## Materials and Methods
### Theoretical Method  
The method implemented is based on an innovative two-dimensional (2-D or 2.5 D, if we take the ground reflection into consideration) formulation founded in UTD (uniform theory of diffraction) to analyze multiple diffraction/reflection of acoustic waves; the authors demonstrated, made comparisons, and validated such formulation in [43]. The implemented model is deterministic (it does not pose any inherent randomness), for which the output is fully determined by the parameter values and the initial conditions. This UTD methodology is enhanced by using graph theory, Fresnel ellipsoids, and funicular polygons, so that consideration is only given to those paths and obstacles in this complex environment that make a contribution to global _IL_. This allows for the provision of swift, accurate, and efficiently computed predictions for sound attenuation, something that cannot be achieved by employing alternative more time demanding techniques (e.g., BEM). Using this technique, a substantial quantity of obstacles (which includes adjacent ones of the same height) may be managed in high-frequency resolution and in a sufficiently short time. We can model the obstacles as cylinders, rectangles, wedges, or knife edges, and also as a number of other polygonal deflecting obstacles (e.g., T-or Y -shaped barriers or trapezoids). In this manner, the total diffracted (and reflected on ground/floor) complex pressure field emanating from a source _Tx_ at any frequency to the receiver _Rx_ will represent the entirety of the rays converging on _Rx_ on every possible path:  
\[\phi_{tRx}=\sum\
olimits_{i=1}^{n}\phi_{tpath\;i}(Tx,Rx), \tag{1}\]  
with \(\phi_{tpath\;i}(Tx,Rx)\) being the complex received field for one ray from _Tx_ to _Rx_, which follows one possible selected path n. It is assumed that _Tx_ is a point source generating spherical wave fronts in the context of a perfect isotropic (uniform) medium (e.g., air), whilst the receiver of this acoustic field _Rx_ is a listener where the isotropic pattern has unity gain. Consideration of the phase occurs for every signal on every path, utilizing the UTD theory combining the diffraction phenomenon with ray representation, permitting the field at the receiver to be calculated as a vector derived from combining all the fields that correspond with each ray. Every frequency bin is then run on a loop from the minimum to maximum to derive the whole sound pressure field for a full set of paths on a specific frequency. With every frequency, the algorithm further permits checking of the sound pressure field levels for every path at the receiver. With sorting of the set of paths, it is predicted that the derived sub-fields will progressively decrease, defining an adjustable pressure condition that can block new fields from being added when the current field does not meet the threshold. We may define the threshold as being the absolute value for the ratio between the signal level received on the shortest path at this frequency and the signal level received for the current path (e.g., \(1\times 10^{-5}\)). When the threshold level is set at 1, this ensures that no path is discarded. It is important to note that the compensation time for this suggested algorithm is independent of frequency and may be restrained or limited at any time, as worst-case scenarios can be predicted through the multiplication of consumed times at any frequency by the quantity of frequency bins under consideration in the simulation. Regarding the above, obtaining the complex field at the receiver for all frequencies and all paths may be achieved using the following expression:  
\[\phi_{tpath\;i}=\frac{\phi_{0}}{s_{T}}\cdot e^{-jb_{T}}\cdot\prod\limits_{n=1} ^{N-2}\left( \begin{array}{c}D_{n}\\ R_{n}\end{array}