Say we have collected data, and from a physical perspective we know that the collected data should increase positively with time. However the data looks more like this: 
This data shown in the figure is generated as fowllows:
We first generate 100 random values sampled from a Gamma PDF with shape parameter = 0.78 and scale parameter = 1.1, lets call this vector of a 100 data points Xgamma
We then generated another 100 random values sampled from a Gaussian PDF with mean = 0 and standard deviation = 10, lets call this vector of a 100 data points Xnoise
Now let's call Xgammapro the cumulative sum of Xgamma, basically the gamma process created by the individual random values that were sampled from the gamma PDF
Data 1 = cumulative sum (Xgamma + Xnoise)
Data 2 = Xgammapro + Xnoise
What should be the best approach if we wanted to estimate the parameters of a Gamma PDF from Data 1 and Data 2 ?
Is there a good way to separate the noise from the real data ? And what should be done in case the noise is not Gaussian ?
Any help would be greatly appreciated