Thanks Ahmadi Murjani . I am looking for the relationship between rainfall and sst in a certain region of the ocean and SST has increased over the years. I thought I needed to get rid of that effect.

I think it is OK to select freqeuncy bands of interest when doing time series analysis. As some freqeuncy components will be of greater interest than others.

You could think of "de-trending" as a high-pass filtering operation.

One way of high-pass filtering the time series is to design and apply a non-causal forward/backward smoother (with a zero group delay) to the data. Then subtract that low-pass filtered signal from the raw data.

You could design your low-pass smoother (e.g. a Butterworth filter) by ensuring that it passes only polynomial components up to some specified degree.

Pre-whiten both time series by taking first differences to avoid spurious regression (Granger and Newbold, 1974). It is not okay to select frequency bands of interest. You can nearly always get a correlation if you do that. Such a correlation is therefore meaningless.

I don't know what means SST but I agree with other participants that cross-correlating two time series is very dangerous. Cross-correlations are severely biased and autocorrelated. Like autocorrelations, cross-correlations can only be statistically interpreted if the data generating process is stationary. It is often recommended (e.g. Box et al. 2016) to transform the two series such that they become realizations from a stationary process, and then prewhiten at least one of them (e.g. by replacing it by the residuals of a suitable ARMA process). In a complement of my book in French, I show the correlation matrix between 15 independent realization of random walks of length 100. Among the 105 (=15*14/2) different cross-correlations, not less that 77 are statistically different from 0 at the 5% probability level.

Guy Melard. I agree. A similar problem occurs with regression against time (Nelson and Kang, 1984). See my 2016 paper doi = {10.1177/0958305X16686447}. What is the citation for your book?

Of course, except that cross-correlation has no meaning then. Sorry for the citation. Mélard, Guy, Méthodes de prévision à court terme, 2e édition, Editions de l'Université de Bruxelles, Bruxelles (ISBN 978-2-8004-1408-9) & Ellipses Edition Marketing, Paris (ISBN 978-2-7298-3766-2) (avec CD-ROM), 2007, 537 pages.

And also Box, G. E. P., Jenkins, G. M., Reinsel G. C. & Ljung G. C. (2016), Time Series Analysis, Forecasting and Control, Wiley, (5th edition).

Thank you very much for those references.