General line integrals for gravity anomalies of irregular 2D masses with horizontally and vertically dependent density contrast

Line integrals (LIs) are an efficient tool in calculating the gravity anomaly caused by an irregular 2D mass body because the 2D surface integral is reduced to a 1DLI. Historically, LIs have been derived for 2D mass bodies of depth-dependent density contrast. I derive LIs for 2D mass bodies with density contrast dependent on (1) horizontal and (2) horizontal and vertical directions. Assuming the density contrast depends only on horizontal position, two types of representative LIs are derived: LIs with logarithmic kernel and density-integrated LIs for any integrable density-contrast function. A general density-contrast model that depends on horizontal and vertical directions is developed to include three components: a function of horizontal position, a function of vertical position, and a sum of crossterms of horizontal and vertical positions. Based on the general density-contrast model defined and proper selection of 2D vector gravity potentials, general LIs are derived to calculate the gravity anomaly. The newly developed LI method is then compared with two cases from the literature in calculating gravity anomaly, and agreement is obtained. However, the new LI method allows for more general 2D density-contrast variations and can be used to calculate the gravity anomaly of a 2D mass body. Such a mass body can have any cross-sectional profile that can be approximated by a polygonal cross section with any density-contrast function that can be approximated by a rich set of basis functions.