Threshold voltage data analysis
The threshold voltage  is the minimum voltage needed to create a  conducting path along the diode. As a real diode is never completely off, one can define the threshold voltage at the point where the diode starts conducting in the exponential fashion. So there are different ways to define the actual threshold. (It is also heavily dependent on temperature.)

Here is one way to get a value for the threshold voltage:

If you increase the voltage applied to the diode and measure the current flowing along the diode, you can use this data to determine the threshold voltage by plotting the sqrt of the current.
I did this some time ago from simulation data for a nmos device, so actually this curve represents the diode characteristic, in which you are interested in.



In the Appendix of the book CMOS Analog Circuit Design from P. E. Allen and D. R. Holberg you find this procedure in more detail.