The effect of rock composition on muon tomography measurements

Abstract. In recent years, the use of radiographic inspection with cosmic-ray muons has spread into multiple research and industrial fields. This technique is based on the high-penetration power of cosmogenic muons. Specifically, it allows the resolution of internal density structures of large-scale geological objects through precise measurements of the muon absorption rate. So far, in many previous works, this muon absorption rate has been considered to depend solely on the density of traversed material (under the assumption of a standard rock) but the variation in chemical composition has not been taken seriously into account. However, from our experience with muon tomography in Alpine environments, we find that this assumption causes a substantial bias in the muon flux calculation, particularly where the target consists of high {Z2∕A} rocks (like basalts and limestones) and where the material thickness exceeds 300 m. In this paper, we derive an energy loss equation for different minerals and we additionally derive a related equation for mineral assemblages that can be used for any rock type on which mineralogical data are available. Thus, for muon tomography experiments in which high {Z2∕A} rock thicknesses can be expected, it is advisable to plan an accompanying geological field campaign to determine a realistic rock model.


1 Energy loss in rocksmass averaging approach To obtain an energy loss equation for rocks, a similar procedure as for forming minerals out of elements can be applied. Starting from Eq. (B5) we consider the energy loss for a rock as mass weighted average of the energy losses of its mineral constituents where are the mass fractions of the j-th mineral within the rock, analogous to Eq. (B6), Analogue to the mineral case we can now define new average energy loss parameters for the rock. The formula for mass density, given the weight fractions, is The average { ⁄ } is given by and similarly, the average { 2 ⁄ } can be calculated according to The rock's mean excitation energy is The only difference between the rock calculation and the mineral calculation enters in the calculation of the plasma energy.
While in the mineral case we were advised to use Eq. (B11) instead of what would naturally follow from the weighted average in Eq. (B5), we prefer to use the weighted average, Eq. (S1), for the case of rocks. The reason for this lies in the fact that the density effect operates on a nanometric scale, whereas minerals, in general have sizes between several micrometres and a few centimetres. In the case of a mineral compound, the molecular structure is also on a nanometric scale.
These parameters can then be rearranged into an ionisation loss term for a rock Like Eq. (B14) the radiative losses can be rewritten as a weighted average of the mineral radiative losses Equations. (S8) and (S9) can then be joined together to form again a similar term to Eqs. (B1) and (B15), the energy loss equation for rocks. 1

Calculations for an alternative flux model
In this supplement, we present the resulting figures of our calculations for a flux model by Tang et al. (2006). Similar to the flux model by Reyna (2006), we attributed to both models a systematic error of ± 15 %. Essentially, the lithology-specific effect on the flux remains the same for all lithologies. Furthermore, the same conclusions can be drawn, i.e. that it is only safe to use the standard rock approximation up to 300 m thickness and that limestone and basalt deviate the most from the density modified standard rock flux. Table 1 Table  1 as a function of rock thickness.    Table 1 as a function of rock thickness. Figure S2-7: Relative error, which is made in the thickness estimation of a block of rock by assuming a density modified standard rock versus the actual rock thickness.

Figure S2-1: Simulated muon intensity vs. thickness of the four igneous rocks from
Most notably this flux model exhibits lower fluxes at low energies. However, this difference does not project into Fig. S2-3, S2-6 or S2-7, as the fluxes for both simulations have been changed equally. Thus, the rock composition effect is not affected by the lower flux from the flux model. The small numerical instabilities, well visible in Fig. S2-7, can be explained by the fact that the flux model by Tang et al. (2006) is a composite function for different energy ranges. At the transitions of one energy domain into another a jump might occur, which is visible as a peak in its ratio.