Intersection Coordination Research Articles

Traveltime inversion for dipping layers, large moveouts, and arbitrary source and receiver positions

The seismic traveltime inversion algorithm presented here requires few limiting assumptions and is computationally efficient. The traveltime for a raypath in a layered earth (uniform or linearly varying sound speeds in each layer) can be written in closed form if the coordinates of the intersections between the raypath and the layer interfaces are known. The traveltime on an actual raypath is minimal with respect to neighboring paths. The raypath can thus be determined by equating the derivatives of the traveltimes with respect to the intersection coordinates to zero. This approach can be applied to the inverse problem, which seeks those layer parameter values which “best fit” the observed traveltimes from a CMP reflection gather. The result is a system of equations in which the intersection coordinates and layer parameters are independent variables, with the constraint that the traveltimes be minimal with respect to the coordinates. When this systemis linearized with a step‐size constraint, an efficient inversion algorithm results. The equations are solved without numerical differentiation or iterative solution of the forward problem at each step in the inversion. The number of operations required in each iteration is proportional to the number of moveouts and the third power of the number of interfaces. The algorithm has been applied to the estimation of sound speed in thin (e.g., 3–15 m) sea‐floor sediment layers from CMP reflection data obtained with a 3 kHz system bandwidth. Errors were reduced by fitting traveltime differences determined by crosscorrelation of events at different moveouts. In favorable conditions, the scatter in estimated sound speeds was 2 percent for a 15 m layer.

A BASIC program for the estimation of Michaelis-Menten parameters by the direct linear plot

The use of manual graphical methods for the estimation of Michaelis-Menten kinetic parameters as recommended by Eisenthal and Cornish-Bowden (Biochem. J. 139 (1974) 715–720) was found to be impractical for large ( n > 5) sample numbers. A BASIC program providing the rank ordered coordinates of intersections, which correspond to estimates of K m and V max in the direct linear plot method, is described.

Aberrations for grazing incidence telescopes

The wavefront aberration polynomial and transverse ray aberration expansions will be derived for grazing incidence two-mirror Wolter telescopes including all paraboloid-hyperboloid and paraboloid-ellipsoid combinations. The reference sphere is determined with the help of the principal surface of the telescope, and the aberration polynomials will be given as functions of the coordinates of the ray intersection with the reference sphere. Third, and some of the fifth- and seventh-order aberration terms will be analyzed. Also, the wellknown relationship between wavefront aberration polynomial and transverse ray aberration polynomials will be verified.

A method of finding invariant values of kinetic parameters

A method of calculating the Arrhenius parameters has been proposed, based on an evaluation of the coordinates of intersection of greatly extended confidence regions determined in solving the inverse kinetic problem. The validity of this method is illustrated by comparison of the Arrhenius parameters found from non-isothermal and isothermal data reported by other investigators.

A System to Digitize Bathythermograph Aperture Cards

This paper describes a system which allows the automatic transformation of bathythermograph (BT) data to their digital equivalent. The BT data now stored on aperture cards, a technique described by Sauer (J. Fish. Res. Bd. Canada, 21(3), 1964), form the input to digitization equipment especially modified for this application. A description of the system and required equipment is followed by the mathematical approach employed in the Cartesian to BT grid coordinate transformation. It comprises a type of double inverse interpolation in the coordinates of the major grid intersections. Firstly, a point P on the BT trace is localized in or near a group of four "cells" in the grid. Secondly, the grid data belonging to this four-cell block are used in an iterative process based on two separate Lagrangian interpolations. The latter, both 3-point formulae, appear more than adequate in coping with BT grid distortions. This, in turn, leads to a two-dimensional fit up to fourth order terms. Any subsequent needs for translation, rotation, and linear amplification or reduction of the BT trace relative to its grid may be dealt with by means of a linear transformation.

New Electronic Wavefront Plotter

An instrument is described which can be used to assess the quality of optical systems by forming a picture of the wavefront shape on a CRT. Ray intersection coordinates are first formed by means of a suitable pupil scanning mechanism and then the wavefront shape is derived by the use of a simple analog computer.