(a) Ellipsoidal and (b) spherical models compared to determine their 3D structural resemblance. (c) Perspective view of four selected vertical slices that show cross-gradient vectors obtained for these models. Note the circular behavior of (d), the cross-gradient vector, and the partial 2D resemblance indicated by the null x-component of all cross-gradient vectors.
(a) Gravity and (d) magnetic noise-corrupted anomalies produced by a cube of density contrast of and induced magnetization of , embedded in a homogeneous media with density and induced magnetization equal to 0. The assumed direction of the inducing magnetic field is and . Anomalies correspond to observed data with added random Gaussian noise. In both data sets, standard deviation of the added noise is 2% of the maximum amplitude of the anomaly. Parts (b) and (e) show calculated anomalies corresponding to Laplacian smoothness models obtained after separate inversion (Figure 4a and b). Parts (c) and (f) correspond to Laplacian with bottom-layer-constrained models of Figure 4e and f. The true position of the cube is outlined by a solid line.
(a) Density and (b) magnetization models obtained after separate inversion; (c) density and (d) magnetization models obtained after cross-gradient joint inversion; (a) through (d) use Laplacian smoothness and an a priori model with large covariances. (e) Density and (f) magnetization models were obtained after separate inversion. (g) Density and (h) magnetization models were obtained after cross-gradient joint inversion; (e) through (h) use Laplacian and constrain the bottom layer towards the a priori model with near-to-zero covariance values. The joint-inversion experiment that produced models (c, d, g, and h) use a truncated SVD such that . The true position of the cube is outlined by a solid line.
Parts (a and b) show the magnitude of cross-gradient vectors for separately inverted models, and (c and d) illustrate jointly inverted models for the two experiments in Figure 4. Parts (a and c) correspond to Laplacian regularization terms and an a priori model with large covariances, and (b and d) correspond to flat bottom. The true position of the cube is outlined by a solid line.
Density and magnetization crossplots obtained from (a) separate and (b) joint 3D inversion of gravity and magnetic data from our synthetic experiment of a buried cube, for the case of homogeneous smoothness in Figure 4a–d. (c) Data misfit and (d) model convergence, corresponding to equations 30 and 31 of misfit and equations 32 and 33 of convergence, achieved at every step of the joint-inversion experiments on the cubic model.
(a) Distribution of the land and marine gravity data, and (b) land, marine, and airborne magnetic data in Todos Santos Bay, Mexico. The assumed direction of the inducing magnetic field is and . Black dashed line delimits the region selected for inversion, and white dotted lines are the position of the cross sections for separated and joint inversion shown in Figure 9. Purple solid line indicates the shoreline and the map coordinates correspond to UTM-11.
(a) Data misfit and (b) convergence for each iteration of the joint-inversion experiment with real data, using . Squares in (a) indicates the rms for the first iteration (separate inversion). Squares in (b) indicate the iteration number for the selected model.
Illustration of the two parallel section models found after separate (a-d) and joint (e-h) inversion of Todos Santos Bay data with their corresponding data profile. Density and magnetization models are illustrated for vertical cross sections across the bay with direction SW-NE, corresponding to profile in Figure 7. In addition, we show parallel cross sections, corresponding to the profile in Figure 7. Red lines are the observed data and blue lines are the calculated data for each profile.