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2 Tomographic Imaging of Volcano Structures with Cosmic‐Ray Muons

       Seigo Miyamoto1,2 and Shogo Nagahara1,3

       1 Earthquake Research Institute, The University of Tokyo, Tokyo, Japan

       2 International Virtual Muography Institute, Global

       3 Graduate School of Human Development and Environment Kobe University, Kobe, Japan

      ABSTRACT

      This chapter introduces two mathematical methods to reconstruct three‐dimensional density images from multi‐directional muographic images. One method is a linear inversion, while the other is a filtered back projection. Linear joint inversion has been studied in recent years to effectively incorporate both gravity and muon flux, which are density‐sensitive observables, as well as other a priori information. Since it can be also applied to multi‐directional muography, the basic theory of this method is described. The filtered back projection method is widely common in the medical field; however, intensity, angle of the beam, and number of detectors are considerably different in observation of large structures such as volcanoes. In order to resolve these differences and to make filtered back projection applicable to volcanic observations, we have introduced an improved geometric approximation and a method to estimate the error due to the statistics of muons. We also evaluate the performance of these two methods with a forward simulation and describe future research avenues.

2.1 INTRODUCTION

      The internal structure of a volcanic body reflects the eruption dynamics and history of activity. The shallow structure of a volcano controls the eruptive activity. For example, Koyaguchi and Suzuki (2018) reported that whether an eruption forms a pyroclastic fall or flow deposit during an explosive eruption depends on the shape of the shallow crater and conduit. Therefore, density imaging of the shallow volcanic structure can provide insights into such processes.

We highlight some of the muographic observations of volcanoes that motivated our study. Tanaka et al. (2007) muographically imaged the shallow part of the Asama volcano, Japan, using high‐energy cosmic‐ray muons. Although the muographic observation was conducted from one direction, the captured density image revealed the structures of the caprock and the shallow conduit underneath the crater. One‐directional muography yields the density length, which is an integration of density and length along the muon pathway. In the case of one‐directional muography, it is possible to uniquely determine the average density of a mountain body along the muon pathway using a priori information on the mountain topography. However, due to the lack of spatial resolution along the muon path, the results include an uncertainty of uninteresting parts. A simple solution for obtaining additional spatial resolution is to increase the number of observation directions. For example, Tanaka et al. (2010) used two muon detectors to determine the three‐dimensional structure of the shallow conduit of Asama volcano. However, two‐ or three‐directional stereographic muography generally cannot avoid including a priori assumptions, and lacks the spatial resolution to resolve the detailed structure of a volcano. Nishiyama et al. (2014a, 2017), Rosas‐Carbajal et al. (2017), and Barnoud et al. (2021) obtained a three‐dimensional image of the internal structure of a volcano by combining muography and gravity techniques.

      The technology for muon detection is improving (e.g., Morishima et al., 2017; Oláh et al., 2018). Nagahara and Miyamoto (2018) studied the feasibility of multi‐directional muon tomography of a volcano based on a simulation using the filtered back projection method. These technological and analytical developments will make multi‐directional muon tomography more feasible. In this chapter, we introduce two methods to reconstruct the three‐dimensional density structure of a body using multi‐directional muographic images. We also evaluate the performance of our methods.

2.2 LINEAR INVERSION

      In this section, we introduce the linear inversion method used by Nishiyama et al. (2014a). The method is based on the theory described in Tarantola (2005). By assuming that the observed number of muons and density probability distribution is Gaussian, the problem can be solved with a least‐squares method by introducing a priori information about the density structure as elements of the covariance matrix.

      If the volume of interest is subdivided into voxels with index j and density ρ ^j (j = 1, 2, …), then the relationship between density length d ⁱ and ρ ^j can be represented as:

(2.1)

where i is the index of the muon pathway for any direction, A ^ij is the length across the j ^th voxel, and d ⁱ is the density length derived from the muon attenuation along the i ^th muon pathway. d ⁱ can represent the number of observed muons rather than the density length, but in this paper, we first treat d ⁱ as the density length. Equation 2.1 can be represented by vectors ,, and the matrix A:

(2.2)

In many cases, multi‐directional muography based on equation 2.2 is an under‐determined system that limits the spatial and density resolution. If is the observational value of , then and form a covariance matrix of and , respectively. In addition, we also define as initial values, set the model expectation as , and denote the posterior covariance matrix as . The relationship between these parameters is (e.g., Nishiyama СКАЧАТЬ