This paper explains two mathematical concepts for data assimilation. The sequential and variational data assimilation consider a stochastic system, which is updated by uncertainty observations. These concepts reduce the simulation uncertainty by using observations. Two case studies show the applications of both concepts in geotechnical engineering problems. The first case study is discussing the possibilities and limitations of the sequential data assimilation concept in a theoretical example, whereas the second case study is demonstrating the combination of settlement measurements and a stochastic subsoil model by means of variational data assimilation. Additionally, the concept of forecast uncertainty quantification is demonstrated in the second case study. At the end a brief review of the data assumption concepts and the given forecast uncertainty quantification approach is given together with conclusions for further research.