I am looking forward to perform a Cross-correlation between Pc5 intensity data and Solar Wind Parameters during Pc5 pulsations. Is it necessary to detrend the data before correlating them? Or detrend is not entirely necessary? What is the difference between the Cross-correlation with dtrend and without dtrend?
Take my reply with a grain of salt since I am not a statistician.
As far as I know, Cross-correlation between two timeseries requires that the timeseries are "stationary" or in simple words that their statistical properties to do not vary in time (aka trend). From that I'd say that non-stationary timeries (which are commonly found in nature, finance etc.) can produce correlations that are not necessarily meaningful.
The answer to your first question is therefore very problem dependent and I am not sure what is the ideal route for your issue. However, maybe the following example will help you understand at least the general issue:
Case 1: Random signal
x=randn(200,2);
crosscorr(x(:,1),x(:,2))
Will provide a very small cross-correlation between the two signals. Specifically, I get ~0.2 at -1 lag. Which is expected, since we have two random signals (aka not correlated).
Case 2 Random signal + linear trend
y = x + [1:200]'
crosscorr(y(:,1),y(:,2))
Now, we simply added a linear trend (numbers from 1 to 200) to our initial signal (x). Now the result of the crosscorrelation provides a correlation of 0.999 at zero lagging. This is again expected since we have basically two linear signals with some noise now, and these are indeed correlated.
Maybe this (old) article has some interesting information for you to begin
Article Detrended Cross-Correlation Analysis: A New Method for Analy...
Also check simple detrend examples (https://se.mathworks.com/help/matlab/ref/detrend.html) and maybe check discussions that are relevant on stats.stackexchange like this:
https://stats.stackexchange.com/questions/120270/how-do-i-detrend-time-series
Generally you should detrend before trying to measure correlation. A classic paper on this subject is by Yule (1926):
https://www.jstor.org/stable/2341482
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