  We investigate the use of bright single pulses from the Crab pulsar to
determine separately the dispersion measure (DM) for the Main Pulse and
Interpulse components. We develop two approaches using cross correlation
functions (CCFs). The first method computes the CCF of the total intensity of
each of 64 frequency channels with a reference channel and converts the time
lag of maximum correlation into a DM. The second method separately computes the
CCF between every pair of channels for each individual bright pulse and
extracts an average DM from the distribution of all channel-pair DMs. Both
methods allow the determination of the DM with a relative uncertainty of better
than 10^-5 and provide robust estimates for the uncertainty of the best-fit
value. We find differences in DM between the Main Pulse, the Low Frequency
Interpulse, and the High Frequency Interpulse using both methods in a frequency
range from 4 to 6 GHz. Earlier observations of the High Frequency Interpulse
carried out by Hankins et al. (2016) resulted in DM_HFIP-DM_MP of 0.010 +-
0.016 pc cm^-3. Our results indicate a DM_HFIP-DM_MP of 0.0127 +- 0.0011 pc
cm^-3 (with DM_comp being the DM value of the respective emission component),
confirming earlier results with an independent method. During our studies we
also find a relation between the brightness of single pulses in the High
Frequency Interpulse and their DM. We also discuss the application of the
developed methods on the identification of substructures in the case of Fast
Radio Bursts.
