inversion

• in the inversion theory, an estimate for the error energy that corresponds to the
  solution of the problem can be made by a taylor expansion for an initial parameter
  set in the neighbourhood of the solution. this provides the representation of the error
  energy in terms of hessian matrix and the gradient vector. the square and symmetric
  hessian matrix contains the second derivatives while the gradient vector consists of
  the derivatives of the error energy with respect to initial guess parameters. the
  hessian and gradient matrices represent the curvature and gradient information,
  respectively. in the parameter estimation problems the hessian and gradient
  matrices should be defined in terms of the theoretical model responses. the hessian
  can be given by the sum of a' a and q matrices. a is a matrix whose columns
  contains the derivative of model response with respect to initial guess parameters
  and is referred to as the jacobian matrix. q consists of the second derivatives of the
  model responses with respect to the initial guess parameters and becomes equal to
  zero for the linear problems. the gradient vector equals to the multiplication of
  transpose of the jacobian by the data differences vector. then the parameter
  correction vector is calculated by the multiplication of the inverse of hessian matrix
  with the gradient vector. this process is referred as the newton inversion method. it
  requires a few numbers of iterations to reach the minimum of the error energy
  function. however, this is only possible if the initial guess is close to the solution and
  provides a linearized problem around the minimum of the error energy. for other
  cases, an approximation for the hessian or only for q is made to obtain stable
  inversion steps. this paper discusses a variety of derivative based parameter
  estimation methods in view of approximations made for the representation of hessian
  matrix.
  
  http://80.251.38.220/…banli_parametre_kestirimi.pdf
  
  http://geop.eng.ankara.edu.tr/