In a laterally homogeneous medium, the traveltime (T) and distance (X) for a ray with horizontal slowness p are linearly related to the depth Z(v) at which the velocity v = 1/p occurs. In order to exploit this linearity, we must infer the inverse velocity p from the observations of X, T pairs. Uncertainty in the determination of p causes correlation between the X and T observations. This correlation can be eliminated by rotation of the data into a coordinate system in which the covariance matrix is diagonal. These independent coordinates are, except for a scaling factor, the well‐known intercept time [Formula: see text] and a new variable [Formula: see text] The derivatives of T and X with respect to a depth‐velocity model contain singularities and so do those for ζ. These singularities can be quelled by representing the model as a stack of layers, each of which has a constant velocity gradient. Depth is then obtained by integration of the gradients. The sharpness of the partial derivatives of ζ w.r.t. the layer gradients indicates that ζ contains information in a more concentrated form than does τ. This manifests itself in smaller error bounds on the solution when ζ observations are used to supplement τ data. In the determination of ζ(p) from X,T data, an uncertainty principle or tradeoff applies. The delta‐like nature of the zeta partial derivatives means that the uncertainty in ζ will be closely related to the solution uncertainty and that we should choose in the parameterization the ζ, p pair which minimizes the uncertainty in ζ. This will avoid degrading the ultimate depth resolution achievable while still in the parameterization stage. We have applied these methods to sea floor hydrophone and surface buoy data from the Bengal Fan, and, we derive a model whose gradient is [Formula: see text] at the surface reaching [Formula: see text] at 500 m and remaining constant to at least 5.5 km.
Imaging the subsurface using induced seismicity and ambient noise: 3-D tomographic Monte Carlo joint inversion of earthquake body wave traveltimes and surface wave dispersion