User Manual
Version 0.5
[go to the
latest online version]
ContenTS
Installation
in MS Windows 64 bit
Installation
in GNU/Linux 64 bit
Installation
in OS X Mac 64 bit
APPENDIX
A: Generation of random models & Requirements for model parameters table
Overview
HV-Inv is a computer code for forward
calculation and inversion of HV spectral ratios of ambient noise (HVSRN) based
on the diffuse field assumption (DFA, Sánchez-Sesma
et al., 2011). It takes advantage of the recently connection between the HVSRN
and the elastodynamic Green's function which arises from the ambient noise
interferometry theory.
The software supports joint inversion of HVSRN and
dispersion curves by using several local and global algorithms: Monte Carlo
sampling simulated annealing, simplex downhill and interior-point.
References:
García-Jerez, A., Piña-Flores, J., Sánchez-Sesma, F.J., Luzón, F., Perton, M. (2016). A computer code for
forward computation and inversion of the H/V spectral ratio under the diffuse
field assumption, Computers
& Geosciences 97, 67–78.
Sánchez-Sesma, F.J.,
Rodríguez, M., Iturrarán-Viveros, U., Luzón, F., Campillo, M., Margerin, L., García-Jerez, A., Suarez, M., Santoyo, M.A., Rodríguez-Castellanos, A. (2011). A theory
for microtremor H/V spectral ratio: application for a layered medium,
Geophysical Journal International 186, 221-225.
García-Jerez A., Luzón F.,
Sánchez-Sesma F. J., Lunedei
E., Albarello D.,
Santoyo M. A., Almendros J. (2013). Diffuse elastic wavefield
within a simple crustal model. Some consequences for low and high frequencies.
Journal of Geophysical Research 118(10), 5577-5595.
Piña-Flores,
J. (2015). Cálculo e inversión del cociente H/V a partir de ruido ambiental. Unpublished M.Sc. Thesis, Universidad Nacional Autónoma de México, México DF,
76 pp. In Spanish.
Hardware: 1GHz CPU, 512 MB RAM, 200 MB of free space on HDD
OS: HV-Inv_Beta has been tested under the following OS configurations:
Windows 8 64 bit, Windows 8.1 64 bit, Windows 10 64 bit
GNU/Linux 64 bit (Ubuntu 16.04 LTS)
OS X Mac 64 bit
Please follow these instructions to install HV-Inv 1.0 Beta on your PC:
Installation in MS Windows 64 bit
Method 1: from compiled version
1. - Before using the compiled version you need to install Matlab Runtime R2015a (v8.5) (free)
2.- Download the latest installation of HV-Inv Beta corresponding to Windows 64 bit operating system from the website: http://www.ual.es/GruposInv/hv-inv/ and save it to your computer.
Extract the downloaded compressed file into a folder. Use WinZip or another uncompressing utility.
3. - Run HVInvBeta.exe.
Note: On Windows 7 or later you might be required to run the HVInvBeta.exe as Administrator.
Method 2: run from source code
If source code is included in your
distribution and you plan to run the program from it, you need to install a
licensed version of Matlab R2015a and run HV_Inv_Beta.m.
Method 1: from compiled version
1. - Before using the compiled version you need to install Matlab Runtime R2015a (v8.5) (free)
2.- Download the latest installation of HV-Inv Beta corresponding to Linux 64 bit operating system from the website: http://www.ual.es/GruposInv/hv-inv/ and save it to your computer.
Extract the downloaded compressed file into a folder.
3. - Run run_HVInvBeta.sh. You
may need to modify the access permissions to the whole installation folder
(chmod –R +rwx <foldername> or sudo chmod –R +rwx <foldername>) and enter from a terminal the following
command (see readme file included in this package):
./run_HVInvBeta.sh
/usr/local/MATLAB/MATLAB_Runtime/v85
or
sudo ./run_HVInvBeta.sh /usr/local/MATLAB/MATLAB_Runtime/v85
Method 2: run from source code
If source code is included in your
distribution and you plan to run the program from it, you need to install a
licensed version of Matlab R2015a and run HV_Inv_Beta.m.
Method 1: from compiled version
1. - Before using the compiled version you need to install Matlab Runtime R2015a (v8.5) (free)
2.- Download the latest installation of HV-Inv Beta corresponding to MAC 64 bit operating system from the website: http://www.ual.es/GruposInv/hv-inv/ and save it to your computer.
Extract the downloaded compressed into a folder.
3. -Run run_HVInvBeta.sh. You may need to modify the access permissions to the whole installation folder (chmod –R +rwx <foldername> or sudo chmod –R +rwx <foldername>) and enter from a terminal the following command (see readme file included in this package):
./run_HVInvBeta.sh /Applications/MATLAB/MATLAB_Runtime/v85
or
sudo ./run_HVInvBeta.sh /Applications/MATLAB/MATLAB_Runtime/v85
Method 2: run from source code
If source code is included in your distribution and you plan to run the program from it, you need to install a licensed version of Matlab R2015a and run HV_Inv_Beta.m.
The Principal Window of HV-Inv 1.0 Beta contains seven panels described below
Links: Menu Toolbar Model Data and current best modelling Select target data Model Parameter bounds Misfit History
Menu
Workers (On /Off). Enables the processors to make the initial random search and some inversion processes in parallel (for “Modified Simulated Annealing” inversion method).
Exit. Close
the application.
Inversion Type. Select the type of inversion algorithm, by default, Montecarlo sampling is selected.
Global Optimization. Global inversion algorithms.
§ Montecarlo sampling.
Number of iterations. Number of iterations for the inversion.
§ Simulated annealing.
- Márkov’s chain length. Number of elements in the Márkov’s chain (Number of iterations for which temperature is kept constant).
- Number of temperatures. Number of Márkov’s chains. The temperature falls from a chain to another.
- Last Márkov’s chain length. Number of additional models evaluated at the lowest temperature reached in order to assess the velocity profile uncertainty.
§ Modified Simulated Annealing
- Number of iterations. Number of iteration for a model series with decreasing temperature.
- Number of Reheatings. Number of times the system is reheated, and a new chain of “Number of iterations” models begins.
- Number of last iterations. Number of additional models evaluated at the lowest temperature reached in order to assess the velocity profile uncertainty.
Local
Optimization Local inversion algorithms
§ Simplex Downhill and Interior-Point methods
- Maximum number of iterations allowed. Maximum number of iterations of the algorithm.
- Maximum number of evaluations allowed. Maximum number of evaluations of the cost function (misfit function).
- Termination tolerance on the function value. Minimum meaningful misfit decrement
- Termination tolerance on parameters. Minimum meaningful variations in model parameters
Settings
Initial Population. Represents the number of random models tested in the whole range of model parameters during a random search stage. The best model found is used as initial model for the subsequent inversion methods. If you enter zero, you will be requested to enter a specific initial model.
Initial Temperature. This parameter directly affects the probability of accepting a model when the cooling schedule starts. (Available only for the “Modified Simulated Annealing” and the “Simulated Annealing” methods). The box asks for the probability of accepting a relative misfit increment (i. e. normalized by the misfit of the initial model).
Cooling Schedule. This value controls the descent of the "Temperature" parameter to progressively reduce the acceptance probability of the models for which the misfit is worse than for the current one. The box asks for Ti+1 / Ti ratio (Available for the Simulated Annealing and the Modified Simulated Annealing only).
Perturbation Range. This value controls the maximum size of the perturbation range around the current model, wherein the next model is generated. The maximum variation of a given parameter around the current model is plus/minus a “Perturbation range” percent of the corresponding full range stated in the “Model Parameters” table. Available only for the Monte Carlo sampling, Simulated Annealing and the Modified Simulated Annealing methods.
Low Parameters Zone. Allows low velocity zones.
Allow low-velocity Vs zones. Allows low velocity zones to exist (LVZ) for Vs.
Allow low-velocity Vp zones. Allows low velocity zones to exist (LVZ) for Vp.
Allow low-density zones. Allows low density zones to exist (Disabled in this version).
Maximum Vs for halfspace. Excludes models for which halfspace does not have the greater Vs.
Wave Parameters. Wave
parameters for the direct calculation of the HV curve.
Ø Rayleigh waves modes. Maximum number of Rayleigh wave modes to be considered in the forward calculation of the HV curve (The number zero indicates no contribution of the Rayleigh waves).
Ø Love waves modes. Maximum number of Love wave modes to be included in the forward calculation of the HV curve (The number zero indicates no contribution of Love waves).
Ø Minimum number of integration points for BW (Body waves). Minimum number of samples to evaluate the body waves contribution.
Ø Maximum number of integration points for BW (Body waves). Maximum number of samples to value the body waves contribution.
Note: Enter zero in this filed to disable calculation of body waves contributions (time consuming).
Ø Regularization factor. Value used to generate small imaginary parts for
frequencies or wavenumbers to stabilize body wave integrals. The default value
0.01 is recommended. A value of zero is also admissible for most of the models.
View
Parameters space. The relationship between the model misfit and particular model parameters is separately studied. Each panel represents a model parameter. The existence of well constrained and poorly constrained properties can be assessed in this way
Models. This sub-menu displays the inversion results.
Help
About. This sub-menu displays information about HV-inv 1.0 Beta.
User Manual. This sub-menu opens the "User Manual"
file.
Toolbar
Print Figure. This tool prints out the main window.
Zoom in, Zoom out. Zoom control buttons.
Pan. This tool allows you to navigate through the graphics.
Data Cursor. This
tool shows the X Y coordinate values of any point of the curves.
Zoom in/out, Pan & Data cursor
These buttons are available for "Model”, “Graphics” and “Misfit History” panels only.
Loads the data of the experimental HV curve from a * .txt file in the following format:
|
Frequency |
Amplitude H/V |
Standard deviation |
|
0.1 |
0.5 |
0.2 |
|
0.2 |
0.52 |
0.15 |
If the file lacks of
standard deviation column, the software let you choose between the following
three methods to generate assumed standard deviation values:
STANDARD DEVIATION GENERATION (STDG) FOR H/V
Constant (Units). The input in the box is assumed to be a constant standard deviation for all frequencies, with same units as the corresponding target.
Proportional (%). The standard deviations are assumed to be proportional to the data value (H/V value or velocity). The input in the box is interpreted as a percentage.
Cancel. The standard deviation is assumed to be one target unit for all frequencies.
Target DC
Loads the data of the experimental
dispersion curve from a * .txt file in the following format:
|
Frequency |
Velocity (m/s) |
Standard deviation |
|
0.1 |
1000 |
10 |
|
0.2 |
1010 |
15 |
To identify the type of dispersion curve selected, you have to enter the following information:
Polarization. Type of polarization (Rayleigh or Love).
Velocity. Type of velocity (Phase or Group).
Mode Index. Number of the mode corresponding to the provided dispersion curve data (only fundamental mode is available in the current version).
Sample weight. Check on this box to equalize the weight of the two observables in joint inversions, regardless the respective number of samples. Weight. This value modulates the relative influence of the observables on the cost (misfit). A 0 value indicates that the dispersion curve is not taken into account. A 0.5 value indicates that all the points have the same weight in the cost function, regardless which target they represent. A value of 1 indicates that the dispersion curve takes the whole weight in the inversion.
Next curve. Adds higher-modes dispersion curves (Disabled).
Open File.
Loads the files of higher-modes dispersion curve (Disabled).
STANDARD DEVIATION GENERATION (STDG) FOR DC
Constant Standard deviation (Units). The input is assumed to be a constant standard deviation for all frequencies.
Proportional Standard deviation (%). The standard deviations are assumed to be proportional to the data value (H/V value or velocity). The input in the box is interpreted as a percentage.
Cancel Standard deviation. The
standard deviations are assumed to be one target unit for all frequencies.
Target HV & DC
Loads the data of the experimental HV and
dispersion curves. This button enables joint inversion.
Forward H/V
Shows the HV forward computation interface.
Start Inversion/Stop Inversion
Starts or stops the inversion process.
This panel contains the elastic parameters limits (a priori information).
Valid parameter ranges should be introduced. The
requirements of the velocity ranges to be considered valid depend on the way of
dealing with possible velocity inversions in Vs and Vp
(Low-velocity zones), that is, on the settings in Inversion
settings menu > Low Parameters Zone.
See APPENDIX A:
GENERATION OF RANDOM MODELS & REQUIREMENTS FOR MODEL PARAMETERS TABLE.
You
should enter the number of layers, including the halfspace.
You
should enter the minimum and maximum value of the elastic parameters. To fix the
value of a parameter to a constant, enter the same value as minimum and
maximum. See APPENDIX A:
GENERATION OF RANDOM MODELS & REQUIREMENTS FOR MODEL PARAMETERS TABLE.
Load Parameters
Loads the elastic parameters limits from a *.para file with the following format:
# Number of layers
3
# Min-thickness Max-thickness Min-Vp Max-Vp Min-Vs Max-Vs Min-Density Max-Density Min-Poisson Max-Poisson
10 50 400 6000 200 3400 2000 2000 0.25 0.4
10 50 400 6000 200 3400 2000 2000 0.25 0.4
0 0 400 6000 200 3400 2000 2000 0.25 0.4
Lines starting with # are not read (comments)
Use 0 for halfspace thickness
Save Parameters
Saves the elastic parameters limits in a *.para file with the format shown in the previous section:
Graphics panel shows the target HV and dispersion curves (black lines). Blue lines are the current best-fitting curves. The horizontal axis represents the frequency (Hz) and the vertical axis represents the amplitude or velocity. The type of axis scales (linear or logarithmic) can be modified by pressing the secondary button and selecting the scale type.
The Model panel shows the current best-fitting velocity profiles. The horizontal axis represents wave velocity and vertical axis represents depth. Vs and Vp profiles are shown with red solid line and the red dashed line, respectively.
The Misfit History panel shows the evolution of the misfit along the inversion process. The magenta line represents the behavior of the cost function. The blue line represents the historical minimum of the cost function. The red line represents the temperature parameter controlled by the cooling schedule. The temperature parameter is shown for the Simulated Annealing and the Modified Simulated Annealing methods only.
The Inversion Report panel shows the final information about the inversion process.
The Forward calculation main Window contains five panels described below. This interface lets the user simulate the H/V ratio for a particular model and check the influence of each model parameter and of the computation settings.
Links: Menu Toolbar
Model Parameters Wave
Parameters Vp/Vs/dens. Profile
HV ratio
# Layers
Enter the number of layers including the
halfspace.
parameters TABLE
Enter the elastic parameters values.
Vp,
Vs and Poisson’s ratio are interdependent. The behavior of the table when one
of these quantities is modified depends on the keep Poisson ratio settings, described here.
LOAD / SAVE MODEL
Loads a model from a file instead of filling the
table manually. Saves the model currently stored in the table to a file. This
model is shown with blue color in the central panel.
LOADED MODEL H/V
Computes the HV curve for the loaded model shown
with black color in the central panel. The result will be shown with black line
in the HV Ratio panel. Uncheck this
box to delete the line.
Freq-min (Hz)
Represents the minimum frequency for calculation of the HV curve.
Freq-max (Hz)
Represents the maximum frequency for calculation of the HV curve.
Samples
Represents
the number of samples between the minimum and maximum frequency values for the
HV curve.
Rayleigh wave modes
Maximum number of Rayleigh wave modes to be included in the forward calculation of the HV curve (Enter zero to ignore the contributions of Rayleigh waves).
Love wave modes
Maximum number of Love wave modes to be included in the forward calculation of the HV curve (Enter zero to ignore the contributions of the Love waves).
BW Integration points
Represents the discretization of the wavenumber
integrands for calculation of the body waves contribution. If you don’t need
the body waves contribution, enter zero in this box.
Sampling
Select the sampling type
- Linear. Computes the HV curve for evenly spaced frequencies
- Log. Computes the HV curve for evenly spaced frequency logarithms.
Compute H/V
Computes the HV curve for the model in the table.
Save HV *.txt
Saves the HV curve values in a *.txt file with the following format:
|
Frequency |
Amplitude H/V |
|
0.1 |
0.5 |
|
0.2 |
0.52 |
|
|
|
Menu & Toolbar
Load HV
data Loads an H/V
curve from a file to be used as reference (e.g. for manual inversion of that
curve). H/V files with or without
standard deviation are both admissible.
Regularization factor.
Value used to generate small imaginary parts for frequencies or wavenumbers to
stabilize body wave integrals. The default value 0.01 is recommended. A value
of zero is also admissible for most of the models.
Exit Close
the window.
Toolbar
Print Figure. This tool prints out the main window.
Zoom in/out, pan & data cursor tools
are also available. Zoom in and return to original view have been
enabled for the click and double-click on the scroll wheel button.
Vs / Vp / Density Profile
- This panel shows the model stored in the
Parameters table with a blue line. The vertexes occupying odd positions (seen
from the top) can be manually moved by using the mouse. To do so, choose a
vertex and then drag and drop with the primary button, while the pointer is
showing a circumference.
- Double-click on the panel background (while the
standard pointer is shown) to compute the theoretical H/V curve for the current
model, just as pressing the Compute HV button.
- After loading a model from a file, a static
copy of that profile is preserved for reference in this panel, together with
the movable blue model. The loaded model is shown with black line. The
corresponding H/V will be shown with black line in the HV Ratio panel after checking the Loaded
Model HV box.
Mouse ButtonS
primary button Double click computes the
theoretical H/V curve for the current model, just as pressing the Compute HV. Use a single
click to exit from the contextual menu.
SCROLL WHEEL button Zoom in and zoom out to original view have been enabled for click and double-click on the scroll wheel button, respectively.
SECONDARY button It shows the contextual menu
described below and turns off the zoom if activated.
The following
menu is shown by clicking on the Profile
Panel background with the secondary button of the mouse. It provides
suitable options for controlling on the model display and modification and the drag and drop method.
Lock X/Y Axis Keeps either the layer thicknesses (case Y)
or the model property shown with the blue profile (case X) unchanged during a drag and drop procedure. It prevents for
undesired variations in the parameters during this manual operation.
Keep poisson
ratio If this option is checked, Vp
or Vs are changed automatically when the user moves the other velocity, in
order to keep de Poisson’s ratio unchanged. If this option is not checked, the
Poisson’s ratio is recomputed when the user acts on a (single) velocity. Note
that this choice also applies if the table is edited directly.
ChangE one Thickness
If this option is checked, the thickness of the layer located immediately above
a movable vertex is changed when the users drags the
vertex up and down. The thicknesses of the remaining layers are preserved.
Move one interface In this case, the user can change the depth of an interface while the positions of the remaining interfaces are preserved. Therefore, this action involves variation in thickness of the layers located immediately above and below the vertex, preserving the sum of their thicknesses.
Vp/Vs/density Profile Use these options to choose
the model property to be shown in the graphic profile(s).
Keep HV ratio If this option is checked,
the modifications made by means of the drag and drop method will preserve the
H/V curve. If Vp or Vs profile are displayed, then
all layer thicknesses and velocities (both Vp and Vs)
will be scaled by the same factor. If density profile is being displayed, the
only change allowed is rescaling the densities by a common factor.
- The HV ratio panel shows last computed
HV curve with a blue line. This curve is updated by pressing the Compute HV button or double-clicking on the Profile panel background.
- If a model has been loaded from file and the Loaded Model HV box has been checked, the
corresponding synthetic ratio will be displayed here with a black curve.
- If an auxiliary (e.g. experimental) H/V curve
has been loaded by means of the Load HV Data option, the
corresponding synthetic ratio will be displayed with a dashed red curve.
- Horizontal axis represents frequency (Hz) and vertical axis represents amplitude. If you prefer, the type of scales can be modified by pressing the secondary button and selecting the scale type.
APPENDIX A: Generation of random models & Requirements for model parameters table
INDEPENDENT MODEL
PARAMETERS
HV-INV
considers the following model parameters as independent variables: layer
THICKNESSES (or interface DEPTHS from version 2.1), VP VELOCITIES, VS
VELOCITIES and DENSITIES. This means that all these parameters or a
subset of them will be passed to the inversion algorithms for optimization (4 x
Number of layers – 1 parameters maximum). ‘Number of layers’ includes the
halfspace.
Whenever the
maximum equals the minimum value for any of these parameters, the parameter
will be kept constant (for the corresponding layer) and removed from the list
of independent variables.
During
operation of Interior Point method or Downhill-Simplex method (see Inversion settings > Inversion Type > LOCAL OPTIMIZATION
menu) the maximum and minimum Poisson’s ratios and the activated conditions on
VS(z) or VP(z) (see Inversion
settings > Low Parameters Zone menu)
are passed to the algorithms as constraint on the model space.
If the
inversion method requires generation of random models, HV-INV assumes UNIFORM
probability distributions for THICKNESSES and DENSITIES as well as for either VS
(default) OR VP, using the ranges in the Model Parameters
Table. Below we describe i) which velocity is
uniformly sampled (VS or VP), ii) the way of random
generation of the remaining velocity type and iii) the requirements for the
velocity limits in the model parameters table, depending on the program
settings.
HV-INV
includes a semi-analytical way for generation of random models with increasing
V(z) as depth increases, that is Vi ≤ Vi+1 where V represents
either VS or VP. This involves a non-trivial algorithm
that results equivalent to the independent uniform sampling of Vi
for the set of layers (within the corresponding limits Vi min Vi max) and the subsequent rejection of
the model if V(z) does not increase downwards. The inefficiency of this direct
approach as the number of layer increases (if velocity ranges overlap) has been
circumvented by computing and sampling the corresponding marginal or
conditional probability distributions for Vi (see Appendix A in Garcia-Jerez et al, 2016 for further details).
CASE #1: LOW VELOCITY ZONES ARE ALLOWED IN
BOTH Vs and Vp
This is the
simplest case, with these two conditions ticked in the Inversion settings > Low Parameters Zone menu. In this case, VS i are sampled with uniform probability in the
respective ranges [VS i min VS i max]. Note that no value of VS i in [VS i min VS
i max] should be forbidden by the settings of VP i
and νi ranges (ν stands for
Poisson’s ratio). That is, VS
i min and VS i
max should fulfil the conditions:
VS i min ≥ VP i min * sqrt((1-2* ν i max)/(2*(1- ν i max)))
VS i max ≤ VP i max * sqrt((1-2* ν i min)/(2*(1- ν i min)))
The program will ask the user for parameter updating if the introduced ranges
do not meet these requirements.
Once VS i has
been sampled, VP i is then sampled with
uniform probability inside the larger subinterval of [VP i min VP i max]
where VP i can be consistent with the
sampled VS i and the extreme values [ν i
min ν i max] written in the table.
Note that this is NOT the default setting.
CASE #2: LOW VELOCITY
ZONES ARE NOT ALLOWED IN VS
This is the
default setting, with the first condition unticked in the Inversion settings > Low Parameters Zone menu. The way of sampling VS(z)
is equivalent to sampling of VS i with
uniform probability in the respective ranges [VS i
min VS i max] and rejection of
models without monotonically increasing VS (that is, VS i ≤ VS
i+1 is required).
In the
current version, we require increasing values of VS i min and VS
i max, that is:
VS i min ≤ VS i+1 min
VS i max ≤ VS i+1 max
for every layer i. In
addition, no value of VS i
in [VS i min VS i
max] should be forbidden by the settings of VP i and νi ranges. That is, VS i min
and VS i max should also fulfil the
conditions:
VS i min ≥ VP i min * sqrt((1-2* ν i max)/(2*(1- ν i max)))
VS i max ≤ VP i max * sqrt((1-2* ν i min)/(2*(1- ν i min)))
The program will ask
the user for parameter updating if the introduced ranges do not meet these
requirements.
Once VS i has
been sampled, VP i is then sampled with
uniform probability inside the larger subinterval of [VP i min VP i max]
where VP i can be consistent with the
sampled VS i and the extreme values [ν i
min ν i max] written in the table.
CASE #3: LOW VELOCITY
ZONES ARE NOT ALLOWED IN VP
This is the
case if second condition is unticked in the Inversion
settings > Low Parameters Zone menu.
In this case VP and Vs are exchanged from Case #2:
The way of sampling VP(z) is equivalent to
sampling of VP i with uniform probability
in the respective ranges [VP i min VP
i max] and rejection of models without
monotonically increasing VP (that is, we require VP i ≤ VP i+1).
In the
current version, increasing values should be introduced for VP i min and VP
i max, that is:
VP i min ≤ VP i+1 min
VP i max ≤ VP i+1 max
for every layer i. In
addition, no value of VP i in [VP i min VP i max] should
be forbidden by the settings of VS i and νi ranges. That is, VP i min and VP i max
should also fulfil the conditions:
VP i min ≥ VS i min * sqrt(2*(1- ν i min) /(1-2* ν i min)))
VP i max ≤ VS i max * sqrt(2*(1- ν i max) /(1-2* ν i max)))
The program will ask the user for parameter updating if the introduced
ranges do not meet these requirements.
Once VP i has
been sampled, VS i is then sampled with
uniform probability inside the larger subinterval of [VS i min VS i max]
where VS i can be consistent with the
sampled VP i and the extreme values [ν i
min ν i max] written in the table.