This study is the first trial to apply the method of filtered back projection (FBP) to reconstruct three-dimensional (3-D) bulk density images via cosmic-ray muons. We also simulated three-dimensional reconstruction image with dozens of muon radiographies for a volcano using the FBP method and evaluated its practicality.

The FBP method is widely used in X-ray and CT image reconstruction but has not been used in the field of muon radiography. One of the merits of using the FBP method instead of the ordinary inversion method is that it does not require an initial model, while ordinary inversion analysis needs an initial model.

We also added new approximation factors by using data on mountain topography in existing formulas to successfully reduce systematic reconstruction errors. From a volcanic perspective, lidar is commonly used to measure and analyze mountain topography.

We tested the performance and applicability to a model of Omuroyama, a monogenetic scoria cone located in Shizuoka, Japan. As a result, it was revealed that the density difference between the original and reconstructed images depended on the number of observation points and the accidental error caused by muon statistics depended on the multiplication of total effective area and exposure period.

Combining all of the above, we established how to evaluate an observation plan for volcanos using dozens of muon radiographies.

Muon radiography is a method that can be used to make a map of the inner bulk density structures of large objects such as volcanoes, archeological targets, and so on, using secondary cosmic-ray muons. These muons are generated by the interactions between high-energy primary cosmic rays (the main component is proton) and nuclei in the atmosphere. The flux, energy spectrum, and the zenith angle dependence of secondary cosmic-ray muons have been well researched (e.g., Dorman, 2004; Honda et al., 2004; Patrignani et al., 2016; Nishiyama et al., 2016). Their behavior including energy loss in various materials has also been investigated (Groom et al., 2001). Therefore, when we assume “density length”, which is the integral of multiplication of density and material thickness, we can evaluate the number of penetrating muons. Muon detectors have been developed in the field of particle physics and cosmic-ray physics. To make a bulk density map, we need to measure not only the counts of penetrating muons from the target, but also the direction. For example, nuclear emulsion films (Morishima et al., 2017), hodoscope by scintillating plastic bars (Jourde et al., 2013, Ambrosino et al., 2015), glass resistive plate chambers (Ambrosino et al., 2015), the multi-wire proportional chambers (Oláh et al., 2018) are capable of doing that. By implementing these muon detectors around the target, we can obtain the penetrating muon flux for each direction from the detector; then by comparing to the initial muon flux, we also obtain the attenuation of muons for each directions. By using the topographic data of the target, it is possible to derive the two-dimensional averaged bulk density from the muon attenuation and the path length of the target material.

The principle of X-ray radiography and muon radiography is very similar. There are two significant differences between these two methods: the first is the attenuation length. Typical X-ray beams can penetrate the material in less than 1 m water equivalent. Conversely, some muons can penetrate 1 km of water equivalent because their kinetic energy is very high. The second difference is the origin of the source. The source of cosmic-ray muons is completely environmental and we cannot control the flux while X-ray beams are generated by accelerating the electron artificially. Typically, the number of observed muons is much smaller than in ordinary X-ray radiography.

The first significant result for volcanology was the two-dimensional bulk density imaging of the shallow conduit in Mt. Asama by Tanaka et al. (2007a). Several observations have been made since this research (e.g., Tanaka et al., 2007b, 2014; Lesparre et al., 2012).

The internal structure of volcanoes gives important information for volcanology. For example, the shape of a shallow conduit affects the eruption dynamics (Ida, 2007). However, muon radiography by only one direction makes just a 2-D image, and this density is the average of material along the muon path direction. Therefore, if we find some contrast in the 2-D density image, we cannot distinguish the actual position of this density anomaly along the muon path direction. To observe the real conduit shape, it is necessary to obtain the density image from different directions to reconstruct the three-dimensional bulk density image.

Tanaka et al. (2010) attempted to observe the target from two directions in Mt. Asama. Nishiyama et al. (2014, 2017) conducted a 3-D density analysis in the Showa-Shinzan lava dome, combined with gravity observation data, which is also sensitive to density. Jourde et al. (2015) evaluated this joint-inversion method between muon radiography and gravity, and they observed and conducted 3-D density analyses by using three-point muon radiography and gravity data (Rosas-Carbajal et al., 2017). These previous studies required prior information on internal density distribution because of insufficient observation data, and they were performed using the inversion technique.

In this study, we propose the application of a 3-D density reconstruction analysis method using filtered back projection (FBP), which does not require prior information. In the FBP method, it is possible to obtain a 3-D density distribution from many projection images only without using the inversion technique (Deans, 2007). This method is applied to X-ray-computed tomography (CT), so it is possible to apply to muon radiography data in principle. However, muon radiography differs from X-ray CT in three points. First, there is a constraint on the number of observation points and position. In X-ray CT, there are hundreds of observation points, and each position is controllable. However, for muon radiography, we can only use several dozen points, and the positions are limited because of topography. Second, the cosmic-ray muon attenuation flux is not a simple exponential. Therefore, the influence of muon statistical error depends on the results of 3-D density, which is not trivial. Third, in the case of muon radiography, typically the amount of signal is much less than for X-ray because the source of cosmic-ray muons is completely environmental. Therefore, it is important to study the features of the FBP method in the case of realistic observations with various numbers of muon radiographies. So we should consider not only the reconstruction error of the FBP method, but also how the error of muon statistics propagates to the final image.

The radon transform is used to obtain projection images from all directions
with respect to a density distribution. In muon radiography, this
corresponds to acquiring observation data on density length from all
directions. For three dimensions, the radon transform

A schematic of radon transform and the definition of parameters

In a 3-D case, if observation data are cone-beam data and observation points
only exist on the circumference, a complete inverse radon transform does not
exist. Therefore, approximation is needed. Feldkamp (1984) proposed one of
the best methods to approximate a solution with a small elevation angle in
two dimensions. This approximation is written as follows:

In this section, we describe the specific components of the simulation
calculation. The simulation calculation is divided into the following four
steps:

parameter setup

simulation calculation of the observed muon counts

reconstruction calculation using data created in step 2

calculations for evaluating the reconstruction results.

Path length schematic and the approximation difference between
the Feldkamp approximation and the path length normalization approximation. In
the Feldkamp approximation, the approximation density length is

We simulated and reconstructed the density structure of Omuroyama, which is located in Shizuoka, Japan. We chose this volcano for two reasons. First, this volcano is easily observable from all directions because there are no large structures around the surrounding muon shields in a topographical view. Second, there are no occurrences of muon radiography for these large scoria hills. Omuroyama is a large scoria hill. We base the internal structural model of the large scoria hill on observations at the time of its formation (Luhr and Simkin, 1993). However, there are currently no direct examples of these observations.

Figure 3 shows the contour map of the Omuroyama model used in the
simulation. We assume that the

We configure the internal density distribution similarly to a checkerboard
with a side length of 100 m and a density of 1000 and
2000 kg m

The field of view was set from

All observation points were assumed to be on the circumference of radius

An example of theoretical muon count simulation:

The simulation calculation of the observed number of muons is mainly
performed with the following procedures.

Calculate the density length

Calculate the theoretical muon flux

Calculate the theoretical muon count observation

It is not suitable to use the muon flux table in the region of the 10 m water
equivalent or less because of small change. To avoid this region, we did not
use these data when the path length

The reconstruction calculation procedure is as follows:

Calculate the muon flux

Calculate the observed density length

In the path length normalization approximation,
calculate path length

Calculate the density reconstructed image

We evaluated the systematic error, which is defined as the density difference between the original and reconstructed images at two points. First, we compared the differences between the methods for approximating the elevation angle (i.e., Feldkamp approximation and path length normalization approximation). Second, we quantified the relationship between the observation points and systematic errors.

An example of the reconstruction results. All plots were
calculated with the results from path length normalized approximation. The
altitude of each section is 490 m. Plots are only from the mountains.

We modeled scenarios with 4, 8, 16, 32, 64, 128 and 256 observation points.
The reconstruction results are shown in Fig. 5. The systematic error

The relationship between the number of observation points and the
systematic error deviation

The relationship between the number of observation points and systematic
error deviation

A comparison of the Feldkamp approximation and path length normalization approximation.

We simulated both the Feldkamp approximation and path length normalization
approximation. Figure 7 shows the reconstruction results of both
approximations. In the Feldkamp approximation, the average value of the
systematic error

We also evaluated the accidental error

The accidental error

We defined the average of

(A). Reconstruction results on a

(B). Reconstruction of results on a

(C). Reconstruction results across

In Fig. 6, the systematic error does not converge to zero even if the
number of observation points increases to more than 200. The observation
point position

Why is the average systematic error value different between the Feldkamp approximation and path length normalization approximation? For a volcano with a structure similar to Omuroyama, which is cone shaped with a crater on the summit, the length of the muon path and the elevation angle tend to be shorter than the path length estimated in the horizontal plane (see Fig. 2). In the path length normalization approximation, given that the approximation is made with the path length as a reference, the difference in path length is not important in the Feldkamp approximation; however, the difference in path length is not taken into consideration and is influenced by the change in the path length. As a result, in the Feldkamp approximation, the average value of the systematic error is negative because of the presence of results with short path lengths.

Why does the accidental error

The relationship among the number of observation points,
systematic error deviation

We performed these simulations under the condition that the total effective
area of the observation device is equal. For a 16-point observation, ST
per point is 62.5 m

This discussion is able to apply for actual observation with any muon
detector type. In the case of an emulsion-type detector, it is easy to divide
the effective area

We summarized systematic error and accidental error for Omuroyama and
ST

In this evaluation, the observation points were arranged in a circular
orbit. In the future, it is necessary to study more realistic observation
point placements. For example, it is difficult to put the observation points
on the same plane or in the same interval of

We simulated the systematic error of the 3-D density structure of Omuroyama
volcano by using several muon detectors via the FBP method with and without
information on mountain topography.

Systematic error, which is defined as the density difference between the original and reconstructed images in each voxel internal mountain, depends on the angular resolution of the muon detectors and the number of observation points.

By comparing the systematic error with and without information on mountain topography, the systematic error deviations are nearly identical. However, the mean value of systematic error becomes more precise in the former case; i.e., the value is more precise when a new method of approximation of path length normalization is used.

Considering the above, we established how to evaluate an observation plan of dozens of muon radiographies.

We assumed that there are tens observation points in this study. The actual observations, which involve many nuclear emulsion muon detectors, were executed by Morishima et al. (2017). Furthermore, Oláh et al. (2018) succeeded in developing a high-quality and inexpensive multi-wire proportional chamber system. Considering such recent advances, the CT volcanic observation of volcanoes by using numerous muon detectors will be possible in the near future.

The muon flux data we used have been published and are accessible through Honda et al. (2004) and Groom et al. (2001). Omuroyama DEM data can download from the Geospatial Information Authority of Japan.

SN led the research, wrote the paper, and made the simulation code. SM supervised the work of SN, contributed to discussions, and edited the paper.

The authors declare that they have no conflict of interest.

The authors appreciate Masato Koyama of Shizuoka University and Yusuke Suzuki of Izu Peninsula Geopark Promotion Council for advice and help with the geological structure in Omuroyama. The support of The University of Tokyo is acknowledged. Edited by: Lev Eppelbaum Reviewed by: two anonymous referees