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| | == Common pre-requisites for array processing - getting ready == | | == Common pre-requisites for array processing - getting ready == |
| | | | |
| − | === Loading signals ===
| + | For any array processing of ambient vibration data there are some |
| | + | common steps / pre-requisites regarding your signal recordings (selection |
| | + | of simultaneous recordings, coordinate insertion, signal grouping). |
| | + | Learn about all these options/preparatory steps in the following |
| | + | sub-chapters (linking to other parts of this wiki) |
| | | | |
| − | === View time series === | + | === Loading and viewing signals (time series) === |
| | | | |
| − | === Group signals ===
| + | Before processing your ambient vibration data, you need to [[Loading and viewing signals|load your signal files]] |
| | + | into geopsy and build a database. For checking the data quality of recordings and potential timing |
| | + | problems while recording you might want to use the [[Geopsy: Graphic|graphic viewer]] and the [[Geopsy: Chronogram|chronogram viewer]]. |
| | + | It is often also a good idea to compute [[Spectral_amplitudes|Spectra]] and or [[H/V_spectral_ratio|H/V spectral ratios]] (in case of 3C data) from your |
| | + | recordings to check the spectral energy content, problematic installations and ''1-D-ness'' of your region. |
| | | | |
| − | === Insert / edit station coordinates ===
| + | links to [[Loading_and_viewing_signals|loading and viewing signals]] |
| | | | |
| − | == F-K Toolbox (conventional f-k) == | + | === Group signals === |
| | | | |
| − | === Opening f-k toolbox - overview ===
| + | An array is defined by grouping a set of stations with common time base and recording simultaneously the properties |
| | + | of the wavefield at different, but closely co-located positions in space. After having built a database with many recordings, you may |
| | + | group individual arrays from the full set of signals. Both single component and three component arrays can be bundled |
| | + | together causing the array processing toolboxes to work on single or three component data, respectively. |
| | | | |
| − | [[Image:LepRing01OnlyVertical.png|thumb|right|400px|Signal viewer for one group of simultaneously recorded waveforms ready for array processing]] | + | For details on how to group signals for array processing take a look in [[Geopsy: Groups|groups]] |
| | | | |
| − | If you have followed all steps above you should have now a set of simultaneously recorded waveforms
| + | === Insert / edit station coordinates === |
| − | building a group and containing coordinates for each waveform. You can view this group of waveforms in the
| |
| − | data viewer and should get a picture like the one shown to the right.
| |
| | | | |
| − | Now you can open the conventional frequency wavenumber toolbox by pushing the plugin icon [[Image:FKToolboxPluginIcon.png|32px]].
| + | The final step in the preparation of your data set consists in attaching the stations coordinates to your |
| | + | recordings. Note that some [[Geopsy:_Supported_file_formats|waveform formats]] store the station coordinates and are eventually already included |
| | + | while importing the signal files. For most formats, however, this will be not the case. So, prepare you coordinate information |
| | + | from yor field book or your measurement devices (theodulite / (D)GPS device / etc.) and prepare some [[SciFigs: Multi-column Parser|multi-column ASCII file]] |
| | + | which can be read into [[geopsy]]. Details how to set coordinates can be found in section [[Geopsy: Set receivers|set receivers]] |
| | | | |
| − | The f-k toolbox should open and is now attached to the [[Geopsy: Signal Viewer|signal viewer]]. No other toolbox will now open with
| |
| − | the signals you have loaded to the signal viewer. In order to process the same data set by another
| |
| − | tool, you have to open them in a different signal viewer window. Note also, that once you close either
| |
| − | the f-k toolbox window or the signal viewer, there is no way to undo this action - you will have to start
| |
| − | again by opening a new signal viewer and attaching the f-k toolbox to it. Alternatively you can drag your
| |
| − | group directly into the f-k toolbox icon. Geopsy will automatically open a new exclusively attached signal viewer
| |
| − | for you.
| |
| | | | |
| − | <br style="clear: both"/>
| + | Since July 2018, high resolution and conventional FK toolboxes have been merged into a single FK toolbox. The text below should be reworked. |
| | | | |
| − | [[Image:FKToolboxTimeTab.png|thumb|left|300px|Conventional f-k toolbox]] | + | FK processing can be achieved inside Geopsy graphical interface or through a command line [[geopsy-fk]]. |
| | | | |
| − | Let's now have a close look to the individual parameters to be set in the f-k toolbox. There are four
| + | == F-K Toolbox (conventional f-k) == |
| − | main groups of parameters to be considered.
| |
| | | | |
| − | * Pre-processing parameters for excluding data from processing.
| + | The conventional frequency wavenumber technique as implemented in '''Geopsy''' is based |
| − | * Processing parameters for selecting time windows on the remaining data.
| + | on the simple idea of [[Wikipedia:Sensor array#Delay-and-sum_beamforming|''delay and sum'' (or ''shift and sum'')]]. |
| − | * Processing parameters for selecting narrow frequency bands for processing
| + | This technique may be effectuated equivalently in time domain or frequency domain. |
| − | * Parameters related to the output of results from f-k processing.
| + | In Geopsy we follow the frequency domain approach, as it is the most convenient and effective |
| | + | way to use this approach for determination of frequency dependent apparent velocity estimation |
| | + | (i.e dispersion curve estimation under the assumption of the wave field being composed of |
| | + | surface waves only). |
| | | | |
| − | These parameters are laid out in different parts of the f-k toolbox widget.
| + | The simultaneous waveform recordings of a group of spatially distributed stations are analyzed |
| − | Note, that the toolbox has three main tabs: ''Time'', ''Processing'' and ''Status''.
| + | in many narrow (mostly overlapping) frequency bands for individual analysis windows cut from the |
| | + | overall recordings. For each analysis window and frequency band, a grid search is performed |
| | + | in the wavenumber domain to effectively find the propagation properties of the most coherent |
| | + | and/or powerful plane wave arrival in the analysis window. Given the assumption of surface |
| | + | waves dominating the wave field, the apparent velocity equals the phase velocity of the |
| | + | surface wave at this particular frequency. |
| | | | |
| − | The pre-processing parameters for excluding data from processing follow the same argumentation as the
| + | Details of the signal processing can be found in the [[Array signal processing|array signal processing]] |
| − | data selection for other toolboxes like [[Geopsy: HV Toolbox|H/V processing]] and can be found in the
| + | page of this wiki or following one of the links below in section [[#See also|see also]] |
| − | ''Time tab''.
| |
| | | | |
| − | E.g. one can apply the so-called [[Geopsy: Anti Trigger|anti-trigger]]
| + | For learning about the detailed use of the f-k toolbox (geopsy plugin) in a tutorial like fashion, please [[FK|follow this link to FK]]. |
| − | approach for removing transient signals from the continuous recordings (i.e. excluding those time chunks from
| |
| − | processing). The anti-trigger approach can be applied to raw or filtered data. For the detailed
| |
| − | settings please look into the description of [[Geopsy: Anti Trigger|anti-trigger]].
| |
| − | Another, second, approach for excluding data from processing is provided via the [[Geopsy: Bad sample|bad sample]] approach.
| |
| − | For details in setting parameters with the bad sample approach check [[Geopsy: Bad sample|this page]].
| |
| | | | |
| − | <br style="clear: both"/>
| + | == High resolution frequency wavenumber Toolbox (Capon's method) == |
| − | | |
| − | === Parameter settings === | |
| − | | |
| − | ==== Parameters for time window settings ====
| |
| − | | |
| − | [[Image:FKTimeLimits.png|thumb|left|250px|Time limits layout group]]
| |
| − | There are two distinct inputs in the f-k toolbox for selecting time windows for processing.
| |
| − | First, it is necessary to specify start and end times of the overall data window for processing
| |
| − | by selecting entries ''From'' and ''To'' in the ''Time limits'' part of the ''Time tab'' of the fk-toolbox.
| |
| − | | |
| − | <br style="clear: both"/>
| |
| − | | |
| − | There are three options each for the
| |
| − | drop boxes ''From'' and ''To''. For the ''From'' entry you can select among [[Image:FKFromDropList.png|100px]].
| |
| − | | |
| − | * this time (entries in form xx '''h''' xx '''m''' xx.xxx '''s''')
| |
| − | * T0 (selects earliest common start time of all traces in group)
| |
| − | * during (selects time duration before end of the signals - either specified with ''End'' or ''this time'' as explained below)
| |
| − | | |
| − | For the ''To'' entry there are the options [[Image:FKToDropList.png|100px]]:
| |
| − | | |
| − | * this time (entries in form xx '''h''' xx '''m''' xx.xxx '''s''')
| |
| − | * End (selects latest common end time of all traces in group)
| |
| − | * during (selects time duration from start time as given above - either specified by ''this time'' or ''T0'')
| |
| − | | |
| − | Note: a selection of time limits as ''From: during'' && ''To: during'' is NOT valid.
| |
| − | | |
| − | [[Image:FKGeneralTab.png|thumb|left|300px|General tab]]
| |
| − | After selecting the overall data window for processing you finally have to specify the window length for
| |
| − | each single analysis window to be processed. You can find this ''Length'' parameter in the ''General'' tab inside the
| |
| − | ''Time tab''. There are again three options provided in a drop box [[Image:FKWindowLength.png|60px]].
| |
| − | | |
| − | * Exactly (takes exactly the time window length as provided in the corresponding editable box - unit is in seconds).
| |
| − | * At least (takes at least the time window length as given in the editable box, if possible it takes more - unit is in seconds).
| |
| − | * Freq. dep. (read: frequency dependent. Scales window length to the center frequency of each frequency band to be processed. The number given in the editable box corresponds to a window length in number of cycles for each frequency band. ''T'' is the center period of the band).
| |
| − | | |
| − | Note: the most common setting for the ''Length'' parameter is ''Freq. dep.'' with values between 20.00 to 50.00 meaning that 20 to 50 cycles
| |
| − | of the central signal period per frequency bands are used as window length. Example: for a frequency band with center frequency 2.5 Hz, the central period is 1/2.5 = 0.4 s. Therefore, with a frequency dependent ''Length'' setting of ''30.00 T'', the window length will be chosen as 12.00 s.
| |
| − | | |
| − | <br style="clear: both"/>
| |
| − | | |
| − | ==== Parameters for frequency band selection ====
| |
| − | | |
| − | [[Image:FKToolboxProcessingTab.png|thumb|left|300px|Processing tab of f-k toolbox]]
| |
| − | The selection of the frequency bands to be processed is driven by parameter inputs inside the ''Processing'' tab of the f-k toolbox.
| |
| − | In the ''Frequency sampling'' layout group you can specify the minimum and the maximum (central) frequencies for processing.
| |
| − | Between these two limits, the frequency axis will then be sampled according to the settings in the next line [[Image:FKFreqSampling.png|250px]].
| |
| | | | |
| − | There are two options to be specified, namely ''Step'' and ''Number of Samples''. For the ''Step'' parameter, there are two options to be selected
| + | The high resolution frequency wavenumber (hrfk) algorithm can be viewed as a '''''generalized''''' beamforming algorithm. It is implemented in '''Geopsy''' following the ideas of Capon (1969) |
| − | from a drop box: ''Log'' or ''Linear''. You might have guessed the meaning right away. When choosing ''Linear'' then the frequency axis is sampled
| + | <ref name="Capon (1969)"> Capon, J., High-Resolution Frequency-Wavenumber Spectrum Analysis, Proceedings of the IEEE, 57, No. 8, 1408-1419, 1969.</ref>, |
| − | linearly between the given limits and exactly ''Number of Samples'' will be distributed along the axis. Choosing ''Log'' accordingly samples the frequency axis logarithmically, i.e. frequency bands are ''close'' spaced at lower frequencies.
| + | using an auto-adaptive (optimal) complex spatial weighting scheme for analysis of narrowband stationary signals. It is one of the most common and preferred frequency wavenumber techniques applied to |
| | + | ambient vibration analysis. |
| | | | |
| − | Note: the sampled frequencies are taken as '''center frequencies''' in the processing scheme. Therefore, it is necessary to specify additionally a bandwidth
| + | The Capon beamformer relies on a very simple formula: |
| − | so that the processing can be performed in a finite but narrow frequency band. The bandwidth parameter is chosen as relative half-bandwidth in the ''FK-gridding'' layout group in the ''processing tab'' [[Image:FKGridding.png|250px]].
| |
| | | | |
| − | The frequency limits for each frequency band are computed as
| |
| | <math> | | <math> |
| − | [(1-bw)\cdot f_c,(1+bw)\cdot f_c]
| + | BP_{Capon}(\omega,\vec{k}) = \frac{1}{\vec{e}^\dagger(\omega,\vec{k})\underline{R}^{-1}(\omega)\vec{e}(\omega,\vec{k})} |
| | </math> | | </math> |
| − | where <math>f_c</math> is the central frequency and <math>bw</math> is the bandwidth parameter.
| |
| − |
| |
| − | Example: a setting of 0.10 for the bandwidth, a ''Linear'' spacing of 10 samples along the frequency axis from 1 Hz to 10 Hz will create the following frequency bands:
| |
| − | {| border="1" cellpadding="2" cellspacing="0"
| |
| − | |+ align="bottom" style="color:#e76700;" |Frequency band creation using ''Step'': Linear, ''Bandwidth'': 0.10, ''From'': 1 Hz, ''to'': 10 Hz
| |
| − | ! Freq. band index
| |
| − | ! Lower frequency limit
| |
| − | ! Center frequency
| |
| − | ! Upper frequency limit
| |
| − | |-
| |
| − | | 1
| |
| − | | 0.9 Hz
| |
| − | | 1.0 Hz
| |
| − | | 1.1 Hz
| |
| − | |-
| |
| − | | 2
| |
| − | | 1.8 Hz
| |
| − | | 2.0 Hz
| |
| − | | 2.2 Hz
| |
| − | |-
| |
| − | | 3
| |
| − | | 2.7 Hz
| |
| − | | 3.0 Hz
| |
| − | | 3.3 Hz
| |
| − | |-
| |
| − | | 4
| |
| − | | 3.6 Hz
| |
| − | | 4.0 Hz
| |
| − | | 4.4 Hz
| |
| − | |-
| |
| − | | 5
| |
| − | | 4.5 Hz
| |
| − | | 5.0 Hz
| |
| − | | 5.5 Hz
| |
| − | |-
| |
| − | | 6
| |
| − | | 5.4 Hz
| |
| − | | 6.0 Hz
| |
| − | | 6.6 Hz
| |
| − | |-
| |
| − | | 7
| |
| − | | 6.3 Hz
| |
| − | | 7.0 Hz
| |
| − | | 7.7 Hz
| |
| − | |-
| |
| − | | 8
| |
| − | | 7.2 Hz
| |
| − | | 8.0 Hz
| |
| − | | 8.8 Hz
| |
| − | |-
| |
| − | | 9
| |
| − | | 8.1 Hz
| |
| − | | 9.0 Hz
| |
| − | | 9.9 Hz
| |
| − | |-
| |
| − | | 10
| |
| − | | 9.0 Hz
| |
| − | | 10.0 Hz
| |
| − | | 11.0 Hz
| |
| − | |}
| |
| − |
| |
| − | ==== Parameters influencing the sampling of the wavenumber space ====
| |
| | | | |
| − | The basic idea of f-k processing consists of delaying the observed recordings at different stations according
| + | where <math>\underline{R}^{-1}(\omega)</math> is simply the inverse of the cross spectral matrix estimate and <math>\vec{e}</math> |
| − | to a particular horizontal wavenumber vector <math>\vec{k}=(k_x,k_y)^T</math> and computing the semblance coefficient <math>Semb(\omega,\vec{k})</math> (coherence measure, see Neiddell and Taner (1971)
| + | is the so-called steering vector summarizing the shift times for the harmonic plane wave with wavenumber vector <math>\vec{k}</math> |
| − | <ref name="Neiddell, N., and Turhan Taner, M. (1971)">Neiddell, N. and Turhan Taner, M.: Semblance and other coherency measures for multichannel data, Geophysics, 36, 482–497, 1971.</ref> or Douze and Laster (1979) <ref name="Douze, E. J. and Laster, S. J. (1979)">Douze, E. J. and Laster, S. J.: Statistics of semblance, Geophysics, 44, 1999–2003, doi:10.1190/1.1440953, short Note, 1979.</ref>) and/or beam power
| + | for each station within the array.The <math>\dagger</math> symbol presents the conjugate transpose operation. In the same notation, |
| − | <math>BP(\omega,\vec{k})</math> of the shifted stacked output of all array stations. | + | the conventional f-k algorithm would be formulated as: |
| − | Beampower and Semblance are defined as follows:
| |
| | | | |
| − | * Beampower:
| |
| − | <math>
| |
| − | BP(\omega,\vec{k}) = \sum_{\omega=\omega_l}^{\omega=\omega_h} \left[\sum_{i=1}^{i=N} X_i(\omega)\exp(j\vec{k}\vec{r}_i)\right]^2
| |
| − | </math>
| |
| − | * Semblance:
| |
| | <math> | | <math> |
| − | Semb(\omega,\vec{k}) = \frac{\sum_{\omega=\omega_l}^{\omega=\omega_h} \left[\sum_{i=1}^{i=N} X_i(\omega)\exp(j\vec{k}\vec{r}_i)\right]^2}
| + | BP_{conv.}(\omega,\vec{k}) = \vec{e}^\dagger(\omega,\vec{k})\underline{R}(\omega)\vec{e}(\omega,\vec{k}) |
| − | {N \sum_{\omega=\omega_l}^{\omega=\omega_h}\sum_{i=1}^{i=N} \left[X_i(\omega)\exp(j\vec{k}\vec{r}_i)\right]^2}
| |
| | </math> | | </math> |
| | + | |
| | + | The required computation of the inverse of the cross-spectral matrix results in some potential instabilities for the HRFK estimates. |
| | | | |
| − | In the above formulas <math>X_i(\omega)</math> is the observed record at station <math>i</math> at frequency <math>\omega</math>. The station coordinates in the array plane (usually earth's surface)
| + | The corresponding toolbox can be opened using the icon [[Image:HRFKPluginIcon.png|32px]]. It is nearly equivalent to the [[FK|conventional frequency wavenumber]] |
| − | are given by <math>\vec{r}_i</math>. Therefore, the time delays of a harmonic plane wave
| + | toolbox and the processing flow is exactly equivalent to the conventional technique. |
| − | propagating along the horizontal plane in direction of <math>\vec{k}</math> and apparent propagation
| |
| − | velocity <math>c</math> related to the absolute length of the wavenumber vector <math>\|\vec{k}\| = 2\pi/\lambda = 2\pi f/c = \omega/c</math> can be computed as <math>\tau = \vec{k}\vec{r}_i</math>.
| |
| − | Thus, the beampower/semblance measure the power/coherence of a plane wave propagating with wavenumber
| |
| − | vector <math>\vec{k}</math>.
| |
| − | By testing many wavenumber vectors in the wavenumber plane one tries to find those wavenumber
| |
| − | vectors which maximize the array output. The wavenumber found corresponds to a plane wave
| |
| − | along the surface crossing the seismometer array.
| |
| − | | |
| − | So, the problem at hand is a general optimization problem trying to find the maximum
| |
| − | of a function of two parameters (here <math>(k_x,k_y)^T</math>). In order to optimize
| |
| − | the grid search approach, the f-k toolbox relies on an iterative refined sampling strategy.
| |
| − | In the first iteration potential maxima are identified within the search limits of the
| |
| − | wavenumber plane using a ''coarse'' grid step in both axis directions <math>(k_x,k_y)</math>.
| |
| − | Note, that ''coarse'' is relative and must be fine enough in order to match the goal: identifying
| |
| − | potential maxima in the wavenumber plane. Therefore, the grid step must be as small enough as to
| |
| − | sample the peaky structure of the [[Wikipedia:Array Response|array reponse]] sufficiently fine
| |
| − | for not missing any small maximum in the optimization procedure. The [[Wikipedia:Array Response|array response]] is closely related to the 2-dimensional [[Wikipedia:Array geometry|array geometry]] (please check also
| |
| − | the tools [[Gpfksimulator|gpfksimulator]] and [[Warangps|warangps]] for this issue). In particular
| |
| − | the width of the main lobe peak - we refer to it as <math>k_{min}</math> - in the array response
| |
| − | (which is the one we are looking for) is approximately related to the overall aperture of the
| |
| − | array <math>D_{max}</math> by <math>k_{min} \approx 2\pi / D_{max}</math>.
| |
| − | | |
| − | In the current version of the f-k toolbox, the grid_step for sampling is automatically calculated as
| |
| − | <math>grid\_step = k_{min}/4</math>. Thus, any potential peak to be indentified in the wavenumber
| |
| − | domain will be sampled in either direction at least with four samples.
| |
| − | | |
| − | In a similar way, the maximum search radius in the wavenumber space (we call it ''grid_size'')
| |
| − | is related on one hand to the minimal wave speed that is expected to be observable in
| |
| − | the wave field. The second limit to be considered is related to the existence of strong side lobes or [[Wikipedia:Grating lobes|grating lobes]] in the [[Wikipedia:Array Response|array reponse]] function.
| |
| − | However, this limit is much more ambiguous than the ''grid_step''. Taking the viewpoint of scanning
| |
| − | the whole wavenumber space up to physical limits of wave propagation velocities will lead often to an
| |
| − | unnecessary computation time for lower frequency bands. On the other hand, restricting the ''grid_size''
| |
| − | too strongly on the apparent side lobes in the array reponse function will not allow for high frequencies to
| |
| − | obtain any result for slowly propagating waves. For an in-depth discussion of this issue [[K_max|look here]].
| |
| − | | |
| − | [[Image:FKGridding.png|thumb|left|300px|Gridding parameters for the wavenumber plane]]
| |
| − | The setting for the parameters regarding the gridding of the wavenumber plane are summarized in the
| |
| − | FK gridding layout group of the ''processing tab'' in the f-k toolbox (see picture to the left).
| |
| − | The values for '''grid_step''' and '''grid_size''' are computed automatically following the comments
| |
| − | provided about the minimum physical propagation velocity is set per default to 100 m/s. Depending
| |
| − | on the knowledge about your site conditions, you may or may not modify this setting. Be aware that
| |
| − | soft soil sites may find [[Wikipedia:Rayleigh wave|Rayleigh wave]] propagation velocities for the fundamental
| |
| − | mode [[Wikipedia:fundamental mode|fundamental mode]] well below 100 m/s!
| |
| − | | |
| − | | |
| − | <!--
| |
| − | In other application fields like e.g. radar
| |
| − | signal processing, the maximum wavenumber in the searhc space is related to half the distance form the
| |
| − | main lobe to the grating lobes. This is called
| |
| − | -->
| |
| − | <br style="clear: both"/>
| |
| − | | |
| − | ==== Parameter settings for output / saving results ====
| |
| − | | |
| − | [[Image:FKOutput.png|thumb|right|250px|Power spectrum maxima group]]
| |
| − | | |
| − | There will be two output files created. The name of the output file and the location (path) for storing the output files is specified in the
| |
| − | lowest layout group of the ''Processing'' tab titled '''Power spectrum maxima''' with the option
| |
| − | '''Output file'''. Pushing the button to the right of the editable textbox entry will open a
| |
| − | file browser and allows you to navigate to some location in your directory tree and specify
| |
| − | an output file name.
| |
| − | | |
| − | The extension of the main output file is '''.max'''. Additionally an other output file is created containing the processing parameters and information created during
| |
| − | the processing run. The extension of this file is set to '''.log'''. Parameters can be reloaded from an existing ''.log'' file in the '''Load parameters''' option form the ''Time tab'' of the f-k toolbox [[Image:FKLoadParams.png|250px]].
| |
| − | | |
| − | '''Note:''' the option '''Maximum number''' allows you to store more than the highest maximum
| |
| − | found in the optimized grid search in the wavenumber plane. In order to identify dispersion curve branches
| |
| − | at higher frequencies where a lot of aliasing may occur, it may prove beneficial to use this option (e.g.
| |
| − | by replacing the defualt number 1 to a value of 2 to 4).
| |
| − | | |
| − | There are two additional output filtering options that can be used to store not every result into the output files. The options are named '''Absolute th.''' and '''Relative th.''' and are explained in detail in
| |
| − | [[max2curve]]. The default values are 0 in both cases meaning that no filtering on the
| |
| − | output results will occur. As one is able to filter the result output at a later stage of the processing
| |
| − | chain using [[max2curve]] we certainly recommend not to change these values. As it is a post-processing
| |
| − | filtering there will be also no gain in processing speed.
| |
| − | <br style="clear: both"/>
| |
| − | | |
| − | === Testing parameter settings and starting f-k processing ===
| |
| − | | |
| − | [[Image:FKTestButton.png|thumb|left|300px|''Test'' button for checking parameter settings
| |
| − | and controlling data quality interactively]]
| |
| − | Once you finished all parameter settings as described above, you may
| |
| − | check the impact of the settings on the f-k processing by using the '''Test''' button
| |
| − | in the ''Time tab'' of the f-k toolbox.
| |
| − | | |
| − | <br style="clear: both"/>
| |
| − | [[Image:FKTestWavenumberDisplay.png|thumb|right|270px|interactive frequency wavenumber window browser]]
| |
| − | Pushing the '''Test''' button will open a frequency wavenumber window browser which allows you to
| |
| − | explore the f-k analysis for individual time windows and different frequency bands.
| |
| − | | |
| − | You can move the '''Time window''' slider to scroll in time over the selected data and
| |
| − | selecting the individually processed data windows. The results of the wavenumber analysis will be displayed
| |
| − | in the browser window as a colored map. The color scale is following the physical ''rainbow'' light spectrum,
| |
| − | associating violet with high and red with low power values. While sliding through time, you will notice
| |
| − | that the wavenumber results will change depending on the direction of wave propagation and
| |
| − | number of identifiable plane wave sources in the wave field.
| |
| − | | |
| − | Note: please be aware of caveats for the interpretation of multiple plane waves propagating at the same time
| |
| − | with distinct propagation properties (check the page on [[Multiple plane waves]] and explore this issue
| |
| − | using [[Gpfksimulator|gpfksimulator]]).
| |
| − | | |
| − | An interesting feature of this wavenumber browser window is the capability to control the consistency of
| |
| − | apparent wave velocity / horizontal slowness estimation. The black circle, which is displayed on top of the
| |
| − | wavenumber plane corresponds to a single propagation velocity / slowness without considering the propagation azimuth. For a wave field completely composed of a mixture of a single mode surface wave
| |
| − | (e.g. Rayleigh for vertical component analysis), we would therefore expect the peak in the wavenumber
| |
| − | domain always to be located at the same distance from the center of the wavenumber map.
| |
| − | Using the black circles helps to locate the relative distance of dominant peak(s) in the wavenumber domain
| |
| − | with respect to the center of the figure. For a single frequeny, we expect only the direction to change, but
| |
| − | finding the peak maximum always below the circle line.
| |
| − | | |
| − | <br style="clear: both"/>
| |
| − | | |
| − | === Contents of output file ===
| |
| − | | |
| − | The contents of the output file are best explained by having a look into the plain ascii file. Header lines are marked with a '''#''' as the first character of the line. The header entries of the max files provide the list of frequency bands that were processed. Other entries are obsolete but are kept for compatibility with former file formats used in other codes during the [http://sesame-fp5.obs.ujf-grenoble.fr/index.htm SESAME project].
| |
| − | | |
| − | # File generated by Geopsy, FK processing
| |
| − | # The process log is saved in file lep_ring01.log
| |
| − | # Header
| |
| − | # GRID_LAYOUT 0
| |
| − | # GRID_TYPE 0
| |
| − | # NPHI 72
| |
| − | # GRID_RESOL 250
| |
| − | # GRID_MAX 10
| |
| − | # NUM_BANDS 100
| |
| − | # Number of freq bands: 100
| |
| − | # Band 0 lower 0.45 center 0.5 upper 0.55
| |
| − | # Band 1 lower 0.468138 center 0.520153 upper 0.572169
| |
| − | # Band 2 lower 0.487007 center 0.541119 upper 0.595231
| |
| − | # Band 3 lower 0.506636 center 0.562929 upper 0.619222
| |
| − | # Band 4 lower 0.527057 center 0.585619 upper 0.644181
| |
| − | # Band 5 lower 0.548301 center 0.609223 upper 0.670146
| |
| − | # Band 6 lower 0.570401 center 0.633779 upper 0.697157
| |
| − | ...............
| |
| − | ...............
| |
| − | # Band 95 lower 19.2104 center 21.3449 upper 23.4794
| |
| − | # Band 96 lower 19.9847 center 22.2053 upper 24.4258
| |
| − | # Band 97 lower 20.7903 center 23.1003 upper 25.4103
| |
| − | # Band 98 lower 21.6282 center 24.0314 upper 26.4345
| |
| − | # Band 99 lower 22.5 center 25 upper 27.5
| |
| − | # seconds from start | cfreq | slow | az | math-phi | semblance | beampow
| |
| − | 30378.9 0.940916 5.15376 21.9177 68.0823 0.491473 63.9522
| |
| − | 30442.7 0.940916 3.18228 38.3304 51.6696 0.210185 65.6729
| |
| − | 30474.6 0.940916 3.85247 121.714 328.286 0.298423 69.8839
| |
| − | 30121.3 0.978841 4.66624 276.785 173.215 0.22663 70.1559
| |
| − | 30366.6 0.978841 6.27949 18.5945 71.4055 0.188424 71.1481
| |
| − | 30397.3 0.978841 6.79274 263.773 186.227 0.177627 63.9104
| |
| − | 30354.7 1.0183 5.79576 65.0153 24.9847 0.192576 71.5275
| |
| − | 30384.2 1.0183 6.28673 265.305 184.695 0.604842 71.942
| |
| − | 30458.6 0.978841 4.57822 4.44118 85.5588 0.23338 69.5334
| |
| − | ...........
| |
| − | ...........
| |
| − | | |
| − | | |
| − | The more interesting lines of the the '''.max''' output file start right below the header lines. The final header line describes the entries in the different columns:
| |
| − | | |
| − | * Time from start of processed data window in seconds
| |
| − | * Center frequency of the processed data window
| |
| − | * Estimated Slowness in s/km (this unit is also kept for compatibility reasons)
| |
| − | * Azimuth of propagation direction (direction of wavenumber/slowness vector) as measured from North via East
| |
| − | * Same angle as before, but now given in mathematical orientation (measured from ''East'' via ''North'')
| |
| − | * Semblance coefficient
| |
| − | * Beampower value given in dB ( <math>10\cdot \log(BP(\omega,\vec{k}))</math>
| |
| − | | |
| − | The Log file contents as displayed below contain three different sections, ''Init''. ''Parameter'' and
| |
| − | ''Process''. The init section reports which signals of which stations are included for processing and
| |
| − | the relative coordinates in units of m are reported for each station. The parameter section contains
| |
| − | a summary of all settings specified in the f-k toolbox setting and which are used for the processing.
| |
| − | The process section finally contains the summary of all windows processed for each frequency bands.
| |
| − | Examples for the three sections are given below:
| |
| − | | |
| − | ### Init Log ###
| |
| − | *********** vertical ***********
| |
| − | Add signal id 3 to component Vertical of station WA_WAU01 at 0.940538 -0.0688223 43.6439
| |
| − | Add signal id 6 to component Vertical of station WA_WAU02 at 5.32643 4.2312 43.7266
| |
| − | Add signal id 9 to component Vertical of station WA_WAU03 at 5.72042 -4.39827 42.5163
| |
| − | Add signal id 12 to component Vertical of station WA_WAU04 at 0.230693 -8.01841 42.9798
| |
| − | Add signal id 15 to component Vertical of station WA_WAU05 at -2.61856 -4.15918 43.5125
| |
| − | Add signal id 18 to component Vertical of station WA_WAU06 at -6.41618 0.965644 43.5362
| |
| − | Add signal id 21 to component Vertical of station WA_WAU07 at -3.53215 4.06034 43.6112
| |
| − | Add signal id 24 to component Vertical of station WA_WAU08 at 0.34882 7.3875 43.6973
| |
| − | Found 8 different stations
| |
| − | ### End Init Log ###
| |
| − | | |
| − | Parameter section - Note: when loading the log file via the ''Load Params'' button, those parameters
| |
| − | will be loaded into the f-k toolbox.
| |
| − | | |
| − | ### Parameters ###
| |
| − | WINDOW MIN LENGTH (s) = 50
| |
| − | WINDOW MAX LENGTH (s) = 30
| |
| − | WINDOW LENGTH TYPE (at least/exactly/freq. dep.) = freq. dep.
| |
| − | DO BAD SAMPLE TOLERANCE (y/n) = n
| |
| − | BAD SAMPLE TOLERANCE (s) = 0
| |
| − | DO WINDOW OVERLAP (y/n) = n
| |
| − | WINDOW OVERLAP (%) = 5
| |
| − | DO BAD SAMPLE THRESHOLD (y/n) = n
| |
| − | BAD SAMPLE THRESHOLD (%) = 99
| |
| − | ANTI-TRIGGERING ON RAW SIGNAL (y/n) = n
| |
| − | USED RAW COMPONENTS = y, n, y, y, y, y, y, y, y, y
| |
| − | RAW STA (s) = 1
| |
| − | RAW LTA (s) = 30
| |
| − | RAW MIN SLTA = 0.2
| |
| − | RAW MAX SLTA = 2.5
| |
| − | MINIMUM FREQUENCY = 0.5
| |
| − | MAXIMUM FREQUENCY = 25
| |
| − | INVERSED FREQUENCY = n
| |
| − | SAMPLES NUMBER FREQUENCY = 100
| |
| − | SAMPLING TYPE FREQUENCY (0=log, 1=linear)= 0
| |
| − | FROM TIME TYPE = 0
| |
| − | FROM TIME TEXT = 8h21m0s
| |
| − | TO TIME TYPE = 0
| |
| − | TO TIME TEXT = 9h29m0s
| |
| − | MIN K = 0.015
| |
| − | MAX K = 0.5
| |
| − | MIN V = 100
| |
| − | FREQ BAND WIDTH = 0.1
| |
| − | N MAXIMA = 3
| |
| − | OUTPUT FILE = /home/mao/GEOPSY_DOC_WORKSHOP/lep_ring01.max
| |
| − | ### End Parameters ###
| |
| − | | |
| − | Process Log section
| |
| − | | |
| − | ### Process Log ###
| |
| − | Process started at 2010-03-08 10:54:47
| |
| − | Adding window from 30060 to 30120 s.
| |
| − | Adding window from 30120 to 30180 s.
| |
| − | Adding window from 30180 to 30240 s.
| |
| − | Adding window from 30240 to 30300 s.
| |
| − | Adding window from 30300 to 30360 s.
| |
| − | Adding window from 30360 to 30420 s.
| |
| − | Adding window from 30420.1 to 30480.1 s.
| |
| − | Adding window from 30480.1 to 30540.1 s.
| |
| − | Adding window from 30540.1 to 30600.1 s.
| |
| − | Adding window from 30600.1 to 30660.1 s.
| |
| − | Adding window from 30660.1 to 30720.1 s.
| |
| − | Adding window from 30720.1 to 30780.1 s.
| |
| − | Adding window from 30780.1 to 30840.1 s.
| |
| − | Adding window from 30840.1 to 30900.1 s.
| |
| − | Frequency 1/100 0.5
| |
| − | Min Window length 60 seconds
| |
| − | Max Window length 60 seconds
| |
| − | 14 Time windows
| |
| − | .......
| |
| − | Adding window from 30919.1 to 30920.3 s.
| |
| − | Adding window from 30920.3 to 30921.5 s.
| |
| − | Adding window from 30921.5 to 30922.7 s.
| |
| − | Frequency 100/100 25
| |
| − | Min Window length 1.2 seconds
| |
| − | Max Window length 1.2 seconds
| |
| − | 713 Time windows
| |
| − | Process run in 00:01:20
| |
| − | Process ended at 2010-03-08 10:56:08
| |
| − | ### End Process Log ###
| |
| − | | |
| − | === Graphical display of f-k results using max2curve ===
| |
| − | | |
| − | For the interpretation of the results of the f-k processing one can use the graphical
| |
| − | tool [[max2curve]] or a combination of [[gphistogram]] and [[max2curve]].
| |
| − | | |
| − | You may call [[max2curve]] on the command line directly with the output file (.max file) produced
| |
| − | from your f-k processing.
| |
| − | | |
| − | $> max2curve your_output_file.max
| |
| − | | |
| − | [[Image:Max2CurveStartWindow.png|thumb|left|250px|Startup of max2curve for selecting histogram parameters]]
| |
| − | The utility program [[max2curve]] reads the contents of the f-k processing output file and computes a
| |
| − | [[Wikipedia:probability density function|probability density function (pdf)]] for slowness values for
| |
| − | each individual frequency band.
| |
| − | In order to produce the pdf from re-normalization of a binned histogram, max2curve needs
| |
| − | to know about the slowness limits and the number of bins to use.
| |
| − | As you have already looked at the picture on the left you will say: stop -
| |
| − | you probably mean an apparent velocity histogram or pdf of apparent velocity
| |
| − | values for a given frequency.
| |
| − | | |
| − | Smart thought, because the entries for the limits to be specified
| |
| − | ('''Minimum velocity''' and '''Maximum velocity''') are specified in
| |
| − | (apparent) velocity. And additionally one has to specify the '''Number of velocity classes'''.
| |
| − | Despite these confusing entries, the computation indeed takes place in the slowness domain.
| |
| − | | |
| − | As you are probably now fully confused, let me explain the thoughts behind it:
| |
| − | the computation takes place in slowness domain, because our estimates are obtained
| |
| − | from measuring time delays. Thus, the error of this measurement goes linear with
| |
| − | the time delay estimates and therefore is linear in slowness.
| |
| − | So, in terms of error propagation the computation makes clearly
| |
| − | sense in slowness domain and not(!) in the inversely related velocity domain.
| |
| − | | |
| − | So why, you may ask, give the limiting values in apparent velocity anyway?
| |
| − | Well, this is a concession to the (still) large part of the community who
| |
| − | this used to think in wave velocities and not the inverse property.
| |
| − | | |
| − | As already explained before in section xxx, it is possible to filter the result output
| |
| − | (.max file(s)) according to two extra criteria. Using the two entries under ''Sample selection'',
| |
| − | i.e. specify percentage values for '''Semblance''' and '''Beam power''',
| |
| − | it is possible to filter individual time window results falling below thresholds for
| |
| − | the semblance coefficients and beam power values. Although semblance is a normalized quantity,
| |
| − | we choose to develop a relative threshold scheme for filtering results for both semblance
| |
| − | and beam power values.
| |
| − | | |
| − | The percentage values should be understood as follows: for all entries in the .max output file,
| |
| − | the minimum and maximum values for the semblance coefficient and the beam power values
| |
| − | are used as lower (0%) and upper (100%) limits. Any given percentage value will compute
| |
| − | the threshold as absolute value in between these two limits by linear interpolation.
| |
| − | Note that the conditions are combined in the sense of a [[Wikipedia:logical and|logical and]].
| |
| − | Only time window that surpass the threshold for both values (semblance and beam power)
| |
| − | will be kept and passed to the histogram computation. Any other analysis window result will be skipped.
| |
| − | | |
| − | <br style="clear: both"/>
| |
| − | [[Image:Max2CurveAfterStartup.png|thumb|right|400px|max2curve main window after startup]]
| |
| − | After having set all necessary values (default values may be a good starting point :-)), [[max2curve]]
| |
| − | will start up with its main window consisting of two main parts. On the left, there is a
| |
| − | [[SciFigs: Curve Browser|curve browser]] element. On the right, the slowness histograms (pdfs)
| |
| − | are displayed as a color image for a log frequency scaling. Per default behaviour [[max2curve]]
| |
| − | computes the sample mean and sample standard deviation for each frequency histogram. Those values are stored in the curve browser as automatically determined mean curve with error estimates.
| |
| − | | |
| − | '''Note:''' Due to the variability in the estimation process, various disturbances in the wavefield
| |
| − | and violations of the [[General: Basic Array|basic array processing assumptions]] (e.g. plane wave arrivals)
| |
| − | and the well-known limitations of the array geometry in terms of aliasing and resolving capabilities,
| |
| − | the automatically determined dispersion curves look often very messy. In order not to get disturbed by this curve plotted over the histogram display, you can hide the curve by toggling the '''visible''' button an the
| |
| − | bottom of the curve browser.
| |
| − | | |
| − | <br style="clear: both"/>
| |
| − | | |
| − | [[Image:Max2CurveGridStatistics.png|thumb|left|250px|Grid statistics window of max2curve]]
| |
| − | | |
| − | | |
| − | | |
| − | <br style="clear: both"/>
| |
| − | | |
| − | === References ===
| |
| − | <references/>
| |
| − | | |
| − | == High resolution frequency wavenumber Toolbox (Capon's method) ==
| |
| − | | |
| − | === Overview ===
| |
| − | | |
| − | The high resolution frequency wavenumber (hrfk) algorithm follows the ideas of Capon (1969)
| |
| − | <ref name="Capon (1969)"> Capon, 1969. xxx ,IEEE Proceedings, , , .</ref>
| |
| − | | |
| − | . The corresponding toolbox that can be called using the following
| |
| − | plugin icon [[Image:HRFKPluginIcon.png|32px]] can be used straightforward in nearly the same way
| |
| − | as the conventional f-k toolbox.
| |
| − | | |
| − | === Parameter settings ===
| |
| − | | |
| − | There is just one more (optional!) parameter that can be set. The calcuation for the hrfk algorithm involves the inversion of
| |
| − | the cross spectral matrix estimate from the data. In order to guarantee a numerically stable solution for the Matrix inversion,
| |
| − | a regularization parameter (damping constant) can be provided to load the diagonal of the cross spectral matrix before inversion.
| |
| − | | |
| − | Note: by doing so (using a damping constant), you will smooth out your wavenumber power spectra and eventually not obtain
| |
| − | much better results compared to the conventional beamforming approach.
| |
| − | | |
| − | === Contents of output file ===
| |
| − | | |
| − | The output files for the high resolution f-k approach are equivalent and compatible to the output
| |
| − | files produced with the conventional approach. For details check [[FK: output files|the following page]].
| |
| − | | |
| − | === Graphical display of f-k results using max2curve ===
| |
| − | | |
| − | The usage of [[max2curve]] is equivalent to using this tool with .max files produced
| |
| − | with the conventional f-k approach. Check [[FK: max2curve|the following page]] for details.
| |
| | | | |
| − | === References ===
| + | For learning about the detailed use of the HRFK toolbox (geopsy plugin) in a tutorial like fashion, please [[HRFK| follow this link to HRFK]] |
| − | </references>
| |
| | | | |
| | == Modified Spatial Autocorrelation (MSPAC) Toolbox == | | == Modified Spatial Autocorrelation (MSPAC) Toolbox == |
| | | | |
| − | === Overview === | + | The Modified Spatial Autocorrelation (MSPAC) was introduced by Bettig et al. (2001) <ref name="Bettig et al. (2001)"> Bettig B., P.-Y. Bard, F. Scherbaum, J. Riepl, F. Cotton, C. Cornou, D. Hatzfeld, 2001. Analysis of dense array measurements using the modified spatial auto-correlation method (SPAC). Application to Grenoble area., Boletin de Geofisica Teoria e Applicata, 42, 3-4, 281-304.</ref> after pioneer paper of Aki (1957)<ref name="Aki (1957)."> Aki, K., 1957. Space and Time Spectra of Stationary Stochastic Waves, with Special Reference to Microtremors, Bull. Earthq. Res. Inst. Tokyo, 35, 415-457. </ref> and allows to compute average spatial autocorrelation coefficients for any arbitary array configurations. As SPAC technique, MSPAC relies on a stochastic ambient noise wavefield stationary in both time and space. Aki (1957) showed, that, given this assumption, the existing relation between the spectrum densities in space and time can be used to derive the following expression for a plane wave narrow-band filtered around <math>\omega_0</math>: |
| − | [[Image:LepRing01OnlyVertical_SPAC_VIEWALL_NEW.png|thumb|right|600px]]
| |
| | | | |
| − | The Modified Spatial Autocorrelation (MSPAC) was introduced by Bettig et al. (2001) <ref name="Bettig et al. (2001)"> Bettig B., P.-Y. Bard, F. Scherbaum, J. Riepl, F. Cotton, C. Cornou, D. Hatzfeld, 2001. Analysis of dense array measurements using the modified spatial auto-correlation method (SPAC). Application to Grenoble area., Boletin de Geofisica Teoria e Applicata, 42, 3-4, 281-304.</ref> after pioneer paper of Aki (1957)<ref name="Aki (1957)."> Aki, K., 1957. Space and Time Spectra of Stationary Stochastic Waves, with Special Reference to Microtremors, Bull. Earthq. Res. Inst. Tokyo, 35, 415-457. </ref>. This method allows computing spatial autocorrelation coefficients for any arbitary array layouts.
| + | <math> |
| | + | \overline{\rho(\omega_0 , r)} = \frac{1}{\pi}\int^{0}_{\pi}\rho(\omega_0,r,\varphi)d(\varphi)= J_0(\frac{\omega r}{c(\omega_0)}) |
| | + | </math> |
| | | | |
| − | After viewing the group of component waveforms in the data viewer, open the MSPAC toolbox by pushing the following plugin icon [[Image:SPACToolboxPluginIcon.png|18px]]. Alternatively you can drag your group directly into the SPAC toolbox icon. The MSPAC toolbox is now attached to the signal viewer.
| + | <math>\overline{\rho(\omega_0 , r)}</math> represents the averaging over azimuth of spatial autocorrelations <math> {\rho(\omega_0 , r, \varphi)} = cos(\frac{\omega_0 r}{c(\omega_0)}cos(\theta-\varphi)) </math> where <math>{\theta}</math> is the wave azimuth and <math>{\varphi}</math> the direction azimuth between stations pairs. |
| − | Four windows should then open as displayed in the figure on the right: (1) a signal time viewer, (2) a map displaying array station locations, (3) a map displaying co-array stations location and (4) the SPAC processing toolbox. This processing toolbox is composed of four tabs:
| |
| − | * ''Rings'' tab allows to design rings from the co-array sensors map;
| |
| − | * ''Time'' tab allows to select the time limits and part of the signals to be processed and the time window length;
| |
| − | * ''Output'' tab is used to set up the frequency band to be processed together with the sampling scale type (linear, log) and number of frequencies, and the output filename
| |
| − | * ''Status'' tab provides information about computing status.
| |
| | | | |
| − | Note that you may process only vertical component or the three components after Bettig et al. (2001) <ref name="Bettig et al. (2001)"> Bettig B., P.-Y. Bard, F. Scherbaum, J. Riepl, F. Cotton, C. Cornou, D. Hatzfeld, 2001. Analysis of dense array measurements using the modified spatial auto-correlation method (SPAC). Application to Grenoble area., Boletin de Geofisica Teoria e Applicata, 42, 3-4, 281-304.</ref> and Köhler et al. (2007) <ref name="Köhler et al. (2007)"> Köhler, A., M. Ohrnberger, F. Scherbaum, M. Wathelet and C. Cornou, 2007. Assessing the reliability of the modified three-component spatial autocorrelation technique. Geophysical Journal International, Volume 168, Issue 2: 779-796. doi: 10.1111/j.1365-246X.2006.03253.x. </ref>. For both cases, spatial autocorrelation coefficients
| |
| − | will be computed and results can be displayed by using [[Max2curve|max2curve]] command tool for three- and vertical components and by using '''spac2disp''' for vertical component. However, computation of Rayleigh and Love waves phase velocities from three-component processing is not yet available. We thus recommend to process only the vertical component.
| |
| | | | |
| − | <br style="clear: both"/>
| |
| | | | |
| − | [[Image:LepRing01OnlyVertical_SPAC_SELECT_RINGS_FIRST_STEP.png|thumb|left|500px|Steps involved in the design of the rings.]]
| + | Application of the SPAC technique (as well as further derived techniques like ESAC) requires perfect shaped arrays (circular, semi-circular, nested triangles) which may be difficult to achieve in urban environment. To overcome these difficulties, Bettig et al. (2001) <ref name="Bettig et al. (2001)"> Bettig B., P.-Y. Bard, F. Scherbaum, J. Riepl, F. Cotton, C. Cornou, D. Hatzfeld, 2001. Analysis of dense array measurements using the modified spatial auto-correlation method (SPAC). Application to Grenoble area., Boletin de Geofisica Teoria e Applicata, 42, 3-4, 281-304.</ref> suggested to use the co-array - which is defined as the set of all possible combinations of two array sensors (Haubrich, 1968 <ref name="Haubrich (1968)"> Haubrich, R.A., 1968. Array Design, Bull. seism. Soc. Am., 58(3), 977–991. </ref>) - to divide the array into several semicircular sub-arrays. Each sub-array (hereafter called ring) is thus composed of several sensors pairs. To account for ring thickness, an azimuthal and radial integration is then needed to compute averaged spatial autocorrelation values <ref name="Bettig et al. (2001)"> Bettig B., P.-Y. Bard, F. Scherbaum, J. Riepl, F. Cotton, C. Cornou, D. Hatzfeld, 2001. Analysis of dense array measurements using the modified spatial auto-correlation method (SPAC). Application to Grenoble area., Boletin de Geofisica Teoria e Applicata, 42, 3-4, 281-304.</ref>: |
| | | | |
| − | === Defining rings ===
| + | <math> |
| − | | + | \overline{\rho_{r_1,r_2}(\omega)} = \frac{2}{r_2^2-r_1^2} \int^{r_1}_{r_2}r.J_0(\frac{\omega r}{c(\omega)})dr = \frac{2}{r_2^2-r_1^2}\frac{c(\omega)}{\omega}[r.J_1(\frac{\omega r}{c(\omega)})]^{r_1}_{r_2} |
| − | Few words about selection of the rings ....
| + | </math> |
| − | | |
| − | In ''Rings'' tab:
| |
| − | * Push on [[Image:SPACToolbox_AddIcon.png|50px]] button to add a ring
| |
| − | * Specify the inner and outer circles radii of the ring (see step (1) in the figure on the left)
| |
| − | * Press on [[Image:SPACToolbox_OptimizeIcon.png|50px]] buttom to adjust the inner and outer circles radii in order to fit at best sensors pairs location (see step (2) in the figure on the left). This ring is composed of 10 pairs of sensors as specified in the "Pairs" column.
| |
| − | * Define the next rings (see step (3) in the figure on the left). Note that you can associate a specific color to each ring.
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| − | * Note that you add or remove specific ring by pushing on the [[Image:SPACToolbox_AddIcon.png|50px]] or [[Image:SPACToolbox_RemoveIcon.png|50px]] buttons
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| − | * Once done, click on [[Image:SPACToolbox_SaveIcon.png|50px]] to save your rings definition.
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| − | [[Image:LepRing01OnlyVertical_SPAC_SAVE_RINGS.png|thumb|center|300px]]
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| − | * The rings definition can next be loaded by pushing on the [[Image:SPACToolbox_LoadIcon.png|50px]] button
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| − | * The rings definition are saved in ascii format with the following format: inner circle radius (in meters), outer circle radius (in meters), RGB color code associated to the ring
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| − | | |
| − | # MinRadius MaxRadius Red Green Blue
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| − | 4.23 6.58 255 0 0 10
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| − | 7.42 9.33 0 255 0 8
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| − | 11.17 15.41 0 0 255 10
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| − | | |
| − | | |
| − | <br style="clear: both"/> | |
| − | | |
| − | === Parameters settings ===
| |
| − | ==== Time limits ====
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| − | | |
| − | In ''Time'' tab:
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| − | [[Image:SPACToolbox_TimeLimits.png|thumb|right|300px]]
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| − | * Select the time limits to be processed. You can either use the begin ('''To''') and end time ('''End''') of the time series, either specify the begin and end time (''' This Time ''') or specify the time duration ('''Duration''')
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| − | | |
| − | <br style="clear: both"/>
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| − | | |
| − | [[Image:SPACToolbox_WindowLength.png|thumb|right|300px]]
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| − | * Specify the time window length for each single analysis window to be processed. You can either choose a window length that is inversely proportionnal to the center frequency of interest ('''Freq. Dep.''') or a fixed time window length ('''Exactly'''). Note that the most common setting if '''Freq. Dep.''' with values ranging from 50 to 100.
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| − | | |
| − | * By pushing on [[Image:SPACToolbox_Parameters.png|80px]] , you can also load the .log file containing all the input parameters saved from a previous computation (see '''Editing spatial autocorrelation curves''' for complementary information).
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| − | | |
| − | <br style="clear: both"/>
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| − | | |
| − | ==== Defining the outputs and starting the computation ====
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| − | | |
| − | In ''Output'' tab:
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| − | [[Image:SPACToolbox_FrequencyLimits.png|thumb|left|300px]]
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| − | * Choose the minimum and the maximum center frequencies for processing.
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| − | * Specify the number of frequencies and the sampling scale type (constant on a linear or logarithmic scale) between the minimum and maximum center frequencies previously defined.
| |
| − | <br style="clear: both"/>
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| − | [[Image:SPACToolbox_OutputName.png|thumb|left|300px]]
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| − | There are two ways for saving computation results:
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| − | * check the ''Dinver target'' box and provide the output file name in the editing box. The ''.target'' output file will contain only the statistics: mean, standard deviation and number of processed windows for each autocorrelation sample. This file contains also the list of selected rings. This file can be further edited by using '''spac2disp''' tool.
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| − | * check the ''All results for each time window'' box and provide the output file name in the editing box. The ''.max'' output file will contain the autocorrelation results for each processed time window. This file can be further edited by using '''max2curve''' tool.
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| − | | |
| − | * Then, click on [[Image:SPACToolbox_StartIcon.png|50px]] to start the computation. Status of the computation can be checked in the "Status" sub-menu as displayed below.
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| − | [[Image:SPACToolbox_ComputationStatus.png|thumb|center|200px]]
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| − | | |
| − | <br style="clear: both"/>
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| − | | |
| − | === Graphical display of spatial autocorrelation curves / contents of output files ===
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| − | | |
| − | The spatial autocorrelation coefficients and the parameters used for the computation are saved in a ''.target'' (or ''.max'') file and a .log ascii file, respectively. The .log file is saved in the same folder as the ''.target'' (''.max'') file and contains all parameters used for the computation (see below). This .log file can be re-used in further computation by pushing on [[Image:SPACToolbox_Parameters.png|80px]] button in the MSPAC toolbox.
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| − | | |
| − | ### Init Log ###
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| − | *********** Selected from All signals ***********
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| − | Add signal id 3 to component Vertical of station WA_WAU01 at 0.940538 -0.0688223 0
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| − | Add signal id 6 to component Vertical of station WA_WAU02 at 5.32643 4.2312 0
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| − | Add signal id 9 to component Vertical of station WA_WAU03 at 5.72042 -4.39827 0
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| − | Add signal id 12 to component Vertical of station WA_WAU04 at 0.230693 -8.01841 0
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| − | Add signal id 15 to component Vertical of station WA_WAU05 at -2.61856 -4.15918 0
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| − | Add signal id 18 to component Vertical of station WA_WAU06 at -6.41618 0.965644 0
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| − | Add signal id 21 to component Vertical of station WA_WAU07 at -3.53215 4.06034 0
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| − | Add signal id 24 to component Vertical of station WA_WAU08 at 0.34882 7.3875 0
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| − | Found 8 different stations
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| − | Found 28 couples
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| − | ### End Init Log ###
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| − | ### Parameters ###
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| − | WINDOW MIN LENGTH (s) = 50
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| − | WINDOW MAX LENGTH (s) = 100
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| − | WINDOW LENGTH TYPE (at least/exactly/freq. dep.) = freq. dep.
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| − | DO BAD SAMPLE TOLERANCE (y/n) = n
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| − | BAD SAMPLE TOLERANCE (s) = 0
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| − | DO WINDOW OVERLAP (y/n) = n
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| − | WINDOW OVERLAP (%) = 5
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| − | DO BAD SAMPLE THRESHOLD (y/n) = n
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| − | BAD SAMPLE THRESHOLD (%) = 99
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| − | ANTI-TRIGGERING ON RAW SIGNAL (y/n) = n
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| − | USED RAW COMPONENTS = y, n, y, y, y, y, y, y, y, y
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| − | RAW STA (s) = 1
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| − | RAW LTA (s) = 30
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| − | RAW MIN SLTA = 0.2
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| − | RAW MAX SLTA = 2.5
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| − | MINIMUM FREQUENCY = 2
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| − | MAXIMUM FREQUENCY = 30
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| − | INVERSED FREQUENCY = n
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| − | SAMPLES NUMBER FREQUENCY = 100
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| − | SAMPLING TYPE FREQUENCY (0=log, 1=linear)= 0
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| − | FROM TIME TYPE = 1
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| − | FROM TIME TEXT = 8h19m55.0000s
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| − | TO TIME TYPE = 1
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| − | TO TIME TEXT = 8h35m22.8400s
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| − | OUTPUT TARGET FILE = y
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| − | OUTPUT STMAP FILE = n
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| − | OUTPUT MAX FILE = n
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| − | OUTPUT FILE = D:/WIKI-GEOPSY/DOCWORKSHOP/PROCESSING/LEP_ARRAY01_SPAC
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| − | ### End Parameters ###
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| − | ### Process Log ###
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| − | --- Ring (4.23 m, 6.58 m)
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| − | WA_WAU03-WA_WAU04 azimuth=33.4024 weight=37.0045
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| − | WA_WAU07-WA_WAU08 azimuth=40.6065 weight=5.5156
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| − | WA_WAU01-WA_WAU02 azimuth=44.4336 weight=3.20576
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| − | WA_WAU06-WA_WAU07 azimuth=47.018 weight=2.26964
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| − | WA_WAU01-WA_WAU05 azimuth=48.9729 weight=39.7101
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| − | WA_WAU04-WA_WAU05 azimuth=126.438 weight=38.7832
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| − | WA_WAU05-WA_WAU06 azimuth=126.539 weight=5.42433
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| − | WA_WAU01-WA_WAU07 azimuth=137.287 weight=5.64569
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| − | WA_WAU01-WA_WAU03 azimuth=137.831 weight=5.16714
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| − | WA_WAU02-WA_WAU08 azimuth=147.621 weight=37.274
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| − | | |
| − | ==== Graphical display of spatial autocorrelation estimates and corresponding phase velocities ====
| |
| − | | |
| − | The .target file is [[Xml files|Xml file]]. Graphical display of this file can be done by importing the .target file in the ''target'' menu of ''Dinver''([[Geopsy:Dispersion_curve_inversion#Importing_the_dispersion_curve_to_fit|Importing the autocorrelation curve to fit]]) or by using a post-processing tool called ''spac2disp'' that can be launched either from the windows Geopsy menu either from command line.
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| − | | |
| − | $ spac2disp LEP_ARRAY01_SPAC.target
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| − | | |
| − | [[Image:SPACToolbox_SPAC2DISP_Overview.png|thumb|left|600px]]
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| − | | |
| − | * Spatial autocorrelation curves are displayed for each ring in the right panel.
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| − | * Histograms of phase velocities derived from overallset of spatial autocorrelation values are displayed in the lower left panel. Histograms are computed within the velocity range [''Vmin - Vmax''] - equidistandly sampled in slowness, however - with a number of cells ''Cells'' specified in the lower panel.
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| − | | |
| − | [[Image:SPACToolbox_SPAC2DISP_Cells.png|center|200px]]
| |
| − | | |
| − | * .....
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| − |
| |
| − | [[Image:SPACToolbox_SPAC2DISP_CursorBottomRightPanel.png|center|400px]]
| |
| − | | |
| − | <br style="clear: both"/> | |
| − | | |
| − | [[Image:SPACToolbox_SPAC2DISP_Snapshots.png|center|500px]]
| |
| − | | |
| − | | |
| − | ==== Graphical display of all spatial autocorrelation estimates ====
| |
| − | | |
| − | The ''.max'' ASCII file contains spatial autocorrelation estimates for all processed windows, component and ring.
| |
| − | Format of this file is the following:
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| − | * ''seconds from start' indicates the begin time of the processed window
| |
| − | * ''cfreq'' indicates (center) frequency in Hz
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| − | * ''icomp'' indicates component index (0: vertical, 1: radial, 2: transverse)
| |
| − | * ''iring'' indicates the ring index (from 0), in the same order as the one specified in the ''Rings'' tab of the MSPAC toolbox.
| |
| − | * ''autocorr'' is the spatial autocorrelation value
| |
| − | | |
| − | # File generated by Geopsy, SPAC processing
| |
| − | # The process log is saved in file D:/WIKI-GEOPSY/DOCWORKSHOP/PROCESSING/LEP_ARRAY01_SPAC.log
| |
| − | # icomp can take values 0 (vertical), 1 (radial), and 2 (transverse)
| |
| − | # seconds from start | cfreq | icomp | iring | autocorr
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| − | 30020 2 0 0 0.718134
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| − | 30020 2 0 1 0.839546
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| − | 30020 2 0 2 0.654078
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| − | 30020 2.05546 0 0 0.681075
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| − | 30020 2.05546 0 1 0.801923
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| − | 30020 2.05546 0 2 0.710762
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| − | 30070 2 0 0 0.839481
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| − | | |
| − | The ''.max'' file can also be edited by using '''max2curve''' tool on a command line or from the Windows Geopsy menu.
| |
| − | | |
| − | $max2curve LEP_ARRAY01_SPAC.max
| |
| − | | |
| − | [[Image:SPACToolbox_SPAC2DISP_Max2curveLoadingParameters.png|thumb|right|200px|Loading parameters when using max2curve to display MSPAC results]]
| |
| − | | |
| − | A popup window will then appear allowing to specify:
| |
| − | * component and ring indices as defined above
| |
| − | * the miminum and maximum mspac values and the number of cells to be used for defining the histogram properties
| |
| | | | |
| − | [[Image:SPACToolbox_SPAC2DISP_Max2curveHistograms.png|thumb|left|500px|Histograms of mspac estimates]]
| + | where <math> r_1 </math> and <math> r_2 </math> are the inner and outer radius of the ring, respectively. |
| | | | |
| − | Histograms are displayed on the right panel. The [[Scifigs:color palette|color scale]] indicates the histogram values. By default,
| + | This method allows computing spatial autocorrelation coefficients for any arbitary array configurations. The design of rings results from a compromise between a number of sensors pair per ring as large as possible and a ring thickness as small as possible. |
| − | histograms are overlaid by the mean spatial autocorrelation +/- standard deviation curves whose corresponding values are indicated in the left panel. This 'by default' curves can be either hidden by unchecking the ''visible'' box on the bottom part of the left panel or remove by using the ''Remove'' option in the [[SciFigs:_Curve_Browser|actions]] tab attached to the curve. Note also that you can perform several [[Scifigs:SciFigs:_Curve_Browser|actions]] on this curve. Similarly to FK or HRFK, the ''Grid Statistics'' window allows you to edit the [[Max2curve|statistics]] in order to filter out spurious or
| |
| − | unwanted estimates.
| |
| | | | |
| − | [[Image:SPACToolbox_SPAC2DISP_Max2curveStatisticsToolbox.png|thumb|right|200px|thumb|center|Statistics toolbox]]
| |
| | | | |
| | + | For learning the usage of the MSPAC Toolbox in detail in a tutorial like fashion, please [[MSPAC| follow this link to MSPAC]]. |
| | | | |
| | + | == See also == |
| | | | |
| − | <br style="clear: both"/>
| + | * [[Wikipedia:Array_signal_processing|Wikipedia page on array signal processing]] |
| | + | * [http://www.geopsy.org geopsy.org] |
| | + | * etc. |
| | | | |
| − | === References ===
| + | == References == |
| | <references/> | | <references/> |
. It is nearly equivalent to the conventional frequency wavenumber
toolbox and the processing flow is exactly equivalent to the conventional technique.
![{\displaystyle {\overline {\rho _{r_{1},r_{2}}(\omega )}}={\frac {2}{r_{2}^{2}-r_{1}^{2}}}\int _{r_{2}}^{r_{1}}r.J_{0}({\frac {\omega r}{c(\omega )}})dr={\frac {2}{r_{2}^{2}-r_{1}^{2}}}{\frac {c(\omega )}{\omega }}[r.J_{1}({\frac {\omega r}{c(\omega )}})]_{r_{2}}^{r_{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af9a2b6c819b058ec778207e51d41a812dfd33ff)
