1) The method which is used in this study shows the climate change effects, which can be applied to other scientific studies.
2) The method used in this study shows the climate change effects, which can be applied to other scientific studies.
Sentence 1 shows the sentence with which. This is a restrictive adjective clause, sometimes called a relative clause. In other words, it modifies the subject (like an adjective). It is essential information specifying the exact method and not separated with commas from the rest of the clause. Adjective clauses of this type may be formed with which, who, or that.
Sentence 2 is the same, except we have dropped the which is, forming a reduced relative clause. This is grammatical.
Such clauses may be reduced when they are modifying the subject. They should not be used after the main verb. So, the third clause cannot be reduced.
I would rewrite the entire sentence as:
The method used in this study shows the climate change effects and can be applied to other scientific studies.
For more information, look for Relative Clauses and Adjective Clauses.