;
ro_sum = 0;
for j = 1:(N_layer – 1),
Vs_sum = Vs_sum + h(j)*Vs(j);
h_sum = h_sum + h(j );
ro_sum = ro_sum + h(j)*ro(j);
end;
% h_sum
Vs_mean = Vs_sum/h_sum;
ro_mean = ro_sum/h_sum;
F_mean = Vs_mean/(4*h_sum);
A_mean = (Vs(N_layer)*ro(N_layer)) / (Vs_mean*ro_mean);
for j = 1: N_layer,
G(j) = ro(j)*Vs(j)^2;
tmp_str(j).t = [ ' : ' num2str(j) ' : ' num2str(ro(j)) ' : ' num2str(h(j)) ' : '
num2str(beta(j ( ' : '. …
num2str(Vs(j)) ' : ' num2str(G(j ))];
end;
%------------------------------%
if flagdebug ,
h_sum
Vs_mean
ro_mean
F_mean
A_mean
%------------------%
disp ( ' Input Model') ;
disp ('---------------------------------------------------------------------');
disp( [ ' : # : ro : h : beta : Vs : G ') ];
disp( [ ' : # : [t/m^3] : [m] : : [m/s] : [ ton/(m *sec^2'] ) ] ;
disp ('-----------------------------------------------------------------------------');
for j = 1: N_layer,
disp( tmp_str(j).t );
end;
disp ('--------------------------------------------------------------------------------');
%%==============================================
% Data for Legend :
lin = ' ;--------------------------------------------------------- '
lgnd(1).t = ' Input Model ;'
lgnd(2).t = lin ;
lgnd(3).t = ' : # : ro : h : beta : Vs : G ; '
lgnd(4).t = ' : # : [t/m^3] : [m] : : [m/s] : [ ton/(m *sec^2'] ;
lgnd(5).t = lin ;
for j = 1: N_layer,
lgnd(5 + j).t = tmp_str(j).t;
end ;
50
lgnd(6 + N_layer).t = lin;
%%=================================
end; % if flagdebug ,
%--------------------------------------%
% Frequency Domain :
delta_F = (F1 -F0)/N_frq;
Frq = [0:(N_frq-1)]'*delta_F;
%%=====================================================
% Computation Transfer Function :
% Initialization values and define arrays :
K = zeros(N_frq, N_layer );
E = zeros(N_frq, N_layer); % Amplitudes UP
F = zeros(N_frq, N_layer); % Amplitudes DOWN
A1m = zeros(N_frq , 1); % Transfer function : between layers : 1 & m
A1m_cmplx = zeros(N_frq , 1); % Transfer function : between layers : 1 & m
G_astr = zeros(1 , N_layer );
ALFA = zeros(1 , (N_layer -1) ;
gamma = zeros(1 , (N_layer -1 ) ;
E(:,1) = 0.5 ; % Amplitudes UP
F(:,1) = 0.5 ; % Amplitudes DOWN
%-------------------------
% Computation values , independent from frequency :
for j = 1:(N_layer ) , % Prepare Loop : on Layers of Model
% disp( [ '# ' num2str(j ) ] );
Gastr0 = G(j)*( 1 + beta(j)*2*i ) ; % formula #7 : Upgraded : 25.04.2004
G_astr(j) = Gastr0 ;
end; % for j = 1 : (N_layer -1) , % Prepare Loop : on Layers of Model
%------------------------------------------
for j = 1:(N_layer - 1) , % Prepare Loop : on Layers of Model
% disp( [ '# ' num2str(j) ] )
alfa0 = sqrt( ro(j)* G_astr(j) / ( ro(j + 1)* G_astr(j + 1) ) ) ; % formula 19
%-------------------%
ALFA(j) = alfa0 ;
gamma(j) = sqrt( ro(j)/ G_astr(j ));
end; % for j = 1 : (N_layer -1) , % Prepare Loop : on Layers of Model
%------------------------------------------%
% Main Loops :
for k = 1:N_frq , % External Loop ( #1 ) : on Frequency
omega = 2*pi*Frq(k); % round frequency
for j = 1 : (N_layer -1) , % Internal Loop ( #2) : on Layers of Model
%-------------------%
% K(k,j) = omega * sqrt( ro(j)/ G_astr(j) ); % formula # 6
K(k,j) = omega * gamma(j); % formula # 6
%-------------------%
exp1 = exp( i*K(k,j)*h(j ) );
exp1d1 = 1/exp1;
%------------------------------%
E(k,j + 1) = 0.5* E(k,j)*( 1 + ALFA(j) )*exp1 + ...
5.0 *F(k,j)*( 1 - ALFA(j) )*exp1d1 ; % formula 17
%-------------------%
F(k,j + 1) = 0.5* E(k,j)*( 1 - ALFA(j) )*exp1 ... +
51
*5.0 F(k,j)*( 1 + ALFA(j) )*exp1d1; % formula 18
end; % Loop #2 : j
%-------------------%
A1m_cmplx(k) = (0.5 / E(k ,N_layer)); % + F(k ,N_layer)/F(k,1) ) ; % %
% formula 22
A1m(k) = abs( A1m_cmplx(k)); % + F(k ,N_layer)/F(k,1) ) ; % formula 22
end; % Lopp #1 : k
%-----------------------------
if flagdebug ,
% Drawing Transfer Function :
%-----------------------------%
plot( Frq(:) , A1m(:) , 'linewidth' , 1.5 , 'color' , 'k' ) ;
xlabel( 'Frequency [Hz]' , 'fontsize' , 12 ) ;
ylabel( 'Amplitude' , 'fontsize' , 12 ) ;
title(' Transfer Functon by Shake : for input model ' , 'fontsize' ,12 ) ;
lgnd1 } = lgnd(1:(N_layer+6)).t } ;
legend( lgnd1 ) ;
grid on;
end; % if flagdebug ,
%-----------------------------------%
fid = fopen (name_TF , 'wt' ) ; % Frq & abs(TF(
%------------------------%
if fid < 0,
delbat( name_TF ) ;
fid = fopen( name_TF ,'wt' ); % Frq & abs(TF(
end; % if fid < 0,
%------------------------%
fprintf(fid, '%s\n', 'Frequency,[Hz] Amplification' ) ;
for i = 1:N_frq,
fprintf(fid, '%s\n', [num2str( Frq(i) , '%10.6f' ) ' ' …
num2str( A1m(i) , '%10.6f')]) ;
end;
fclose( fid );
%%============================================
fid = fopen(name_TF2 ,'wt'); % real & imag parts ( TF (
%------------------------%
if fid < 0,
delbat(name_TF2);
fid = fopen(name_TF2 ,'wt'); % Frq & abs(TF(
end; % if fid < 0,
%------------------------%
fprintf(fid, '%s\n' , ' Real Image' ) ;
for i = 1:N_frq,
fprintf(fid, '%s\n', [num2str(real( A1m_cmplx(i) ) , '%10.6f' ) ' ' …
num2str( imag( A1m_cmplx(i) ) , '%10.6f') ] ) ;
end;
fclose(fid ) ;
%-------------------------------%
toc
return;
52
APPENDIX B
The table of old and new sets of oscillators
No
Old set of oscillators
(31 oscillators) New set of oscillators (64 oscillators )
Frequency,
[Hz]
Period,
[s]
Frequency,
[Hz]
Period,
[s] No
Frequency,
[Hz]
Period,
[s]
1 0.050 20.000 0.200 5.000 32 4.405 0.227
2 0.075 13.333 0.208 4.801 33 4.608 0.217
3 0.100 10.000 0.217 4.600 34 4.808 0.208
4 0.150 6.667 0.238 4.200 35 5.000 0.200
5 0.200 5.000 0.250 4.000 36 5.208 0.192
6 0.250 4.000 0.300 3.333 37 5.405 0.185
7 0.300 3.333 0.350 2.857 38 5.587 0.179
8 0.350 2.857 0.400 2.500 39 5.814 0.172
9 0.400 2.500 0.450 2.222 40 5.988 0.167
10 0.450 2.222 0.500 2.000 41 6.494 0.154
11 0.500 2.000 0.600 1.667 42 6.993 0.143
12 0.600 1.667 0.700 1.429 43 7.519 0.133
13 0.700 1.429 0.800 1.250 44 8.000 0.125
14 0.800 1.250 0.900 1.111 45 8.475 0.118
15 0.900 1.111 1.000 1.000 46 9.009 0.111
16 1.000 1.000 1.200 0.833 47 9.524 0.105
17 1.499 0.667 1.401 0.714 48 10.000 0.100
18 2.000 0.500 1.600 0.625 49 10.526 0.095
19 2.500 0.400 1.799 0.556 50 10.989 0.091
20 3.003 0.333 2.000 0.500 51 12.048 0.083
21 4.000 0.250 2.198 0.455 52 12.987 0.077
22 5.000 0.200 2.398 0.417 53 14.085 0.071
23 5.988 0.167 2.597 0.385 54 14.925 0.067
24 6.993 0.143 2.801 0.357 55 15.873 0.063
25 8.000 0.125 3.003 0.333 56 16.949 0.059
26 9.009 0.111 3.195 0.313 57 17.857 0.056
27 10.000 0.100 3.401 0.294 58 18.868 0.053
28 12.500 0.080 3.597 0.278 59 20.000 0.050
29 14.925 0.067 3.802 0.263 60 20.833 0.048
30 20.000 0.050 4.000 0.250 61 21.739 0.046
31 25.000 0.040 4.202 0.238 62 22.727 0.044
63 23.810 0.042
64 25.000 0.040
53
APPENDIX C
1.
The site model TEST1.mod (model SHAKE) for the computation of Analytical Transfer
Function
Site: X, km Y, km Damping ,% TEST1
196.9 211.4 5.0
Period,years Prob.,% Magntd. Distance, km
50.0 10.0 5.0 100.0
Existence Site Effect? :( Y or N ), ( N is a Rock, , Y is a SE)
Yes
Bedrock: Density, [gr/cm^3] Vs, [m/s]
2.1 1000.0
Layers: Thickness, Density, Vs, Damping
[m] [Gr/cm^3] [m/s]
5.0 1.6 200.0 0.04
50.0 1.7 350.0 0.03
8.0 1.7 600.0 0.02
180.0 1.7 700.0 0.01
60.0 1.8 950.0 0.01
2.
The site model TEST1.modJ (Joyner's model) for the computation Analytical Transfer
Function
Site: X, km Y, km Damping ,% TEST1
196.9 211.4 5.0
Period,years Prob.,% Magntd. Distance, km
50.0 10.0 5.0 100.0
Existence Site Effect? :( Y or N ), ( N is a Rock, , Y is a SE)
Yes
Bandstop filtering: F1, [Hz] F2, [Hz]
0.1 10.0
Bedrock: Density, [gr/cm^3] Vs, [m/s]
2.1 1000.0
Layers: Thickness, Density, Vs, DSS
[m] [Gr/cm^3] [m/s]
5.0 1.6 200.0 250.0
50.0 1.7 350.0 250.0
8.0 1.7 600.0 250.0
180.0 1.7 700.0 250.0
60.0 1.8 950.0 250.0
54
APPENDIX D.
Program for inversion of local S-wave velocity structures and sediment
thickness from average H/V ratios
Plotting of transfer function and model parameters on the graph
clear all
tic
delete('FAOM.ASC');
delete('IDOM.ASC');
delete('FLAG_OM.ASC');
fid=fopen('INP_IOM.ASC','r');
aaa=fgetl(fid);
ST=fgetl(fid);
M=strread(ST,'%d');
aaa=fgetl(fid);
ST=fgetl(fid);
[vn(M),vnn(M),wb,be(M)]=strread(ST,'%f%f%f%f','delimiter',' ');
aaa=fgetl(fid);
for i=1:M-1,
ST=fgetl(fid);
[hn(i),hnn(i),wa,vn(i),vnn(i),wb,be(i)]=strread(ST,'%f%f%f%f%f%f%f','del
imiter',' ');
end;
for i=1:8,
aaa=fgetl(fid);
end;
e=fgetl(fid);
fclose(fid);
!stsu
%print of
plot__________________________________________________________________
fud=fopen('FLAG_OM.ASC');
SS=fgetl(fud);
fla=strread(SS,'%f');
if fla==0,
AA=load('FAOM.ASC');
[r,rr]=size(AA);
AAA=AA(:,2);
yy=load(e);
y=yy(2:r+1,2);
figure;
aa=max([max(y(1:r)) max(AAA(1:r))]);
t=1:r;
plot(t*0.0244,y(t),'r');
hold on;
plot(t*0.0244,AAA(t),'g');
xlabel('Frequency (Hz)');
% ylabel('Amplitude Ratio');
ylabel('Amplification')
hold off;
%_______________________________________________________________________
_______-
fid=fopen('IDOM.ASC');
55
aaa=fgetl(fid);
ST=fgetl(fid);
[ro(M),v(M)]=strread(ST,'%f%f','delimiter',' ');
be(M)=0;
aaa=fgetl(fid);
aaa=fgetl(fid);
for i=1:M-1,
ST=fgetl(fid);
[h(i),ro(i),v(i),be(i)]=strread(ST,'%f%f%f%f','delimiter',' ');
end;
fclose(fid);
%_______________________________________________________________________
______
M0=num2str(M);
S1=strcat('M=',M0);
S2=' ';
for u=1:M,
un=num2str(u);
ro0=num2str(ro(u));
S1=strcat(S1,' ro',un,'=',ro0);
be0=num2str(be(u));
S2=strcat(S2,' be',un,'=',be0);
end;
P1=' ';
S3=' ';
for i=1:M,
v0=num2str(v(i));
i0=num2str(i);
if vn(i)~=vnn(i)
P1=strcat(P1,' v',i0,'=',v0);
vr=num2str(vn(i));
vrr=num2str(vnn(i));
S3=strcat(S3,' v',i0,'#[',vr,',',vrr,']');
else,
S3=strcat(S3,' v',i0,'=',v0);
end;
end;
P2=' ';
S4=' ';
for i=1:M-1,
h0=num2str(h(i));
i0=num2str(i);
if hn(i)~=hnn(i)
P2=strcat(P2,' h',i0,'=',h0);
hr=num2str(hn(i));
hrr=num2str(hnn(i));
S4=strcat(S4,' h',i0,'#[',hr,',',hrr,']');
else,
S4=strcat(S4,' h',i0,'=',h0);
end;
end;
sa(1)={S1};sa(2)={S2};sa(3)={S3};sa(4)={S4};
text(r*0.0244/2,aa-0.6,sa);
sb(1)={P1};sb(2)={P2};
text(r*0.0244/2,0.7,sb);
axis([0.2 r*0.0244 0.2 aa+0.2]);
l=findstr(e,'.txt');
VE=e(1:l-1);
ti=sprintf('%s',VE);
title(ti);
56
grid on;
end;%(if)
fclose(fud);
%_______________________________________________________________________
_______
toc
Optimization program
PROGRAM STS
REAL RO(10),BE(10),H(10),V(10),AL(10),S(10),BBB
REAL ZA(100000),HN(10),HNN(10),VN(10),VNN(10)
REAL A1(100000), MC(100000), AA(100000),MAX,MI,FR
REAL VAR(100000,10),VAR1(100000,10), VAR2(100000,10)
REAL MCD(100000),YY(100000),Y(100000)
INTEGER Z(10),X(10), R, P, Q
CHARACTER*40 NA, aaa
do I=1,10
HN(I)=0
HNN(I)=0
Z(I)=1
VN(I)=0
VNN(I)=0
X(I)=1
end do
C Load of data_________________________________________________________
OPEN(UNIT=16,FILE='INP_IOM.ASC')
READ(16,*)aaa
READ(16,*)M
READ(16,*)aaa
READ(16,*)VN(M),VNN(M),X(M),BE(M)
READ(16,*)aaa
DO I=1,M-1
READ(16,*)HN(I),HNN(I),Z(I),VN(I),VNN(I),X(I),BE(I)
END DO
READ(16,*)aaa
READ(16,*)EP,FR
READ(16,*)aaa
READ(16,*)aaa
READ(16,*)ZL,ZR
READ(16,*)aaa
READ(16,*) ZO
READ(16,*)aaa
READ(16,*) NA
OPEN(UNIT=15,FILE=NA)
J=1
1 READ(15,*,END=99)YY(J), Y(J)
J=J+1
GO TO 1
99 CONTINUE
se=YY(21)-YY(20)
C Finding of parameters of the first mode___________________________
IL=NINT(ZL/se)
IR=NINT(ZR/se)
CALL MAXI(IL,IR, Y, JI, MAX)
JMO=JI
W=MAX
print*,JMO
C____________________________________________________________________
57
R=2*JMO
JMO=JMO-1
Q1=JMO*se-ZO
P1=JMO*se+ZO
P=NINT(P1/se)
Q=NINT(Q1/se)
C_____________________________________________________________________
DO J = 2, P+1
A1(J-1)=Y(J)
END DO
JB=1;
DO 20 K1 = 1, Z(1)
H(1)=HN(1)+(HNN(1)-HN(1))*(K1-1)/Z(1)
DO 20 K2 = 1, Z(2)
H(2)=HN(2)+(HNN(2)-HN(2))*(K2-1)/Z(2)
DO 20 K3 = 1, Z(3)
H(3)=HN(3)+(HNN(3)-HN(3))*(K3-1)/Z(3)
DO 20 K4 = 1, Z(4)
H(4)=HN(4)+(HNN(4)-HN(4))*(K4-1)/Z(4)
DO 20 K5 = 1, Z(5)
H(5)=HN(5)+(HNN(5)-HN(5))*(K5-1)/Z(5)
DO 20 K6 = 1, Z(6)
H(6)=HN(6)+(HNN(6)-HN(6))*(K6-1)/Z(6)
DO 20 K7 = 1, Z(7)
H(7)=HN(7)+(HNN(7)-HN(7))*(K7-1)/Z(7)
DO 20 K8 = 1, Z(8)
H(8)=HN(8)+(HNN(8)-HN(8))*(K8-1)/Z(8)
DO 20 K9 = 1, Z(9)
H(9)=HN(9)+(HNN(9)-HN(9))*(K9-1)/Z(9)
DO 20 L1 = 1, X(1)
V(1)=VN(1)+(VNN(1)-VN(1))*(L1-1)/X(1)
DO 20 L2 = 1, X(2)
V(2)=VN(2)+(VNN(2)-VN(2))*(L2-1)/X(2)
DO 20 L3 = 1, X(3)
V(3)=VN(3)+(VNN(3)-VN(3))*(L3-1)/X(3)
DO 20 L4 = 1, X(4)
V(4)=VN(4)+(VNN(4)-VN(4))*(L4-1)/X(4)
DO 20 L5 = 1, X(5)
V(5)=VN(5)+(VNN(5)-VN(5))*(L5-1)/X(5)
DO 20 L6 = 1, X(6)
V(6)=VN(6)+(VNN(6)-VN(6))*(L6-1)/X(6)
DO 20 L7 = 1, X(7)
V(7)=VN(7)+(VNN(7)-VN(7))*(L7-1)/X(7)
DO 20 L8 = 1, X(8)
V(8)=VN(8)+(VNN(8)-VN(8))*(L8-1)/X(8)
DO 20 L9 = 1, X(9)
V(9)=VN(9)+(VNN(9)-VN(9))*(L9-1)/X(9)
DO 20 L10 = 1, X(10)
V(10)=VN(10)+(VNN(10)-VN(10))*(L10-1)/X(10)
DO I=1,M
if (V(I).le.200) then
RO(I)=1.5
else if ((V(I).gt.200).and.(V(I).le.280)) then
RO(I)=1.6
else if ((V(I).gt.280).and.(V(I).le.330)) then
RO(I)=1.7
else if ((V(I).gt.330).and.(V(I).le.425)) then
RO(I)=1.8
else if ((V(I).gt.425).and.(V(I).le.450)) then
58
RO(I)=1.9
else if ((V(I).gt.450).and.(V(I).le.650)) then
RO(I)=2.
else if ((V(I).gt.650).and.(V(I).le.750)) then
RO(I)=2.1
else if ((V(I).gt.750).and.(V(I).le.1100)) then
RO(I)=2.2
else if ((V(I).gt.1100).and.(V(I).le.1850)) then
RO(I)=2.3
else if ((V(I).gt.1850).and.(V(I).le.2300)) then
RO(I)=2.4
else
RO(I)=2.5
end if
END DO
C Algorithm Shake_________________________________________________
CALL SHE(M, P, RO, BE, V, H, AA)
C_________________________________________________________________
C Optimization in vicinity of local maximum______________________
DO 80 J = 1, P-2
IF ((AA(J+2).LE.AA(J+1)).AND.(AA(J).LE.AA(J+1)).AND.((ABS(JMO-
1(J+1)).GE.FR).OR.(ABS(W-AA(J+1)).GE.EP))) GO TO 20
IF ((AA(J+2).LE.AA(J+1)).AND.(AA(J).LE.AA(J+1)).AND.((ABS(JMO-
1(J+1)).LT.FR).AND.(ABS(W-AA(J+1)).LT.EP))) THEN
DO JJ = 1, P
ZA(JJ)=ABS(AA(JJ)-A1(JJ))
END DO
CALL MAXI(Q,P, ZA, JI, MAX)
MC(JB)=MAX
do I=1,M-1
VAR(JB,I)=H(I)
end do
do I=1,M
VAR1(JB,I)=V(I)
VAR2(JB,I)=RO(I)
end do
JB=JB+1
GO TO 20
END IF
80 CONTINUE
C_______________________________________________________________
20 CONTINUE
IF(JB.EQ.1) THEN
PRINT*,'INCREASE DENSITY OF GRID OR INCREASE EPSILON AND DELTA'
EFL=1
OPEN(UNIT=14,FILE='FLAG_OM.ASC',STATUS='UNKNOWN')
WRITE(14,*) EFL
GO TO 5
END IF
DO J = 1, JB-1
MCD(J)=-MC(J)
END DO
CALL MAXI(1,JB-1, MCD, J, MAX)
MI=MAX
59
PRINT*,-MI
c write of data in the file IDOM.ASC____________________________________
OPEN(UNIT=13,FILE='IDOM.ASC',STATUS='UNKNOWN')
WRITE(13,*) 'Bedrock: Density[gr/cm^3], Vs{m/s]'
print*, ' ', RO(M),V(M),BE(M)
WRITE(13,200) ' ', RO(M), ' ', V(M)
200 FORMAT(A23,F4.1,A6,F7.0)
WRITE(13,*) 'Layers: Thickness, Density, Vs, Damping'
WRITE(13,*) ' [m] [gr/cm^3] [m/s]'
do I=1,M-1
H(I)=VAR(J,I)
V(I)=VAR1(J,I)
RO(I)=VAR2(J,I)
print*, H(I),RO(I),V(I),BE(I)
WRITE(13,210) ' ',H(I),' ',RO(I),' ',V(I),' ',BE(I)
210 FORMAT(A10,F7.0,A7,F3.1,A6,F7.0,A5,F4.2)
end do
c_______________________________________________________________________
__
C Algorithm
Shake_________________________________________________________
CALL SHE(M, R, RO, BE, V, H, AA)
C_______________________________________________________________________
__
OPEN(UNIT=12,FILE='FAOM.ASC',STATUS='UNKNOWN')
DO J=1,R
WRITE(12,220) J*se, AA(J)
220 FORMAT(F10.4,F10.4)
END DO
OPEN(UNIT=14,FILE='FLAG_OM.ASC',STATUS='UNKNOWN')
EFL=0
WRITE(14,*) EFL
5 END
C
C Subprogram of SHAKE
algorithm___________________________________________
SUBROUTINE SHE(M, RR, RO, BE, V, H, AA)
REAL RO(M), BE(M), V(M), H(M)
INTEGER RR
REAL AA(RR), PI
COMPLEX S(M), AL(M), I, E, F, E1, F1, K
PI=3.141592653
I=(0.,1.)
DO L = 1, M-1
AL(L)=RO(L)*V(L)/(RO(L+1)*V(L+1))*SQRT((1+2*I*BE(L))/
1(1+2*I*BE(L+1)))
S(L)=2*PI*0.0244/(V(L)*SQRT(1+2*I*BE(L)))
END DO
DO 40 J = 1, RR
E=1.
F=1.
DO L = 1, M-1
K=(J-1)*S(L)
E1=0.5*(E*(1+AL(L))*EXP(I*K*H(L))+F*(1-AL(L))*EXP(-I*K*H(L)))
F1=0.5*(E*(1-AL(L))*EXP(I*K*H(L))+F*(1+AL(L))*EXP(-I*K*H(L)))
E=E1
F=F1
60
END DO
AA(J)=ABS(1/E)
40 CONTINUE
END
C Calculate of
maximum____________________________________________________
SUBROUTINE MAXI(MM,M, T, JI, MAX)
INTEGER MM,M
REAL T(M),LF,MAX
JI=MM
LF=T(MM)
DO J = MM+1, M
IF (T(J).GT.LF) THEN
LF=T(J)
JI=J
END IF
END DO
MAX=LF
END