The issue of uncertainty estimation is important to full waveform inversion (FWI) but still left behind. In our research, we apply a quasi-Newton method name Square-Root Variable Metric (SRVM) to FWI. To make it memory-affordable, we modify SRVM into a vector version. This approach allows us to retrieve the information about Hessian after the inversion is done. We validate our method on the elastic Marmousi model. The variance map is drawn to quantify the uncertainty, and the prior and posterior distributions are visually compared. The application of SRVM to elastic seems encouraging to have results of inversion and uncertainty estimation.