Difference between revisions of "Dispersion curve target"

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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.
  
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== Averaging or merging curves ==
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== Velocity standard deviations ==
  
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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:
+
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).
  
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* Build a vector with X values from the two curves.
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<math>\begin{array}{lll}
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* Resample both curves with this common X sampling.
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dS & = & \frac{(\frac{1}{V-dV}-\frac{1}{V})+(\frac{1}{V}-\frac{1}{V+dV})}{2}\\
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* For each X value and for both curves, we have mean, variance, and weight (number of values used to compute statistics)
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  & = & \frac{dV}{(\frac{1}{S})^2-dV^2}
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<math>
 
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\begin{array}{ll}
 
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\mu^{(1)}, \sigma^{(1)}, w^{(1)} \\
 
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\mu^{(2)}, \sigma^{(2)}, w^{(2)}
 
 
\end{array}
 
\end{array}
 
</math>
 
</math>
  
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Where:
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Where,
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* V and dV are the input velocity and its standard deviation;
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<math>
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* S and dS are the slowness and its standard deviation;
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\begin{array}{lll}
 
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\mu^{(j)}    & = & \frac{\sum_{i=1}^{w^{(j)}}{s^{(j)}_i}}{w^{(j)}} \\
 
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\sigma^{(j)} & = & \frac{\sum_{i=1}^{w^{(j)}} {(s^{(j)}_i-\mu^{(j)})^2}}{w^{(j)}-1} \\
 
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            & = & \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}
 
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\end{array}
 
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</math>
 
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<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.
 
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The mean and variance computed over the whole population is then:
 
−
 
 
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<math>
 
−
\begin{array}{lll}
 
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\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>
 

Latest revision as of 14:49, 29 March 2010

Dinverdc dispersion target.png

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.

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.

Modes

Set the mode of the current curve. In most situations, there MUST be only one entry in this table, meaning that the current curve is identified as one particular mode. The mode is considered in its general sense:

  • Phase or Group
  • Rayleigh or Love
  • A positive index, '0' means fundamental mode, '1' for first higher mode,...

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 curve browser, when loading new curves from text files, the 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).

Where,

  • V and dV are the input velocity and its standard deviation;
  • S and dS are the slowness and its standard deviation;
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