The discovery of the muon (Neddermeyer and Anderson, 1937) entailed experiments to characterise its propagation through different materials. The fact that muons lose energy proportionally to the mass density of the traversed matter (see Olive et al., 2014) inspired the idea of using their attenuation to retrieve information on the traversed material. This was first done by George (1955) for the estimation of the overburden upon building of a tunnel, and then later by Alvarez et al. (1970) to search for hidden chambers in the pyramids in Giza (Egypt). In a related study, Fujii et al. (2013) employed this technology to locate the reactor of a nuclear power plant. Recently, Morishima et al. (2017) successfully accomplished quest of Alvarez's team in the Egyptian pyramids.

Besides these applications, which have mainly been designed for archaeological and civil engineering purposes, scientists have begun to deploy particle detectors to investigate and map geological structures. In recent years, this has been done for various volcanoes in Japan (Nishiyama et al., 2014; Tanaka et al., 2005, 2014), including the Shinmoedake volcano (Kusagaya and Tanaka, 2015), the lava dome at Unzen (Tanaka, 2016) and most recently the Sakurajima volcano (Oláh et al., 2018). Further experiments have been conducted in the Caribbean, in France (Ambrosino et al., 2015; Jourde et al., 2013, 2015; Lesparre et al., 2012; Marteau et al., 2015) and in Italy on Etna (Lo Presti et al., 2018) and Stromboli (Tioukov et al., 2017). Recently, Barnaföldi et al. (2012) used this technology to examine karstic caves in the Hungarian mountains. Our group is presently carrying out an experimental campaign in the central Swiss Alps for the purpose of imaging glacier–bedrock interfaces (Nishiyama et al., 2017).

Inferences about subsurface structures from observed muon flux (i.e. the number of recorded muons normalised by the exposure time and the detector acceptance) necessitate a comparison of the measurement data with muon flux simulations for structures with various densities. Such a simulation consists of a cosmic-ray muon energy spectrum model and a subsequent transportation of these muons through matter. The former describes the abundance of cosmic-ray muons for different energies and zenith angles at the surface of the Earth. This has been well documented in literature (see, for example, Lesparre et al., 2010). The differences between models and experimental data, and hence the systematic model uncertainty, can be as large as 15 % for vertical muons (Hebbeker and Timmermans, 2002). On the other hand, the attenuation of the muon flux is assumed to depend only on the density of the traversed material. In this context, however, potential effects of its chemical composition have not been taken into account specifically. Instead, previous works employ a certain representative rock, so-called “standard rock”, for which the rate of muon energy loss has been tabulated (e.g. Groom et al., 2001).

The origin of this peculiar rock type can be traced back to Hayman et al. (1963), Miyake et al. (1964), Mandò and Ronchi (1952) and George (1952), who gave slightly different definitions of its physical parameters (mass density ρ, atomic weight A and atomic number Z). A comprehensive compilation thereof can be found in Table 1 of Higashi et al. (1966). Various corrections to the energy loss equation were then added in the framework of follow-up studies, which particularly include a density effect correction (see, for example, Sternheimer et al., 1984). Richard-Serre (1971) listed data relevant for muon attenuation for (i) soil from the CERN (European Organization for Nuclear Research) premises near Geneva (Switzerland), (ii) molasse-type material (e.g. Matter et al., 1980) and (iii) a “rock” that is equivalent to the one from Hayman et al. (1963). These latter authors assigned additional energy loss parameters to this particular rock type, which were similar to those of pure quartz. Lohmann et al. (1985) then adjusted these parameters to energy loss variables for calcium carbonate (i.e. calcite) and gave the standard rock its present shape. In summary, this fictitious material consists of a density of crystalline quartz (i.e. ρqtz=2.65 gcm-3), a Z and A of 11 and 22, respectively (which is almost sodium), and density effect parameters that have been measured on calcium carbonate.

However, when the material's Z and A differ greatly from standard rock parameters as for carbonates, basalts or peridotites, a substantial bias would be introduced to the calculation of the muon flux. Such a situation is easily encountered in geological settings such as the European Alps where igneous intrusions, thrusted and folded sedimentary covers and recent Quaternary deposits are found in close vicinity (e.g. Schmid et al., 1996). Currently, our collaboration is performing a muon tomography experiment in the Jungfrau region, in the central Swiss Alps, aiming at imaging the glacier–bedrock interface (Ariga et al., 2018; Nishiyama et al., 2017). There, we face a variety of lithologies ranging from gneissic to carbonatic rocks that have a thickness larger than 500 m (Mair et al., 2018). In this context, it turned out that the analyses based on the standard rock assumption might cause an over- or an underestimation of the bedrock position in the related experiment. Such an uncertainty arising from the chemical composition of the actual rock has to be reduced at least to the level of the statistical uncertainty inherent in the measurement as well as in the systematic uncertainty of the muon energy spectrum model.

To achieve this, we investigate how different rock types potentially influence the results of a muon tomographic experiment. We particularly compare the lithologic effect on simulated data with standard rock data to estimate a systematic error that is solely induced by a too-simplistic assumption on the composition of the bedrock.

Figures 2 and 3 show the muon flux simulations as a function of rock thicknesses up to 2 km for igneous and sedimentary rocks, respectively. The depth–intensity relation is described by a power law, as it is the integration of the differential energy spectrum of muons, which also follows a power law.

To better visualise the difference between the fluxes after having passed these 10 rock types and the standard rock, we report the ratio between fluxes calculated after the different materials and that after the standard rock in Fig. 4: frrock=Fcalc,rockFcalc,SR. The attenuation of the muon flux expectedly depends predominantly on the rock density, as we can see in Figs. 2 to 4. Rocks exhibiting a high material density result in a larger muon flux attenuation than lithologies with a lower density. This, however, only depicts the overall differences, including density and compositional variations, between real and standard rocks. In this regard, Groom et al. (2001) apply an explicit treatment of density variations of known materials. Thus, the flux data can be simulated for a standard rock with the exact density as its real counterpart. Such a density normalisation enables us to isolate the compositional influence on the computed data. Figures 5 and 6 show the muon flux simulations for each rock compared to a density-normalised standard rock, and Fig. 7 summarises this information by representing the ratio between muon fluxes after passing through real rocks and the muon flux after passing through a density-normalised standard rock. It is important to note that the standard rock muon flux in each flux ratio has been normalised with respect to the density of the original rock (i.e. the peridotite is compared to a standard rock of density ρ=3.340 gcm-3; the limestone is compared to a standard rock of density ρ=2.711 gcm-3). One notices that the flux ratios are rather close together, mainly within 2.5 % of the standard rock flux, before they start to diverge towards larger (dolomite, shale and arenite) and smaller (igneous rocks, arkose, limestone and aragonite) flux ratios beyond 300 m thickness of penetrated rock. Even though the errors on the fluxes are relatively large and sometimes even overlap with the standard rock fluxes, the propagated errors on the flux ratios remain well bounded near their means. This effect is due to the correlation of the errors in the numerator and the denominator in Eq. (6). A detailed discussion of how uncertainties have been propagated is presented in Appendix C.

The differences in the calculated muon flux illustrated in Figs. 2 and 3 become even more pronounced in Fig. 4, where the fluxes are compared to the case where cosmic fluxes are attenuated by a standard rock. One notices a direct correlation with material density. This is reinforced by the fact that the granite (Fig. 2) has the same density as the standard rock (2.650 gcm-3) and shows an overall similar flux magnitude as the standard rock, i.e. a flux ratio of 1. This can be explained by Eq. (4), as the energy loss is almost directly proportional to the density, while the presence of density in the ionisation loss term (i.e. {a(E,ρ,A,Z)}) is negligible compared to this factor. Thus, if the rock flux data are compared to a standard rock with equal density, this effect should be removed, and one is left with the composition difference only.

A closer look at Fig. 7 reveals that the muon fluxes for every rock below 300 m do not depart more than 2.5 % from their respective density-modified standard rock flux. The chemical composition effect can thus be considered negligible when compared to the systematic uncertainty originating from the muon flux model. We explain this through the dominance of the ionisation energy loss in this thickness region. Muons that penetrate down to 300 m of rock are still slow enough to predominantly lose their kinetic energy for the ionisation of the rock's electrons. As the number of electrons per unit volume is given by the product, ρrock⋅{Z/A}, ionisation losses are proportional to this term. When comparing a density-normalised standard rock with a real rock, the only difference can emerge from the second part, i.e. {Z/A}. According to Table 1, these values do not change more than 1 % with respect to each other.

When the rock thicknesses become larger than 300 m, the flux ratios start to exceed ±2.5 % and the ratio patterns diverge. This corresponds to the point where radiative losses start to become the dominant energy loss processes. The latter are interactions of the muon with the nuclei of the atoms within the rock and its cross-section is mainly proportional to the square of the nucleus' charge (i.e. {Z2/A}). Hence, rocks that exhibit a lower {Z2/A} value than a standard rock (e.g. dolomite, arenite and shale) attenuate the muon flux less (i.e. flux ratio >1), while all igneous rocks as well as limestone, aragonite and arkose, that have a higher {Z2/A} value, attenuate the muon flux more, which results in a lower flux ratio.

The above results reflect only the most striking connections to the chemical composition of a rock. In reality, however, the nature of muonic energy loss processes is much more complex than the shape of the flux ratios in Fig. 4 below 300 m suggests. The actual ionisation energy loss, Eq. (B27), is an interplay of the mean excitation energy {I}, i.e. the mean energy needed to ionise a material's electrons, the material density ρrock, {Z/A} and various correction terms that depend on these parameters. These additional factors are also responsible for the nonlinear behaviour of the flux ratios between 100 and around 600 m, as effects from radiative losses start to become significant. However, as the resulting differences due to these processes remain smaller than 2.5 %, a detailed discussion of these matters falls beyond the scope of this paper.

As we see above, the muon flux calculation is significantly biased when one employs the standard rock assumption and thus neglects the effect of the chemical composition, especially when the thickness of the rock is beyond 300 m. This systematic error would then later turn into an over- or an underestimation in the assessment of density structures. We can roughly estimate the error on a thickness estimation of a certain structure by employing the following formula: εdxroF=xSRF-xRoFxRoF. Here, xSRF and xRo(F) denote the thickness of standard rock and a real rock, respectively, needed to attenuate the cosmic-ray muon flux to F. This is possible because the flux, as a function of rock thickness, is a strictly decreasing function. The domain of this function ranges from zero to infinite thickness, where its image takes the values from the initial flux, F0, to zero. On these two sets, the function is a bijection, and therefore an inverse function, x(F), exists. Although its functional form might be unknown, it is still possible to interpolate between the simulated points. For our rocks, this is shown in Fig. 8.

As an example, in the case where the target is 600 m thick and made of limestone (ρ=2.711  gcm-3), the standard rock assumption underestimates the flux by 7 %–8 % and thus overestimates the thickness by around 15 m or 2.5 %. The same is valid for basalt and aragonite.

The above discussion concentrates on calculations of the mean values of model parameters. A full description encloses also the propagation of their uncertainties. The rather large error bounds on single flux calculations stem from the uncertainties in the flux model and in the interaction cross-sections. However, by taking a ratio, i.e. Eq. (6), of quantities with correlated errors, the resulting uncertainty on the ratio tends to cancel out. If the errors were propagated by linear operations, they would even cancel out perfectly. The small error bars which are still present in Figs. 4, 7 and 8 can be seen as effects of the nonlinearity in the differential equation, Eq. (5).

Because this is a pure sensitivity study, we cannot offer distinct measurements to verify our predictions. The reason for this is mainly because dedicated experimental campaigns have not yet been conducted, and thus such data are not available. We suggest that future studies in this field will address the composition issue and try to experimentally constrain this theoretical model. Nevertheless, our inferences are based on the same conceptual framework that has already been used for other materials, including standard rock. As a result of this, we find significant differences if the rock parameters are changed, especially for rock thicknesses larger than 300 m.

Our results suggest that it is safe to use the standard rock approximation for all rock types up to thicknesses of ∼300 m, as the flux ratio will mainly remain within 2.5 % of the standard rock flux, which generally lies within the cosmic-ray flux model error. However, we also find that beyond these thicknesses the use of the standard rock approximation and its density-modified version could lead to a serious bias. This mainly concerns basaltic and carbonate rocks. The flux error for these rock types increases with growing material thickness. It can be extrapolated that the errors grow even further beyond 600 m of material thickness up to a point where any inference based upon this approximation becomes difficult. This is, however, a thickness range where muon tomography becomes increasingly hard to perform, as lower fluxes have to be counterbalanced by larger detectors and longer exposure times.

In order to account for the composition of rock, it is advisable to undertake a geological study of the region alongside the muon tomography measurements, especially when faced with basaltic rocks or carbonates, which includes at the least the analysis of local rock samples. Auxiliary data could comprise rock density measurements (i.e. helium pycnometer or buoyancy experiments), chemical composition and mineralogical information (i.e. X-ray diffractometry/fluorescence measurements) as well as microfabric analyses (i.e. mineral and fabric identification on thin sections). This additional information may help to constrain solutions of a subsequent inversion to a potentially smaller set. The use of additional information, such as spatial information in the form of a geological map or a 3-D model of the geologic architecture, is strongly encouraged, because it might greatly improve the state of knowledge about the physical parameters that are to be unravelled.