## 1 Introduction  
Coda waves recorded from small local earthquakes are interpreted as body waves of backscattering from heterogeneities that are distributed randomly but uniformly in the Earth's crust and upper mantle. Coda waves are predominantly composed of scattered S-waves and, in order to relate coda wave characteristics to seismic attenuation, a scattering model is necessary. In the early 1970s, [PERSON] [PERSON] [1] and [PERSON] [2] developed a single-scattering model in the Born approximation, where the primary wave encounters a scatterer (inhomogeneity) only once, producing a secondary wave reaching the receiver with no other impact. This model parametrizes the seismic attenuation using a single parameter, \(Q_{c}\), called Q-coda, which is easily calculated once the decay rate of the coda envelope is measured from data. Worldwide, Q\({}_{c}\), estimates generally report low values in regions with active tectonics, and high values in stable regions [3, 4, 5, 6]. The physical interpretation of \(Q_{c}\) is a long-standing problem in seismology. Years after the development of the early single-scattering models, it was demonstrated by [PERSON] et al. [7] that the single-scattering model is a first-order approximation of the general solution to the energy transport equation (ET) ([8], which accounts for multiple scattering and is parametrized in terms of two parameters, intrinsic \(Q_{i}\) and scattering \(Q_{s}\). New interpretations of coda wave envelopes, based on the Paasschens solution of the ET equation,then demonstrated that single-scattering models are unrealistic for most of the experiments carried out through the Earth (see, e.g., [9; 10; 11] for a review), while the other asymptotic approximations of the ET equation, the so-called diffusion model [12], are much more realistic in cases showing high geological heterogeneity, like volcanoes. Despite this evident limitation, \(Q_{c}\), or, which is the same in practice, the coda envelope decay parameter, is still used, due to the easy application of the single-scattering model to experimental data. In the single-scattering model, \(Q_{c}\) is a combination of scattering \(Q_{s}\) and intrinsic \(Q_{i}\) attenuation (\(Q_{c}^{-1}=Q_{s}^{-1}+Q_{i}^{-1}\)) [1]. It is interesting to note that the diffusion model predicts that \(Q_{c}\) asymptotically approaches the intrinsic attenuation \(Q_{i}\) at increasing times in the coda wave train (assuming a uniform half-space [12]). However, single-source-receiver couple estimates of \(Q_{i}\) and \(Q_{s}\) can only be obtained in strongly heterogeneous media, where the diffusion model is applicable. For all other cases, the separation is almost impossible and the only measurable (unbiased) parameter is \(Q_{c}\)[11]. [PERSON]'s [2] derivation of the single-isotropic-scattering model, and the numerical computations of [PERSON] and [PERSON] [13], suggest that \(Q_{c}\) is an intrinsic attenuation parameter. [PERSON]'s [7] simulations imply that \(Q_{s}^{-1}\) must be approximately from 5 to 15 times as \(Q_{i}^{-1}\) to have a comparable effect on the coda decay rate. To achieve a correct interpretation of the measured \(Q_{c}\), the association of the attenuation within volumes crossed by the coda radiation with the structural, tectonic, or geodynamic maps must be performed. Starting from the early and approximative \(Q_{c}\) images [14], many attempts were made to spatially confine the attenuation anomalies measured through coda waves (as reviewed by [PERSON] et al. [11]), in order to reveal heterogeneous Earth structures and interpret them in terms of rock quality. Among the imaging methods, sensitivity kernels are essential to obtain an attenuation image from coda waves; the theoretical approach, in the framework of both radiative transfer theory and the diffusion model, focuses on the spatial changes of \(Q_{c}\) from distributed sources recorded at a seismic network [15; 16].