Conjugate complex variables method for the computation of gravity anomalies

Abstract

Using conjugate complex variables, a generalized method is presented to derive formulas to calculate first-and higher-order derivatives of the gravity potential due to selected mass models. Double integrals in the computation of gravity-gradient anomalies are transformed into complex contour integrals. Analytical expressions for higher-order derivatives of the gravitational potential in arbitrary directions due to two-dimensional (2-D) polygonal mass models are derived. The method is extended to 2-D polygonal bodies whose density contrasts vary with depth and horizontal distance and can be generalized to deal with 2-D bodies of any shape. The vertical gravity field and its first derivatives due to a homogeneous radially symmetric body are also computed using conjugate complex variables. Derivation of gravity and gravity gradient formulas generally is greatly simplified by the use of complex variables.Complex variables have been an effective mathematical tool in geophysical gravimetry research for Soviet geophysicists (Antonov, 1975; Devitsyn, 1981a,b;Sretensky, 1946;Strakhov, 1974; Tsirulsky, 1963;Zhdanov, 1980). My paper presents a unified method using conjugate complex variables to compute first-and higher-order derivatives of the gravity potential for 2-D polygonal bodies. The spatial variables x and z are replaced by the conjugate complex variables w = x + iz and w = x -iz. Using complex-variables theory, double integrals in the computation of gravity anomalies are transformed into complex contour integrals. The method is then extended to 2-D polygonal bodies whose density contrasts vary with depth and lateral distance; the technique can be generalized to deal with 2-D bodies of any shape. I also compute, using conjugate complex variables, the vertical gravity field and its first-order derivatives due to homogeneous, radially symmetric bodies. This formulation significantly simplifies the procedures for the derivation of the computational formulas for gravity anomalies.

Singularities In Gravity Computation For Vertical Cylinders and Prisms

“…It is therefore important to seek the most generalized and efficient algorithm for the integration procedure. Success has been obtained with the use of a conjugate complex variables formulation to compute the vertical gravity and its higher order derivatives of 2-D and symmetric gravity anomalies (Kwok 1989). This paper shows that the same formulation can also be applied to derive efficient computational formulae to obtain the vertical gravity and its higher order derivatives for vertical cylindrical bodies under a variety of circumstances.…”

“…The complex form of the Green theorem (Kwok 1989) facilitates the conversion of a double integral into a complex contour integral. The theorem is stated as…”

“…This approach has been considered recently by Van Baak (1989). This paper presents the use of the conjugate complex variables formulation (Kwok 1989) to derive formulae to compute the vertical gravity and its higher order gradient tensors for cylindrical mass models for a variety of situations. Residue theory and t 2 Figure 1.…”

Vortex dynamics in the studies of looping in tropical cyclone tracks

“…In the Zabusky et al CD method, the above integrals are transformed into line integrals, which are then integrated approximately. However, following the conjugate complex variables formulation proposed by Kwok (1989) on calculation of the gravitational potential due to a two-dimensional mass anomaly, the above integrals can be evaluated exactly. The exact analytic formulas for evaluating induced velocities due to the polygonal vortex patch are…”

Gravimagnetic similarity in anomaly formulas for uniform polyhedra

“…Goodacre (1973) also notes closure of the transcendental functions under differentiation in the context of rectangular prismatic targets and comments on the possibility of reuse in gravity-and magneticfield computations within the same program. Strakhov (1982) and Kwok (1989Kwok ( , 1991 independently generalize the complex variable theory to arbitrary uniform polyhedra. These two approaches again recover a common set of transcendental functions in which to express the gravimagnetic anomalies.…”