I wanted to check the completeness of an earthquake catalog. I saw the paper "Probabilistic Seismic Hazard Analysis for Rock Sites in the Chennai City" by G. Kalyan Kumar and it used the Visual Cumulative Method (CUVI). But I do not know how to identify mathematically or via excel the first point "when the slope of the cumulative fitting curve can be well approximated by a straight line"
Julian, If you are using Excel for your computations, please consider the value of R-squared coefficient. It is estimated routinely, but you should select the appropriate box, if you want to see its value. A value of R-squared close to 1, like 0.9 or 0.85 will indicate a good fitness by a straight line. I suppose you will obtain such a value for the whole data set in your Figure1 above. So, you may evaluate that first point by following what is happening with the R-squared values. If continuously decreasing the number of points used in your computations just makes R-squared remain high (close to 1), with fluctuations in the range +/_ 5%, it means your data set may be indeed approximated by a single straight line. At some point, the R-squared will be (maybe) out of that range and that is what you are looking for. Yet, the main problem is here, i. e. IF your data set may be indeed fitted by such a single line or not. So, first of all, try to fit the whole set with a single straight line and see R-squared value here. If the obtained value is low, as I expect with the data set in your Figure 2 above, you should first identify the reasons for such a low value. Myself, I estimate at least two straight lines, with quite different slopes, may be fitted to the data set in Figure 2. So, if you are looking for the point where the whole data set may be fitted by two straight lines (with different slopes), R-squared may be also of use. R-squared value for the first data subset will be high, it will remain also high for the second data subset, but it will be low for the whole data set.
You can use Zmap software for more seismicity analysis!!
Otherwise you can plot cumulative number (log scale) vs magnitude ) graph ,and identify that point where the graph trends start to follow the straight line,in some cases there will be multiple trends you can observe in the plot,depending upon the incompleteness,heterogeneity,etc.
https://www.researchgate.net/publication/285715153_Estimating_the_magnitude_of_completeness_for_earthquake_catalogs
or use Gismo
http://geoscience-community-codes.github.io/GISMO/
Julian, If you are using Excel for your computations, please consider the value of R-squared coefficient. It is estimated routinely, but you should select the appropriate box, if you want to see its value. A value of R-squared close to 1, like 0.9 or 0.85 will indicate a good fitness by a straight line. I suppose you will obtain such a value for the whole data set in your Figure1 above. So, you may evaluate that first point by following what is happening with the R-squared values. If continuously decreasing the number of points used in your computations just makes R-squared remain high (close to 1), with fluctuations in the range +/_ 5%, it means your data set may be indeed approximated by a single straight line. At some point, the R-squared will be (maybe) out of that range and that is what you are looking for. Yet, the main problem is here, i. e. IF your data set may be indeed fitted by such a single line or not. So, first of all, try to fit the whole set with a single straight line and see R-squared value here. If the obtained value is low, as I expect with the data set in your Figure 2 above, you should first identify the reasons for such a low value. Myself, I estimate at least two straight lines, with quite different slopes, may be fitted to the data set in Figure 2. So, if you are looking for the point where the whole data set may be fitted by two straight lines (with different slopes), R-squared may be also of use. R-squared value for the first data subset will be high, it will remain also high for the second data subset, but it will be low for the whole data set.
Marian Ivan How we can interpret the different trends in the FMD plots?
So, you mean I have to plot whole data first and then one by one delete until I get the starting point at which R-squared becomes highest? Marian Ivan
Jelian Reyes For excel it will be better if you use two plots. 1st plot will be for your whole data. After your 1st plot it will be obviously visible for which range your data follows approximate linear relationship. I mean it will be visible for you that you can avoid (first) several data which don't follow linear trend. Then edit your data of 1st plot and add 2nd series to make your 2nd plot with that range which approximately follow linear trend, draw linear trend line with those data and see the R squared value. To get the starting point you can increase or decrease your data for which get better plot i.e higher R-squared value.
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