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The seismological inverse problem has much in common with the data assimilation problem found in meteorology and oceanography. Using the data assimilation methodology, I will formulate the seismological inverse problem for estimating seismic source and Earth structure parameters in the form of weak-constraint generalized inverse, in which the seismic wave equation and the associated initial and boundary conditions are allowed to contain errors. The resulting Euler-Lagrange equations are closely related to the adjoint method and the scattering-integral method, which have been successfully applied in full-3D, full-wave seismic tomography and earthquake source parameter inversions. I will review some recent applications of the full-wave methodology in seismic tomography and seismic source parameter inversions and discuss some challenging issues related to the computational implementation and the effective exploitation of seismic waveform data