Below is the scatter plot of Brightness temperature on Y axis and corresponding Rainfall rate from TRMM satellite image on the x-axis. I want to know if there is a relationship between the two. If yes, how to get a regression equation from it and if not, how to know the reason for it?
Just from eyeballing the scatter plot, it doesn't look like there is much of a relationship. You could improve the plot by choosing a different symbol for the dots - one that takes up less room or is translucent would be good. You might also want to transform the rainfall rate, perhaps using log (if the rate is always positive) or square root.
Then you could try fitting a loess line or other smooth line to the data.
Then you can try regression; how to do regression depends on the package you are using (and questions about code are off topic here) but this plot looks like Excel - I, personally, would avoid using Excel and use a statistics package instead, because they have more options and so on.
If you decide not to transform the data, then you are likely to have some influential points. You might try a robust regression or a quantile regression.
Given that it is chronological data the scatter plot between two series may not be inferential. It is well known that if a series is auto-correlated the cross correlation is of limited value. When someone views the scatter plot trying to extract "guidance" time series data may not be amenable as the cross-correlation is functionally related to the slope of a Least Squares model.
Initially "torturing the data until it confesses" by attempting to power transform the data based upon visualization may not only be anachronistic but inappropriate when dealing with chronological data.
As an example of this consider the GASX problem from Box & Jenkins we have the following XY plot which is somewhat despiriting as there is only a hint of possible success due to obfuscations. With thorough time series analytics , we can develop a model like this with statistics like this with an eye-popping r-squared .. all without unnecessary and unwanted power transformations (data torture !) .
In summary the things we were taught in the first course in statistics , i.e. eye the data first ... may only be applicable to certain types of data like cross-sectional data without gaussian violations. Chronological data analysis presents not only problems but opportunities to develop what might be a useful modelala GEP Box.
Certainly influential observations, changes in variance , changes in parameters etc. also need to be seamlessly incorporated into the model formulation which is an iterative process.