Modelling and parameterization errors in body wave seismology

When body wave seismic data are analysed under ray theory assumptions it is common to transform the observations so that p, the horizontal slowness, is the independent variable to allow use of a linear formulation of the inverse problem. Modelling errors, such as the presence of inhomogeneities in the real Earth but not in the model, introduce correlated errors in distance X(p) and time T(p), as does the uncertainty in estimating p, the derivative of the travel-time curve. The propagation of errors into the model space is most easily done if the covariance matrix of the observations is diagonalized. For the error sources listed above, the transformation producing the desired result is a transformation to the variables |Gt(p)=T-pX and X(p). The coordinate rotation used by Dorman & Jacobson diagonalized parameterization errors alone but is slightly suboptimal in the presence of modelling noise.