Dispersion curve target - Revision history

Marc: /* Modes */

2010-03-29T14:49:10Z

Modes

Alexandros Savvaidis at 09:49, 8 March 2010

2010-03-08T09:49:55Z

Alexandros Savvaidis at 09:48, 8 March 2010

2010-03-08T09:48:25Z

Alexandros Savvaidis at 09:47, 8 March 2010

2010-03-08T09:47:15Z

Alexandros Savvaidis: /* Loading curves */

2010-03-08T09:45:28Z

Loading curves

Marc: /* Averaging or merging curves */

2009-11-06T12:53:03Z

Averaging or merging curves

Marc: /* Averaging or merging curves */

2009-11-05T20:31:12Z

Averaging or merging curves

Marc: /* Averaging or merging curves */

2009-11-05T20:21:23Z

Averaging or merging curves

Marc: /* Averaging or merging curves */

2009-11-05T20:16:15Z

Averaging or merging curves

Marc: /* Averaging or merging curves */

2009-11-05T20:13:30Z

Averaging or merging curves

← Older revision		Revision as of 14:49, 29 March 2010
Line 13:		Line 13:

	If you want to use the '''Mode Guess''' feature, you can specify several modes. For each sample, the misfit is then computed for all modes given in the table. The best match is kept in the global curve misfit.		If you want to use the '''Mode Guess''' feature, you can specify several modes. For each sample, the misfit is then computed for all modes given in the table. The best match is kept in the global curve misfit.
		+
		+	== Velocity standard deviations ==
		+
		+	In all computations, slowness is considered rather than velocity. In a [[SciFigs: Curve Browser\|curve browser]], when loading new curves from text files, the [[SciFigs: multicolumn file\|parser]] accepts input curves in both domains: velocity or slowness. For the average value, the input values are just inverted. For the standard deviation a slightly more complex transformation is carried on but only an approximate standard deviation in the slowness domain can be obtained (arithmetic average).
		+
		+	<math>\begin{array}{lll}
		+	dS & = & \frac{(\frac{1}{V-dV}-\frac{1}{V})+(\frac{1}{V}-\frac{1}{V+dV})}{2}\\
		+	& = & \frac{dV}{(\frac{1}{S})^2-dV^2}
		+	\end{array}
		+	</math>
		+
		+	Where,
		+	* V and dV are the input velocity and its standard deviation;
		+	* S and dS are the slowness and its standard deviation;

← Older revision		Revision as of 09:49, 8 March 2010
Line 13:		Line 13:

	If you want to use the '''Mode Guess''' feature, you can specify several modes. For each sample, the misfit is then computed for all modes given in the table. The best match is kept in the global curve misfit.		If you want to use the '''Mode Guess''' feature, you can specify several modes. For each sample, the misfit is then computed for all modes given in the table. The best match is kept in the global curve misfit.
−
−	~~== Averaging or merging curves ==~~
−
−	Once you have at least two curves loaded, you can select the following action to average or merge several curves. A dialog box lets you select the curve to average or merge. Merging applies to non-overlapping frequency ranges, average applies to overlapping ranges. Both actions are automatically performed. If more than two curves are selected, curves are averaged or merged one by one using the same process:
−
−	* Build a vector with X values from the two curves.
−	* Resample both curves with this common X sampling.
−	* For each X value and for both curves, we have mean, variance, and weight (number of values used to compute statistics)
−
−	~~<math>~~
−	~~\begin{array}{ll}~~
−	~~\mu^{(1)}, \sigma^{(1)}, w^{(1)} \\~~
−	~~\mu^{(2)}, \sigma^{(2)}, w^{(2)}~~
−	~~\end{array}~~
−	~~</math>~~
−
−	~~Where:~~
−
−	~~<math>~~
−	~~\begin{array}{lll}~~
−	~~\mu^{(j)} & = & \frac{\sum_{i=1}^{w^{(j)}}{s^{(j)}_i}}{w^{(j)}} \\~~
−	~~\sigma^{(j)} & = & \frac{\sum_{i=1}^{w^{(j)}} {(s^{(j)}_i-\mu^{(j)})^2}}{w^{(j)}-1} \\~~
−	~~& = & \frac{\sum_{i=1}^{w^{(j)}} {[(s^{(j)}_i)^2+(\mu^{(j)})^2-2s^{(j)}_i\mu^{(j)}]}}{w^{(j)}-1} \\~~
−	~~& = & \frac{\sum_{i=1}^{w^{(j)}} {(s^{(j)}_i)^2}+w^{(j)}(\mu^{(j)})^2-2 \mu^{(j)}\sum_{i=1}^{w^{(j)}} {s^{(j)}_i}}{w^{(j)}-1} \\~~
−	~~& = & \frac{\sum_{i=1}^{w^{(j)}} {(s^{(j)}_i)^2}+w^{(j)}(\mu^{(j)})^2-2w^{(j)}(\mu^{(j)})^2}{w^{(j)}-1} \\~~
−	~~& = & \frac{\sum_{i=1}^{w^{(j)}} {(s^{(j)}_i)^2}-w^{(j)}(\mu^{(j)})^2}{w^{(j)}-1}~~
−	~~\end{array}~~
−	~~</math>~~
−
−	<math>w^{(j)}</math> is the weight or equivalently the number of items in the statistical population used for the computation of <math>\mu^{(j)}</math> and <math>\sigma^{(j)}</math>. <math>s^{(j)}_i</math> is the ith item in population j.
−
−	~~The mean and variance computed over the whole population is then:~~
−
−	~~<math>~~
−	~~\begin{array}{lll}~~
−	~~\mu & = & \frac{\sum_{i=1}^{w^{(1)}}{s^{(1)}_i}+\sum_{i=1}^{w^{(2)}}{s^{(2)}_i}}{w^{(1)}+w^{(2)}}=\frac{w^{(1)}\mu^{(1)}+w^{(2)}\mu^{(2)}}{w^{(1)}+w^{(2)}} \\~~
−	~~\sigma & = & \frac{\sum_{i=1}^{w^{(1)}} {(s^{(1)}_i)^2}+\sum_{i=1}^{w^{(2)}} {(s^{(2)}_i)^2}-(w^{(1)}+w^{(2)})\mu^2}{w^{(1)}+w^{(2)}-1} \\~~
−	~~& = & \frac{\sigma^{(1)}(w^{(1)}-1)+w^{(1)}(\mu^{(1)})^2+\sigma^{(2)}(w^{(2)}-1)+w^{(2)}(\mu^{(2)})^2-(w^{(1)}+w^{(2)})\mu^2}{w^{(1)}+w^{(2)}-1}~~
−	~~\end{array}~~
−	~~</math>~~

← Older revision		Revision as of 09:48, 8 March 2010
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	The valid flag attached to each sample indicates whether it is considered or not during the misfit computation. Adding invalid samples lets you plot the dispersion curves variability outside the constrained range.		The valid flag attached to each sample indicates whether it is considered or not during the misfit computation. Adding invalid samples lets you plot the dispersion curves variability outside the constrained range.
−
−	~~== Loading curves ==~~
−
−	~~Click on '''Load''' button~~
−
−	''Note'': you can load as many curve as you want. However, you MUST keep only the curves that you want to invert. PLEASE do not forget to REMOVE the Data Sets you do not want to be used in the Inversion.
−
−	~~== Browsing curves ==~~
−
−	All curves loaded or created so far are displayed on the attached plot. But only the current curve is shown in the table. To change the current curve, play with the horizontal scroll bar. The title of the frame just below give the index and the total number of available curves. All items within this frame apply to current curve:
−	* values in the table
−	* menu [[#Actions\|Actions]]
−	* table [[#Modes\|Modes]]
−	* curve name
−	* visible button
−
−	~~To identify curves on the plot, it is handy to play with the '''Visible''' check box. It does not remove the curve but just hides it on the plot.~~
−
−	~~You can also associate a special color, pen and symbol to each curve with the '''Legend''' button. Naming curves with unique names might be also useful.~~
−
−	~~== Actions ==~~
−
−	~~[[Image:Dinver_dc_actions.png\|thumb\|right\|400px\|List of available actions for the current curve]]~~
−
−	~~=== Save ===~~
−	~~Saves current curve to a text file made of 4 columns: frequency, slowness, standard deviation and weight.~~
−	~~The log is save as comments before the curve samples.~~
−	~~=== Log ===~~
−	~~Shows a log of all actions performed on the current curve.~~
−	~~=== Remove ===~~
−	~~Removes the current curve from the plot~~
−	~~=== Resample ===~~
−	Reorganize sampling of the current curve. Samples can be distributed on a log, linear or curvilinear scale. You can define any frequency range even outside the current available range. In this later case, samples are extrapolated and set as invalid. If the defined range is smaller than the current range, outside samples are not affected. To remove them use '''Cut'''.
−	~~=== Cut ===~~
−	~~Removes all samples outside a defined frequency or period range.~~
−	~~=== Smooth ===~~
−	~~Smooths the curve... currently not implemented.~~
−
−	~~== Manual editing ==~~
−
−	~~You can edit all values like in a usual work sheet.~~
−
−	~~If you want to set the same value to several rows:~~
−	* Selected the rows (by clicking on the left header to select the full row)
−	* Edit one cell of the selection
−	* Hit enter or move to another cell to validate
−	* A dialog box should pop up asking if you want to extend the edition to all other selected rows.

	== Modes ==		== Modes ==

@@ Line 1: / Line 1: @@
 [[Image:Dinverdc_dispersion_target.png|thumb|right|600px]]
-Dispersion curve target is a graphical tool to manipulate curves. Before starting an inversion, the target MUST contain only the curves to invert. All other intermediate curves must be removed. Leaving undesired curves may lead to erroneous inversion results.
+Dispersion curve target is a graphical tool to [[SciFigs: Curve Browser|manipulate curves]]. Before starting an inversion, the target MUST contain only the curves to invert. All other intermediate curves must be removed. Leaving undesired curves may lead to erroneous inversion results.
 The valid flag attached to each sample indicates whether it is considered or not during the misfit computation. Adding invalid samples lets you plot the dispersion curves variability outside the constrained range.

@@ Line 9: / Line 9: @@
 Click on '''Load''' button
-''Note'': you can load as many curve as you want. However, you MUST keep only the curves that you want to invert.
+''Note'': you can load as many curve as you want. However, you MUST keep only the curves that you want to invert. PLEASE do not forget to REMOVE the Data Sets you do not want to be used in the Inversion.
 == Browsing curves ==

@@ Line 79: / Line 79: @@
 <math>
-  \begin{array}{ll}
+  \begin{array}{lll}
-\mu^{(j)}=\frac{\sum_{i=1}^{w^{(j)}}{s^{(j)}_i}}{w^{(j)}} \\
+\mu^{(j)}    & = & \frac{\sum_{i=1}^{w^{(j)}}{s^{(j)}_i}}{w^{(j)}} \\
-\sigma^{(j)}=\frac{\sum_{i=1}^{w^{(j)}} {(s^{(j)}_i)^2}-w^{(j)}(\mu^{(j)})^2}{w^{(j)}-1}
+\sigma^{(j)} & = & \frac{\sum_{i=1}^{w^{(j)}} {(s^{(j)}_i-\mu^{(j)})^2}}{w^{(j)}-1} \\
 \end{array}
 </math>

@@ Line 92: / Line 92: @@
 \begin{array}{lll}
 \mu    & = & \frac{\sum_{i=1}^{w^{(1)}}{s^{(1)}_i}+\sum_{i=1}^{w^{(2)}}{s^{(2)}_i}}{w^{(1)}+w^{(2)}}=\frac{w^{(1)}\mu^{(1)}+w^{(2)}\mu^{(2)}}{w^{(1)}+w^{(2)}} \\
-\sigma & = & \frac{\sum_{i=1}^{w^{(1)}} {(s^{(1)}_i)^2}+\sum_{i=1}^{w^{(2)}} {(s^{(2)}_i)^2}-(w^{(1)}+w^{(2)})*(\mu)^2}{w^{(1)}+w^{(2)}-1} \\
+\sigma & = & \frac{\sum_{i=1}^{w^{(1)}} {(s^{(1)}_i)^2}+\sum_{i=1}^{w^{(2)}} {(s^{(2)}_i)^2}-(w^{(1)}+w^{(2)})\mu^2}{w^{(1)}+w^{(2)}-1} \\
         & = & \frac{\sigma^{(1)}(w^{(1)}-1)+w^{(1)}(\mu^{(1)})^2+\sigma^{(2)}(w^{(2)}-1)+w^{(2)}(\mu^{(2)})^2-(w^{(1)}+w^{(2)})\mu^2}{w^{(1)}+w^{(2)}-1}
 \end{array}
 </math>

@@ Line 67: / Line 67: @@
 * Build a vector with X values from the two curves.
 * Resample both curves with this common X sampling.
-* For each X value and for both curves, we have mean, standard deviation, and weight (number of values used to compute statistics)
+* For each X value and for both curves, we have mean, variance, and weight (number of values used to compute statistics)
 <math>
@@ Line 87: / Line 87: @@
 <math>w^{(j)}</math> is the weight or equivalently the number of items in the statistical population used for the computation of <math>\mu^{(j)}</math> and <math>\sigma^{(j)}</math>. <math>s^{(j)}_i</math> is the ith item in population j.
-The mean and standard deviation computed over the whole population is then:
+The mean and variance computed over the whole population is then:
 <math>
@@ Line 93: / Line 93: @@
 \mu    & = & \frac{\sum_{i=1}^{w^{(1)}}{s^{(1)}_i}+\sum_{i=1}^{w^{(2)}}{s^{(2)}_i}}{w^{(1)}+w^{(2)}}=\frac{w^{(1)}\mu^{(1)}+w^{(2)}\mu^{(2)}}{w^{(1)}+w^{(2)}} \\
 \sigma & = & \frac{\sum_{i=1}^{w^{(1)}} {(s^{(1)}_i)^2}+\sum_{i=1}^{w^{(2)}} {(s^{(2)}_i)^2}-w^{(1)}(\mu^{(1)})^2-w^{(2)}(\mu^{(2)})^2}{w^{(1)}+w^{(2)}-1} \\
-        & = & \sigma^{(1)}(w^{(1)}-1)+w^{(1)}(\mu^{(1)})^2+\sigma^{(2)}(w^{(2)}-1)+w^{(2)}(\mu^{(2)})^2-\frac{(w^{(1)}+w^{(2)})m^2}{w^{(1)}+w^{(2)}-1}
+        & = & \sigma^{(1)}(w^{(1)}-1)+w^{(1)}(\mu^{(1)})^2+\sigma^{(2)}(w^{(2)}-1)+w^{(2)}(\mu^{(2)})^2-\frac{(w^{(1)}+w^{(2)})\mu^2}{w^{(1)}+w^{(2)}-1}
 \end{array}
 </math>