What are the empirical techniques to show causation?
Natural experiments are usually a setting for causal inference rather than a causal inference tool per se. You often need to employ something like difference-in-difference or instrumental variables anyway even when you have a natural experiment.

 Here a list of statistical causal inference approaches (Approach: Lay description) 


Instrumental Variables: Randomly assigned variable X influences Z only through Y 
Difference in Differences:  If two groups have a common trend and only one group is treated then the change in the difference between the groups is the treatment effect
Regression Discontinuity: If a hard threshold determined treatment, look at difference right around that threshold
Propensity score matching: Create a control group by matching untreated observations that were likely to be treated (but not in fact treated) with treated observations with a similar probability of treatment. 
Manhalobis distance matching:   Create a control group by matching untreated observations that look similar to treated. Another notable distance measure is Coarsened Exact Matching.  
Synthetic control: When you have only one treated observation, create a composite of untreated observations that individually are imperfect controls but collectively act as a good control.
Synthetic Cohorts - Treat repeated observations of groups like a panel of individuals and employ panel techniques. 
Selection bias modeling like Heckman correction: Assume a parametric form for selection bias and remove it, so the corrected regression results can be interpreted causally.
Sample weighting more broadly - fix bias resulting from endogenous participation and un-modeled heterogeneity by weighting sample units to look more like the true population of interest.