Abstract

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 tau (p) = T - pX and a new variable zeta (p) = T + pX. The derivatives of T and X with respect to a depth-velocity model contain singularities and so do those for zeta . 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 zeta w.r.t. the layer gradients indicates that zeta contains information in a more concentrated form than does tau . This manifests itself in smaller error bounds on the solution when zeta observations are used to supplement tau data.In the determination of zeta (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 zeta will be closely related to the solution uncertainty and that we should choose in the parameterization the zeta , p pair which minimizes the uncertainty in zeta . 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 1.8 sec (super -1) at the surface reaching 0.6 sec (super -1) at 500 m and remaining constant to at least 5.5 km.

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