upper A Superscript upper T Baseline bold-italic upper C Subscript bold-italic d Superscript negative 1 Baseline left-parenthesis bold-italic d Subscript bold-italic o b s Baseline minus bold-italic upper A rho 0 right-parenthesis EndLayout"/>
and
(3.12)
This solution is not robust enough yet. It varies greatly depending on how the target volume is gridded. One strategy is to introduce a smoothing constraint on the prior covariance matrix C ρ , such as
Figure 3.5 Three‐dimensional representation of density distribution inside Showa‐Shinzan lava dome. The red points and blue lines indicate the gravity stations and the lines of sight of the muography detector, respectively.
Redrawing Figure 6 of Nishiyama et al. (2017).
where σ ρ is the allowance of deviation from the initial guess density, d(i,j) is the distance between the i‐th and j‐th prisms, and λ is the correlation length, which controls the correlation of neighboring prisms. When λ is large, the smoothing effect propagates to a longer distance and the inversion tends to be over‐determined, whereas it becomes ill‐posed when λ is small.
Nishiyama et al. (2017) applied the above linear inversion to a lava dome located in Usu volcanic region, Japan (Showa‐Shinzan lava dome). The surveyed target is characterized by an uplifted plateau with a diameter of ~ 1 km and a dacitic lava dome with a diameter of 200 m on the plateau. This target had been once surveyed by muography in the pioneering era of muography (Tanaka et al., 2007). The first muography shows a two‐dimensional density map that suggests that the lava and surrounding plateau have distinct contrast between high and low density. Additional muography and gravity surveys were then performed to constrain the exact shape of the underground lava block (Nishiyama et al., 2017). Gravity surveys were performed at thirty stations on/around the dome and the muography observation was performed from 500 m west of the summit using emulsion films. Fig. 3.5 shows the obtained three‐dimensional density distribution, which was recovered from one muography and thirty gravity data. The result indicates the high‐density lava block extends vertically in a cylindrical shape within 200 m depth from the surface.
Cosburn et al. (2019) applied a similar linear joint inversion to muography data taken from an underground tunnel and gravity data on the surface and inside the tunnel in Los Alamos canyon, where a regional stratigraphy is very well‐studied. Above the tunnel, there is a layer of high‐density beds (deposits from volcanic surge) spreading horizontally. This configuration provides a unique opportunity to verify the resolution of the joint inversion and to tune the smoothing methods and related parameters. To properly reconstruct such a layered structure in the target volume, they introduce an anisotropic function with two correlation lengths, λ xy and λ z , to allow for independent depth and lateral variations. Specifically, instead of a single correlation length (equation 3.13), they introduce
(3.14)
where x ij , y ij , and z ij are horizontal distances and vertical distance of i‐th and j‐th prisms, respectively. By assigning smaller values for vertical correlation length (λ z < λ xy ), stronger constraints are imposed on prisms at the same elevation, and then a layered structure is preferably reconstructed.
3.4 MORE EXACT FORMULATION
The formulation of the linear inversion described in the previous section works fine as a first‐order approximation; however, it is not precise from the viewpoints of muography data. For instance, when one imagines a bumpy topography crossed by a certain field of view of the muography detector, the approximation used in equation 3.8 is no longer valid, because the muon flux attenuates nearly exponentially as a function of density length. Describing the field of view as a representative line of sight would cause significant biases on estimation. Jourde et al. (2015) derived a comprehensive formulation using the concept of acquisition kernel, which relates a tiny density fluctuation at a certain point in the space to the resulting variation on muon flux (or gravity data). Since such a relationship is not linear, the inversion becomes inherently non‐linear. Jourde et al. (2015) propose linearization for small density fluctuation around the initial guess. When one solves such a problem, the inversion would be sequential; it starts from the initial density model and it is iteratively improved by multiple Bayes updates until convergence.
Lelièvre et al. (2019) employed tetrahedral meshes instead of rectangular meshes, to reproduce significant topography without requiring large numbers of mesh cells. They demonstrate that such meshes could be produced by the free software TetGen (Si, 2015).
3.5 DENSITY BIAS
Rosas‐Carbajal et al. (2017) increased the quantity of dataset significantly, with muography data taken from three directions and more than a hundred gravity data on La Soufrière de Guadeloupe lava dome (Lesser Antilles). They performed a linear inversion, taking a formulation similar to equation 3.9 but with a different approach for regularization. The reconstructed three‐dimensional density profile (Gibert et al., 2021) presents a low‐density region located below the southern part of the lava dome near the summit. The position of this low density corresponds to most presently active fumaroles on the dome. Considering that the electrical conductivity model poses a conductive region there, they conclude that the region is porous and highly altered due to the presence of thermal fluids. In this region of the lava dome, a temporal variation of muon flux associated with hydrothermal activities was also observed (Jourde et al., 2016).
Contrary to the success of 3D imaging, Rosas‐Carbajal et al. (2017) point out a non‐negligible offset between the density estimated from muography and that from gravity. In the case of La Soufrière de Guadeloupe lava dome, such offset amounts to 0.47 g/cm3. Specifically, muography tends to yield lower density compared to the apparent density estimated from gravity
