© The Author(s), Published by EDP Sciences, 2026

1 Introduction

Auralization of urban environments is emerging as a significant area of research for designing, evaluating, and communicating noise mitigation measures of (future) urban spaces. Yet, generating auralizations in dynamically changing, arbitrarily built urban environments presents unique challenges. Urban noise pollution can have significant effects on human health [1, 2]. For that reason, noise control in cities is becoming ever more important. This trend is also evident in the growing body of research on the subject [3–6]. In addition, traffic noise has become a rising concern, addressed by progressively stricter national (e.g., RLS19 for Germany) and European regulations [7, 8]. Due to the complexity of the geometry in urban environments, with buildings, noise barriers, and other obstacles, the prediction of noise and thus the informed proposal of noise mitigation measures is a major challenge. For this reason, engineering methods that account for the geometry have been developed [8–10]. However, noise perception is subjective and can vary significantly between individuals [4, 11, 12]. As a result, a purely numerical prediction of noise levels is not sufficient to assess the impact of noise mitigation measures. Also, the communication of noise mitigation results can be difficult, as the results often need expert knowledge to be interpreted correctly. Moreover, the individuals who are affected by noise are usually non-experts for whom the numerical results are not intuitive. To address the subjective aspects of noise perception and to effectively communicate predictions and future designs of urban spaces, auralizations can be employed. They can be used alongside psychoacoustic metrics or listening tests to evaluate the effectiveness of noise mitigation measures. Moreover, non-experts can be presented with auralizations to communicate the results of noise mitigation measures in an intuitive way [11, 13]. It is already used in architectural acoustics [14–16]. However, the auralization of urban environments poses distinct challenges compared to room acoustics. Rooms are typically concave spaces with minimal or no movement of the source and receiver. For these scenarios, ray-tracing, the Image Source Model (ISM) or hybrid approaches can be used to simulate the sound propagation. In urban environments, the source and receiver can move, sometimes at high speeds, and thus Doppler effects need to be considered. Additionally, the geometry is often convex, which limits the number of reflections. In the context of this work, convex geometry refers to environments where not all face normals point towards each other, as also depicted in Figure 1. Furthermore, as buildings and other obstacles can interfere with the sound propagation, diffraction becomes more significant and needs to be included [11, 14, 17].

Figure 1. Schematic definition of convex and concave geometries. In convex geometries, not all face normals (red arrows) point towards each other, while in concave geometries, all face normals point towards each other.

Engineering-based auralization methods that account for reflections, diffraction, and Doppler effects have been proposed [10, 18–20]. The scope, fidelity, and computational cost of these methods are determined by their underlying modeling assumptions and numerical formulations. For example, Viggen et al. [18] considers propagation in a vertical cross-section, which may omit lateral propagation effects and associated reflections and diffraction. Many engineering models provide third-octave-band or magnitude-only outputs, reducing access to narrow-band spectral and phase information. Several tools target specific source types, such as aircraft [21–25], wind turbines [26], railway noise [27, 28], or road traffic [17, 29]. In such tools, propagation phenomena are often simplified (e.g., flat ground or a limited number of paths). Within their intended scope, these approaches can produce plausible auralizations; generalization to other sources or scenes may be limited. Auralizations can also be derived from impulse responses measured in existing environments [30, 31] or obtained via numerical simulation [32, 33]. This strategy can capture detailed propagation effects for the specific configurations represented; however, it is tied to those configurations and, for numerical simulations, can be computationally demanding. In time-varying scenarios, cross-fading between impulse responses may introduce comb-filtering, and Doppler effects are typically not represented. Real-time methods [34–37] prioritize computational efficiency and hence adopt approximations. For instance, some formulations model diffraction only in the geometric shadow region, which can lead to discrepancies at reflection boundaries [34]. When considering Higher Order Diffraction (HOD), interactions between edges can be significant [38, 39]. In some real-time implementations, however, HOD are approximated by treating each edge as sequentially independent first-order diffractions [35], neglecting these interactions.

To the authors’ knowledge, no open-source auralization tool exists that can efficiently simulate physically accurate sound propagation without efficiency approximations in dynamically changing, arbitrarily built environments, with no limitations on the geometry or the sound source type. In this work, an open-source implementation of such a method is presented. The proposed method is based on a Geometrical Acoustic (GA), path-based approach that simulates reflections, diffractions and Doppler effects. By combining established propagation models, it aims to provide a comprehensive solution for auralization in complex, dynamic environments. Consequently, the accuracy of the framework is limited by the underlying assumptions of these models, such as the high-frequency validity of GA and the infinite-edge approximation adopted in the employed diffraction formulation (see Sect. 2.2). Within this scope, the framework itself does not introduce further heuristic simplifications of the sound propagation. The presented method is implemented as a modular, flexible framework with open interfaces and open exchange formats. These design choices enable integration with existing toolchains, straightforward addition of simulation methods, and module exchange, allowing adaptation to diverse use cases and requirements. For example, if the diffraction model is not suitable for a specific scenario, it could be replaced with a more accurate, albeit computationally more expensive, model. In this contribution, we evaluate the proposed method to assess its capabilities and limitations across various scenarios, as well as to determine whether the established objectives are achieved.

2 Auralization method

The auralization approach presented in this work is a GA, path-based propagation approach [40–42]. The method can be separated into three main steps, as depicted in Figure 2:

Figure 2. Schematic overview of the proposed auralization scheme. The process consists of three main steps: determining propagation paths, calculating propagation parameters, and rendering the audio. The process is repeated for each audio block to adapt to dynamic scenarios.

1. Propagation pathfinder: Based on a given geometry, and the source and receiver positions, a set of propagation paths are determined. This is done via geometric constructions.

2. Propagation parameter calculation: Based on the determined paths, propagation parameters like spreading loss, propagation delay, air absorption, reflections and diffractions are calculated. This is done for each path separately.

3. Digital signal processing: The calculated propagation parameters are applied to the source signal to create the auralization. This is done for each path separately as well. This approach allows for increased flexibility for spatialization.

In the following sections, each step is described in more detail.

2.1 Propagation path determination

As a general concept, a propagation path defines a (geometric) connection between a source and a receiver through which acoustic propagation occurs. The propagation path can be determined via geometric constructions based on the source and receiver positions and the geometry. The simplest example is the direct sound path, which is a straight line between the source and receiver. When considering a given geometry, additional paths can exist. For instance, reflections from the walls, ground, or other surfaces generate reflected paths. However, when the geometry is convex, diffraction needs to be considered. Depending on the geometry and the source and receiver positions, these paths can be blocked, e.g. if the source is behind a wall. However, diffraction should be considered not only if a path is blocked, but further details will also be given in Section 2.2. Consequently, diffraction can introduce additional paths. Moreover, there may be paths that involve a combination of reflections and diffractions. Note that all path segments are straight lines.

If only reflections are considered, the ISM [43, 44] can be used to determine the paths. In urban environments, where the geometry is often convex, the ISM can be insufficient as it cannot identify diffracted paths. As a solution, the Image Edge Model (IEM) [45] can be used to determine the paths.

The IEM [45] is an extension of the ISM that can determine propagation paths including reflections and diffractions. For reflections, the IEM uses the same approach as the ISM. For diffractions, based on the law of edge diffraction [46] an Equation System of Equal Angle (ESEA) proposed by Tsingos et al. [34] can be used to determine the diffraction paths geometrically. The IEM combines and extends these approaches by also mirroring the edges on the faces of the geometry, as done with the source in the ISM. These image edges are included in the ESEA. This allows for the determination of heterogeneous paths, i.e. paths with a combination of reflections and diffractions.

2.2 Propagation parameter calculation

In line with the work of Schäfer et al. [42], propagation parameters describe how the sound propagates from the source to the receiver. These parameters are calculated based on each of the determined paths from the previous stage. The parameters considered in this work are spreading loss, propagation delay, air attenuation, reflections and diffractions. The latter three are also frequency dependent. The parameters are kept separate, which allows for increased flexibility in the signal processing step and is essential when simulating the Doppler effect. The spreading loss is calculated based on the path length and the current wave type. Similarly, the propagation delay is calculated based on the path length and the speed of sound. The air absorption is calculated based on the path length and ISO 9613 [47].

In order to model reflections, the reflection factor, R, needs to be determined for each material on which the sound reflects. If a surface is characterized by absorption coefficients, R is obtained from

$$\begin{array}{c}\hfill R=\sqrt{1-\alpha },\end{array}$$(1)

where α denotes the (frequency dependent) absorption coefficient. If the material specifies the surface impedance Z, the reflection coefficient can be calculated as

$$\begin{array}{c}\hfill R=\frac{Zcos(\theta )-Z_0}{Zcos(\theta )+Z_0},\end{array}$$(2)

with Z ₀ the air impedance and θ the angle of incidence.

Surface parameters are usually specified in 1/3 octave bands; for further processing, they are interpolated to the desired frequency resolution. Because phase information is crucial for the diffraction modeling (described next), magnitude only responses are converted to their minimum phase equivalents. It was shown that this is a valid approximation for reflection modeling [48]. For paths that contain multiple specular reflections, the reflection spectra are combined via frequency domain multiplication.

To simulate diffraction within a GA context, multiple approaches exist. For the presented method, the Uniform Theory of Diffraction (UTD) [49] with the Fresnel approximation by Kawai [39] is used to determine the diffraction filter. This model is chosen for its computational efficiency compared to analytical models like Biot–Tolstoy–Medwin–Svensson (BTMS) [50]. Additionally, in the context of urban auralizations, the underlying assumption of infinite edges is assumed to be valid for the frequency range of interest and the typical, large geometrical dimensions. The UTD is an extension of the Geometrical Theory of Diffraction (GTD) [46] and can be used for [38, 39, 49]

$$\begin{array}{c}\hfill r>\frac{\lambda }{4}\Rightarrow f>\frac{c}{4r}·\end{array}$$(3)

Here, f is the frequency, c the speed of sound, λ the wavelength and r the radial distance with the diffraction edge as the origin. The required geometrical parameters for the diffraction filter are contained within the propagation path definition. Coming back to the importance of diffraction in a GA-based simulation context, Kouyoumjian and Pathak [49] state that the sound field around a diffraction edge can be described as

$$\begin{array}{c}\hfill E=E_i{u}_i+E_r{u}_r+E_d,\end{array}$$(4)

where E _i is the incident (or direct) sound field, E _r the reflected sound field and E _d the diffracted sound field. u _i , u _r are unit step functions that describe where the respective sound field is present. These functions can be determined via GA methods. From equation (4), Kouyoumjian and Pathak [49] stated that the diffracted sound field compensates for the discontinuity of the incident and reflected sound field at their boundaries and must be present all around the considered edge. Based on this, it can be concluded that for an accurate GA simulation, diffracted sound paths must be considered all around the diffraction edge. The perceptual importance of this was also found by Torres et al. [51].

The work of [49] focused on diffraction on a single edge. In realistic scenarios, however, multiple edges can be present. One approach to model these HOD is to approximate them via sequential, first-order diffractions where each edge is treated independently and the final diffraction coefficients are computed via frequency-domain multiplication [35]. If we define D ⁽¹⁾(f) as the diffraction coefficient for a first-order diffraction on a single edge as defined by Kouyoumjian and Pathak [49], the total diffraction coefficient D _total(f) for N sequential edges would be approximated as

$$\begin{array}{c}\hfill D_{}\text{total}(f)=\prod _{i=1}^N{D}_i^{(1)}(f).\end{array}$$(5)

This approach neglects interactions between edges, which can be a significant error. Interaction means that the edges cannot be treated one by one, each being located in the far field of the other. Instead, the incident sound field at one edge is influenced by the diffracted waves from other edges. For this, Kawai [39] extended the UTD to HOD with inter-connected edges. Kim et al. [38] then extended the model further to include HOD for polygonal shapes with arbitrarily connected edges.

In this work, the model from Kim et al. [38] is adapted to also separate the propagation delay, spreading loss, and diffraction to fit into the presented framework. Additionally, the model is enhanced to accommodate heterogeneous paths identified by the IEM. This is feasible because of the linear characteristics of the propagation parameters, allowing the previously mentioned reflection factor R to be combined with the diffraction parameters during the signal processing phase. To achieve this, equation (18) from Kim et al. [38] for the total diffraction field ϕ is rearranged to:

$$\begin{array}{c}\hfill \varphi =\underset{\text{Propagation delay}}{\underbrace{e^{-jkL}}}·\underset{\text{Spreading loss}}{\underbrace{\frac{1}{L}{\prod }_{Q=1}^N\frac{1}{\sqrt{A_Q}}}}·\underset{\text{Diffraction}}{\underbrace{{\left(\frac{1}{2}\right)}^M{\prod }_{Q=1}^N\frac{1}{\sqrt{B_Q}}{D}_Q}},\end{array}$$(6)

with N being the number of diffractions and M the number of connected edges. Two diffraction edges are considered connected if they share a common face, for example, the left and right edge of a wide barrier. In order to define the other parameters, we define the sequence of diffraction path lengths via a sequence $P={(P)}_{i=1}^{(N+1)}$, where each element P _i is the total length between the source, diffraction edges or the receiver. If the path contains reflections, the diffraction path length is defined as the sum of the partial path lengths. For the example in Figure 3, the diffraction path lengths are defined as

Figure 3. Example for two HOD propagation paths over a wide barrier with a ground. Due to the reflection on the ground, a second path Path 2 exists. More paths exist for this scenario, but are omitted for clarity.

$$\begin{array}{cc}\hfill P_{}\text{Path}\text{1}& =(S,W,R)\hfill \end{array}$$(7)

$$\begin{array}{cc}\hfill P_{}\text{Path}\text{2}& =(S,W,R_1+R_2),\hfill \end{array}$$(8)

where the partial path lengths are shown in the Figure 3. This is in line with the IEM where diffraction edges are also mirrored on faces. With ${(P)}_{i=1}^{(N+1)}$, we can define the total path length L as

$$\begin{array}{c}\hfill L=\sum _{i=1}^{(N+1)}{P}_i.\end{array}$$(9)

For the example in equations (7) and (8), this results in

$$\begin{array}{cc}\hfill L_{}\text{Path}\text{1}& =S+W+R\hfill \end{array}$$(10)

$$\begin{array}{cc}\hfill L_{}\text{Path}\text{2}& =S+W+(R_1+R_2).\hfill \end{array}$$(11)

Similarly, the coefficients A _Q can be defined as

$$\begin{array}{c}\hfill A_Q=\frac{{\sum }_{i=1}^Q{P}_i·{\sum }_{Q+1}^{(N+1)}{P}_i}{L}·\end{array}$$(12)

This formulation is a generalization of equation (32) from Kouyoumjian and Pathak [49] for diffractions with cylindrical wavefront incident. For Path 1 this results in

$$\begin{array}{cc}\hfill A_1& =\frac{P_1·(P_2+P_3)}{L}=\frac{S·(W+R)}{L},\hfill \end{array}$$(13)

$$\begin{array}{cc}\hfill A_2& =\frac{(P_1+P_2)·P_3}{L}=\frac{(S+W)·R}{L}·\hfill \end{array}$$(14)

For N = 2 as in Figure 3,

$$\begin{array}{c}\hfill B_Q=\{\begin{array}{cc}1\hfill & \text{if}{X}_1^{-}(·)>X_2^{-}(·)\hfill \\ \rho \hfill & \text{otherwise}\hfill \end{array}\end{array}$$(15)

where X _Q ⁻(⋅) is defined in equation (6) from Kim et al. [38], and the coefficient ρ is defined as

$$\begin{array}{c}\hfill \rho =\frac{L{P}_2}{(P_1+P_2)(P_2+P_3)}·\end{array}$$(16)

For N >  2, the definition of ρ and B _Q is similar, but becomes more complex. As a side note, if we set B _Q  = 1 for all Q, the model reduces to the approximation via sequential, first-order diffractions mentioned earlier (Eq. (5)). Lastly, D _Q in equation (6) is defined as in equation (3) from Kim et al. [38]. Appendix A provides the full definitions of D _Q for completeness. For further details on the definitions of L, A _Q , B _Q , D _Q and ρ, please refer to the definitions in [38]. An implementation of the model is also provided via the open-source software PELICAN [52].

2.3 Signal processing

Similar to the path determination and the propagation parameter calculation, the signal processing is done for each path. The Digital Signal Processing (DSP) structure for a single path is depicted in Figure 2. The propagation delays are applied to the source’s audio signal via a block-based Variable Delay Line (VDL). Schäfer et al. [42] showed that this approach delivers good results if the delay is sampled at a sufficient rate or is interpolated appropriately. As a result of the use of a VDL and the path based approach, per-path Doppler shifts are applied [40, 53]. Frequency independent parameters like the spreading loss are applied via a simple multiplication of the signal. Due to the linearity of the propagation parameters, the frequency dependent parameters air attenuation, reflections and diffractions are combined into a single filter via frequency domain multiplication. This combined filter can then be applied to the audio signal, for example via Finite Impulse Response (FIR) convolution. For the implementation presented in this work, this structure has an added advantage: Since the frequency dependent parameters are calculated in the frequency domain and the FIR convolution is also performed in the frequency domain, the filters do not need to be transformed back to the time domain, reducing the number of Fast Fourier Transform (FFT) operations required.

At this stage, an additional benefit of the path-based approach can be observed: Since the paths are not combined, comb-filter effects due to similar lengths of the paths do not have to be considered. Furthermore, since the propagation delay is applied via a VDL, the filters do not inherit a linear phase shift, further reducing the comb-filter effects. As a result of these two facts, the size of the filter can be reduced, which in turn reduces the computational complexity. It has to be noted though that due to the additional path processing, the overall computational complexity could still be higher than a single high-resolution filter. Lastly, all paths are summed up to create the final auralization. This highlights another benefit of the path based approach: Since the paths are processed separately, instead of a simple summation, spatialization can be applied at this stage. This could be done via a convolution with Head Related Transfer Functions (HRTFs), ambisonic processing or other methods. Though, as this is not the focus of this work, this will not be discussed further.

3 Implementation

A fully open-source implementation for the proposed method in Section 2 is available. The implementation is split into four main parts, reflecting the three steps in Figure 2, with an additional part to tie them together. The propagation paths are determined in PIGEON [54], which implements the IEM [45] as described in Section 2.1. The propagation parameters are calculated in PELICAN [52] as described in Section 2.2. The application allows to configure different propagation models like the UTD [49] or the HOD model by Kim et al. [38]. The signal processing as described in Section 2.3 is done with the DOVELET [55] library/application. Again, the package allows to configure different signal processing methods, like different filtering methods. Lastly, the three parts are combined in pynamic-pigeon-auralization, pynamic for short [56]. pynamic is a Python package that provides a simple interface for the user to configure the auralization process. The scenes are defined via Blender [57], including the geometry, source, receiver, their trajectories and the material properties. A detailed parameter-set controlling the full auralization process can be configured. With the connection to Blender, a visualization including the paths can be created as well. The results presented in Section 4 can be generated with this implementation and are included as examples in the pynamic repository.

4 Results

In the following section the proposed simulation method is evaluated. For this, multiple scenarios are considered with the aim to test a wide range of scenarios from simple test cases to simplified urban environments. First, a simple wedge scenario is used (Sect. 4.1). This can be considered as the fundamental validation scenario for diffraction, because if the method would not work here, it would not work in more complex scenarios. The next scenario is a wide barrier with a ground plane (Sect. 4.2). This can be considered both as a validation scenario for higher order diffractions and combination with reflections, as well as a typical urban scenario like terraced housing. The third scenario is a 95° interior wedge scenario (Sect. 4.3). This scenario is used to test the behavior of the method for non right angles. This can be a critical scenario as the edges of the volume where the Image Sources (ISs) are audible do not perfectly align in space as they would for right angles. In the fourth scenario, a building corner is considered (Sect. 4.4). This scenario is used to test the behavior of the method for more complex and urban-like scenarios as the previous scenarios are rather simplified (only two-dimensional). The scenarios, including the trajectories, will be presented briefly in the following sections. The maximum orders used in these scenarios are given in Table 1. For a detailed definition of the scenarios, please refer to the configuration and Blender files available in the pynamic repository [56]. Additional scenarios are included in the supplementary material to show the flexibility of the method, however, these scenarios are not evaluated in detail here.

In general, no directivities are considered in the scenarios, still, the presented method does allows for directivities to be included. Furthermore, all surfaces are assumed to be rigid and have a reflection factor of 1. These simplifications are made to focus on the propagation effects in scenarios where the propagation paths are changing over time. In these scenarios, diffraction becomes important and is a main focus of this work. For all scenarios, two source signals are used. The first is a triangle wave with a fundamental frequency of 440 Hz. This tonal signal is used to evaluate the method against a signal with a significant tonal frequency content. The triangle wave is used instead of a sine wave to include a broader frequency spectrum. Due to the tonal frequency content, it can be used to evaluate the Doppler effects and possible artefacts in the signal. Furthermore, the triangle wave can represent typical tonal sources that can be found in urban environments, like Acoustic Vehicle Alerting System (AVAS) or multi-rotor drone sounds. The second source signal is pink noise which can be considered as an example for tire noise. The spectrograms of the results were created with the pyfar [58] package, with a 2048 Sample window length, 75% overlap and with the Root Mean Square (RMS) FFT normalization. The result signals were normalized to have a maximum value of 1.

4.1 Wedge scenario

The wedge scenario validates that the diffraction implementation works correctly and that the resulting sound field, and thus the auralization, is contiguous. Figure 4 shows the geometry of the scenario. Consider a cylindrical coordinate system where the longitudinal axis is aligned with the edge of the wedge. The main face of the wedge will be aligned with the φ = 0 axis. The opposite face of the wedge will be aligned with the $\phi =270°=\frac{3\pi }{2}$ axis. The source trajectory is a spiral that can be described as

Figure 4. Considered geometry for the wedge scenario. The wedge has an exterior angle of 270°. The source (in blue) rotates around the edge and moves along the z-axis of a cylindrical coordinate system. The receiver (in red) is static. Both positions are described in detail on equations (17) and (18). The yellow line indicates the source trajectory. The origin of the coordinate system and a 1 m grid are also depicted.

$$\begin{array}{c}\hfill S(t)=\left(\begin{array}{c}\varrho \\ \phi \\ z\end{array}\right)=\left(\begin{array}{c}5\\ \frac{3\pi }{2}(1-\frac{t}{4.7})\\ 5-\frac{10t}{4.7}\end{array}\right)\end{array}$$(17)

for t ∈ [0, 4.7]. Within this coordinate system, the receiver is located at

$$\begin{array}{c}\hfill R(t)=\left(\begin{array}{c}\varrho \\ \phi \\ z\end{array}\right)=\left(\begin{array}{c}4.1\\ 0.172\pi \\ 0\end{array}\right).\end{array}$$(18)

As a result, the source starts in the shadow zone, moves into the illuminated zone (at 1.033 s), and then into the reflection zone (at 2.11 s). For the diffraction model, the wedge is assumed to be infinite.

The outcomes of the described scenario are illustrated in Figure 5. Please note that all the resulting auralizations can be listened to in the supplementary materials. The results for the triangle wave source signal can be found in Figure 5b. While no significant artefacts are noticeable, minor artefacts can be observed around 1.033 s and 2.11 s. Yet, these artefacts are more than 50 dB quieter than the primary signal. The times correspond to the instances where the direct sound and the reflected sound paths are added. To further investigate the artefacts, Figure 5c shows a detailed view of the first artefact with the diffraction and direct sound path separated. It can be seen that the total signal has a single sample artefact at the time where the direct sound is included. The cause of this artefact can be explained by equation (4) where the direct (and reflected) sound field needs to inherit a unit-step-like behavior when they are included. These discontinuities are then compensated by the diffracted path. This behavior can be seen in Figure 5c as well. Nonetheless, in the results presented, the direct sound does not exhibit a perfect step function. This can be seen as the first sample of the direct sound (marked with a black circle) has a slightly lower amplitude than expected. The reason for this discrepancy is currently unknown. Based on tests with the individual DSP elements, it is assumed that the error is caused by the interplay of the DSP elements as the elements separately do not show this behavior. This makes it difficult to determine the exact cause of the artefact. Additionally, low-frequency artefacts (below 40 Hz and around −60 dB) can be observed. These are caused by the abrupt switching of the filters at each block. Figure 5a shows the results for the pink noise source signal. Compared to the triangle wave signal, no obvious artefacts can be seen. The low-pass characteristic of the diffraction can be observed in the beginning of the simulation. Furthermore, comb-filter effects are visible towards the end of the simulation, when the source is in the reflection zone.

Figure 5. Results for the wedge scenario. Figure 5a and 5b show the spectrograms for the pink noise and triangle wave signals, respectively. Figure 5c shows a detailed view of the triangle wave signal to investigate the discussed artefact. Figure 5d shows the Doppler effect for the triangle wave signal.

Lastly, to evaluate the Doppler effect, in Figure 5d the fourth harmonic of the triangle wave signal, which is at 2200 Hz, is depicted. Two distinct lines can be seen in the spectrogram. One is the diffracted path that then combines with the direct sound path, the other is the reflected path. As the propagation paths are not the same length, their propagation delays are different and the Doppler effect is different for each path. This highlights the benefit of the path based approach, as each path can be processed separately. Furthermore, even though the Doppler effect is considered on a block-based approach, each path exhibits smooth Doppler effects as was also shown in Schäfer et al. [42].

4.2 Wide barrier scenario

The next test scenario is a wide barrier with a ground plane. Figure 6 shows the geometry of the scenario. For testing purposes, this is a good extension compared to the wedge scenario for two reasons: (a) due to the addition of a ground, heterogeneous propagation paths can be tested that include a diffraction from the barrier and a reflection from the ground, (b) the barrier is wide enough to investigate the effect of second-order diffractions (and thus HOD). As can be seen in Figure 6, the barrier is 6 m wide and 3 m high. In Cartesian coordinates with the origin on ground level at the center of the barrier, the elliptical source trajectory can be described as

Figure 6. Geometry of the wide barrier scenario. The barrier is 6 m wide and 3 m high. The source (in blue) is on an elliptical trajectory around the barrier. The source trajectory, indicated by the yellow line, is also described in detail in equation (19). The static receiver (in red) position is described in equation (20). The origin of the coordinate system and a 1 m grid are also included.

$$\begin{array}{c}\hfill S(t)=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}0\\ -7cos\left(\frac{\pi }{4.17}t\right)\\ 4.9sin\left(\frac{\pi }{4.17}t\right)+1\end{array}\right)\end{array}$$(19)

For t ∈ [0, 4.17]. The receiver is placed at

$$\begin{array}{c}\hfill R(t)=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}0\\ 5\\ 2\end{array}\right).\end{array}$$(20)

Like the wedge scenario, the source begins in the double shadow zone and moves into the single shadow zone before entering the illuminated zone. Towards the end of the auralization, reflections from both the barrier and ground plane are also included. Note that due to the ground, also reflections of second order must be considered. As with the wedge scenario, this scenario can be considered infinite in the x-axis and for the diffraction calculation.

This scenario can be used to validate HOD, in order to show why a correct handling of HOD is important for dynamic auralizations. First the simplified simulation approach is considered (mirroring the approach taken in [35]), where diffractions of higher order are approximated by a sequential first-order diffraction approach. Figure 7 shows the results for this approximation. At around 0.45 s, when the source transitions from the double shadow zone to the single shadow zone, a very strong change in amplitude can be seen where a smooth transition is expected. The reason for this is that the additional first-order diffraction path cancels out the second-order diffraction path. This can be explained by a simplified example. To aid in this, Figure 8 shows a schematic illustration of the scenario. For this example, we will assume that a transition from the shadow zone to the illuminated zone the phase of the diffraction path flips 180°. In the figure and in the following this is denoted as e^j0 = +1 in the shadow zone and e^jπ = −1 in the illuminated zone. After the transition into the single shadow zone of the barrier (Section I to II in Fig. 8), two primary diffraction paths are present: Path A, a first-order diffraction path that is diffracted over the right edge of the barrier (cf. Fig. 8, petrol, dotted) and Path B, a second-order diffraction path that is diffracted over the left edge and then diffracted over the right edge (cf. Fig. 8, violet, dashed). For Path A, the source is in the shadow zone of the right edge of the barrier, thus the phase is +1. For Path B, in the sequential approach, we need to consider the two edges separately. For the first diffraction (left edge), the right edge is considered as the receiver. Thus, the source is in the illuminated zone relative to the right edge, and the phase is −1. For the second diffraction (right edge), the left edge is the new source position and is in the shadow zone of the actual receiver, thus the phase is +1. Combining the two diffractions for the second-order diffraction path gives −1 ⋅ +1 = −1. In the final auralization the two paths A and B are combined, giving +1 + ( − 1)=0. This is a very simplified example, but it highlights the problem with the sequential approximation and the importance of the HOD model as it is designed to handle this case correctly. In the context of static simulations, this approach can lead to reasonable results if the errors at the transition boundaries are accepted. In dynamic simulations, however, this leads to significant problems as can be seen here.

Figure 7. Time domain view of the triangle wave signal using a sequential first-order diffraction approximation. At around 0.45 s, where the source transitions from the double shadow zone to the single shadow zone, a significant change in amplitude can be seen.

Figure 8. Schematic illustration showing the cause of the error in the sequential HOD approach. The source (blue) rotates around the wide barrier as in Figure 6. Also indicated by the arrow denoted with t. The gray dashed lines show the receiver’s (red) shadow boundaries for the two edges of the barrier. These boundaries separate the space into multiple regions, here only two are denoted: (I) double shadow zone, (II) single shadow zone. The petrol dotted arrow indicates the regions where the propagation path A with first-order diffraction on the right edge of the barrier exists. Indicated by the violet dashed arrow are the regions where the propagation path B with second-order diffraction exists. The two paths also exist further to the right (indicated by the three dots), but are omitted for clarity.

The results of the scenario utilizing the HOD model are shown in Figure 9. For both source signals, the outcomes demonstrate very good performance with regards to artefacts due to discontinuities (cf. Figs. 9a and 9b). The pink noise signal depicted in Figure 9a appears to be free from artefacts once again. The triangle wave signal in Figure 9b shows some small artefact (most notably around 2.7 s and 3.3 s), the cause of this is the same as for the artefacts in the wedge scenario in Section 4.1. Here these artefacts are even smaller than in the wedge scenario. A reason for this reduction in artefact level is additional paths exist in this scenario. Some of these paths have a higher amplitude and thus partially mask the artefacts. Figure 9c shows a time domain view of the triangle wave signal. Compared to the result without the HOD model (cf. Fig. 7), the expected smooth transitions are now present.

Figure 9. Auralization results for the wide barrier scenario using the HOD model.

Due to the ground, this scenario can also be used to validate both paths with a combination of diffraction and reflection as well as the interplay of these paths with the second-order reflection paths. At around 2.7 s the source moves into the reflection zone where the second-order reflection paths are present. Aside from the previously mentioned artefacts, no additional artefacts can be seen for any transitions, even second-order reflection transitions. It has to be noted that the interior angle of the barrier is 90° where the second-order reflections from the two faces transition seamlessly. In Section 4.3 this will be investigated further with a 95° interior corner scenario.

4.3 95° interior corner scenario

The third scenario involves a corner with an opening angle of 95°, as illustrated in Figure 10. This scenario is utilized to evaluate the method’s performance with non-right angles. Considering a cylindrical coordinate system with the longitudinal axis aligned with the edge of the corner, the source moves on a circular trajectory with a radius of 5 m around the corner. One face of the corner is aligned with the φ = 0 axis, while the opposite face is aligned with the $\phi =95°=\frac{95\pi }{180}$ axis. In the coordinate system, the source trajectory can be described as

$$\begin{array}{c}\hfill S(t)=\left(\begin{array}{c}\varrho \\ \phi \\ z\end{array}\right)=\left(\begin{array}{c}5\\ \frac{95\pi }{180}(1-\frac{2}{4.17}|t-\frac{4.17}{2}|)\\ 0\end{array}\right)\end{array}$$(21)

for t ∈ [0, 4.17]. This trajectory can be interpreted as the sound source bouncing back and forth between the two faces of the corner. Within this coordinate system, the receiver is located at

$$\begin{array}{c}\hfill R(t)=\left(\begin{array}{c}\varrho \\ \phi \\ z\end{array}\right)=\left(\begin{array}{c}3.16\\ 0.1615\pi \\ 0\end{array}\right).\end{array}$$(22)

Figure 10. Geometry of the interior corner scenario with an opening angle of 95°. The source (in blue) moves on a circular trajectory with a radius of 5 m around the corner. The source trajectory, indicated by the yellow line, is also described in detail in equation (21). The static receiver (in red) position is described in equation (22). The origin of the coordinate system and a 1 m grid are also included.

The results of the described scenario are shown in Figure 11. For both source signals, the results show minimal to no artefacts (cf. Figs. 11a and 11b). The triangle wave result in Figure b again has the first sample issue, though it is much quieter compared to the wedge scenario (Sect. 4.1) at around −70 dB. For the pink noise signal in Figure 11a, the comb-filter effects due to the reflections can be seen. Notable here is that for 0.4 s to 0.84 s and 3.3 s to 3.75 s the second-order reflection cannot be determined via the ISM but is compensated by the first-order diffraction path. This highlights another benefit of including diffraction in the auralization as it compensates for the missing second-order reflection path.

Figure 11. Results for the interior corner scenario.

4.4 Building corner scenario

The next scenario considers a more urban-like environment, as shown in Figure 12. In addition to representing a more realistic setting of a building with a flat roof and ground, this scenario also tests two important use cases. First, this scenario features a three-dimensional geometry, in contrast to the previous scenarios that can be regarded as two-dimensional. Second, this scenario includes geometries with finite-length edges. As can be seen in Figure 12, here the source moves on a linear trajectory from the left side of the building to the right side. In a Cartesian coordinate system with the origin on the ground at the building corner, the source trajectory can be described as

Figure 12. Considered geometry for the building corner scenario. The source (in blue) moves on a linear trajectory from the left edge to the right edge of the building corner. The source trajectory, indicated by the yellow line, is also described in detail in equation (23). The static receiver (in red) position is described in equation (24). The origin of the coordinate system and a 1 m grid are also included.

$$\begin{array}{c}\hfill S(t)=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}8\\ -15+30\frac{t}{4.17}\\ 1\end{array}\right),\end{array}$$(23)

for t ∈ [0, 4.17]. The receiver is placed at

$$\begin{array}{c}\hfill R(t)=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}-5\\ 3\\ 1.5\end{array}\right).\end{array}$$(24)

Figure 13 shows the initial results for the triangle wave source signal. A very strong artefact can be seen at around 2 s. It was found that the paths that produce this issue are the second order diffraction paths over the building that converge towards the upper corner of the building. One of these paths is depicted in Figure 14. In this case, the distance between the two edges of the roof becomes smaller as the path converges toward the corner. This can also easily be imagined geometrically or as a simple triangle where two sides are shortened, the third side must also be shortened if one of the angles is kept constant (as is the case with the geometry). As a result of this, the distances involved in the calculation of the HOD are no longer larger than $\frac{\lambda }{4}$ (see Eq. (3)) which is a precondition for the HOD model [38, 39]. This leads to diffraction coefficients with values larger than one, especially for low frequencies, which in turn cause the artefact as seen in Figure 13. This is in line with the fact that the UTD is a high frequency model.

Figure 13. Initial results for the building corner scenario. At around 2 s a strong artefact can be seen.

Figure 14. Highlighted propagation path that causes the artefact as seen in Figure 13 for different points in time.

To overcome this issue, two solutions were investigated in this work. First, a fading approach, where the erroneous diffraction paths are faded in and out. Second, the diffraction coefficients are clamped to a maximum absolute value of one, under the assumption that a diffraction should not amplify the sound and sound waves are less affected by obstacles much smaller than their wavelength. However, it must be stated that, to the knowledge of the authors, an exact solution for this UTD deficiency does not exist and that both approaches are physically informed approximations.

For the fading approach, the ρ parameter (Eq. (16)) from Kawai, Kim et al. [38, 39] is used as the fading parameter. With this parameter a linear fading function can be defined as

$$\begin{array}{c}\hfill f_{}\text{fade}(\rho )=\{\begin{array}{cc}1\hfill & \text{if}\rho \ge 0.35\hfill \\ \frac{\rho -0.3}{0.35-0.3}\hfill & \text{if}0.3<\rho <0.35\hfill \\ 0\hfill & \text{if}\rho \le 0.3\hfill \end{array}.\end{array}$$(25)

The fading limits of ρ = 0.3 and ρ = 0.35 were determined empirically and provide good results. It has to be noted that this approach cannot work when diffractions of first order are causing errors.

Figure 15 shows the results of the scenario with the fading applied. As with the previous scenarios, the pink noise signal in Figure 15a shows no artefacts. With these results we can also validate the heterogeneous paths which are even more important as in the wide barrier scenario (Sect. 4.2). Up to around 1.5 s the source is fully in the shadow zone of the building corner. Still, the comb-filter effects due to the combination of two propagation paths are observable. One is the diffraction on the building corner, the other path first reflects on the ground and then diffracts on the building corner. This can also be interpreted as the diffraction being mirrored on the ground. This shows that the heterogeneous paths are working as intended. In the range of 1.5 s–2.7 s, the source moves into the illuminated zone. It can be seen that the comb-filter effects that were present in the shadow zone transition seamlessly into the illuminated zone. This indicates that also the compensation effects discussed in Section 4.1 are working as intended with heterogeneous paths. Lastly, starting at around 2.7 s, the source moves into the reflection zone where the reflection from the building’s side wall is present. Here, both first and second-order reflections are included. This can be seen through the additional comb-filter effects.

Figure 15. Results for the building corner scenario with the fading applied to reduce artefacts due to model limitations.

The results for the triangle wave signal are shown in Figure 15b. In comparison to the previous results, more pronounced artefacts are observed. Most notably, these occur around 2.6 s, although other artefacts are also present. Unfortunately, the exact cause of these artefacts has not yet been fully determined. Multiple reasons can be assumed, however due to the complexity of the scenario, it is not easy to determine the exact cause. One reason for this can be that in the considered geometry, the edges can no longer be assumed to be infinitely long. As the UTD, and its HOD extension, assumes the edges to be infinitely long the diffraction model is not valid anymore. Another possible cause for the artefacts can be found in the limitation of the path orders utilised by the IEM [45]. Depending on the scenario and the order limitations, the IEM could find paths for which the compensating diffraction paths are not present. Or vice versa, a diffraction path is present for which the incident or reflection path is not present (E _i and E _r in Eq. (4)). This problem was also discussed in Stienen [40] in Section 3.5. Lastly, the previously mentioned problem with the non ideal step function behavior when new paths are added could also be a reason for the artefacts. Especially in this scenario for the artefacts at around 2.6 s two paths are added at the same time. Compensating diffraction paths exists, but the step function of the new paths are not ideal. To further complicate the explanation, these causes can also happen in combination. Based on the results of this scenario the two latter explanations, being the limited path order and non-ideal step function behavior, are the more likely causes for the artefacts. However, in the additional scenarios from the supplementary material, the first explanation with the limited edge lengths also appears to influence the artefacts.

From the results of the two source signals (cf. Fig. 15) it can be seen that the fading approach described above works well to reduce the discontinuity found in the initial result for this scene. The artefacts due to the $\frac{\lambda }{4}$ condition are reduced significantly and the fading itself appears to not introduce any additional artefacts visible in the presented results.

The results for the clamping approach (cf. Fig. 16) show a similar trend with the artefacts due to the $\frac{\lambda }{4}$ condition being reduced. However, for the triangle wave, at 2 s the artefact is stronger as the path over the building is not compensated correctly by another path. In the fading approach, this issue is less pronounced as the path does not contribute to the sound field at that time anymore. For the pink noise signal, no difference is observed. Otherwise, the same findings as the fading approach hold.

Figure 16. Results for the clamping approach with the triangle wave signal. Compared to the fading approach, at around 2 s an additional artefact can be seen. This is likely caused by the path over the building not being compensated correctly by another path.

4.5 Discussion

The presented results demonstrate the applicability of the presented method for auralizing urban environments. The results of the simpler scenarios show that the method is able to auralize the scenarios with minimal to no artefacts. The more complex scenarios show that the method is able to handle these scenarios. Yet, some artefacts and limitations of the method are identified.

One artefact that was identified is the first-sample issue when certain paths are added. This was noted in Section 4.1 and is also present in the other scenarios. The cause for this issue is not yet fully identified. This artefact highlights an important finding of the presented method. As stated in equation (4), for diffraction to be applied correctly the direct sound and reflected sound need to be included in a step like behavior. In extension, for dynamic scenarios where the considered paths change due to movement, the direct sound and reflected sound should be included in a step like behavior. Furthermore, the change of the diffraction filter, especially at the boundaries of direct and reflected sound, must also be applied in a step like behavior for the compensation to work correctly. An example of this can be seen in Figure 17. This is in contrast to the common techniques of GA auralization, where most effects are faded in, out or cross-faded to avoid artefacts. When diffraction is included, it performs the compensation for the discontinuity of the direct sound and reflected sound. A fading approach was also evaluated in this work. This did reduce the first-sample artefact, however it also introduced additional artefacts that were caused as the fading worked against the compensation of the diffraction. The best results with the least notable artefacts were achieved with the step like behavior of the filters. If very long cross-fades are used, as for example in Schissler et al. [35], the artefacts discussed here will also be reduced or even eliminated. This approach, however, can lead to problems for tonal sounds and suffers from the comb-filter effects of cross-fading two filters. If the propagation paths are combined into a single filter, i.e. not a fully path based rendering, the artefacts might also be reduced, but again, this will lead to comb-filter effects.

Figure 17. Simplified illustration of the diffraction field around an edge via the angle θ as described in equation (4) [49]. From top to bottom the different sound field components are depicted: incident field E i , reflected field E r , diffraction field E d and total sound field E. The incident field E i and reflected field E r are added in a step like behavior. The diffraction field compensates for the discontinuities at the boundaries of the incident and reflected fields such that the sound field E is continuous. The same logic applies if instead of the angle θ, the incident and reflected field are considered to be included in the time domain.

Another consequence of the results is that the diffraction path must be present all around the edge, otherwise the diffraction filter cannot compensate for the discontinuities. This was also noted by Kirsch and Ewert [59] and is different to the approach of Schissler et al. [35] where the diffraction filter is only applied in the shadow zone.

At this point, it should also be noted that during the development of the method a strong influence of the utilized filter length was observed. At 44 100 samples per second, a filter length of less than 129 bins resulted in an increase of artefact strength at the shadow and reflection boundaries. The reason for this is likely the low frequency resolution of the filter. Conversely, a filter length of more than 129 bins resulted in little benefit.

Another artefact that can be seen in the results is due to the combination of the block-based approach and the step like behavior of the filters. Since in the presented method the filters are switched, small errors can occur at the block boundaries. This can be reduced by decreasing the block size, however this will increase the computational complexity. There is also the option for a hybrid approach where the filters are only switched on shadow and reflection boundaries and cross-faded in between. This would however require a more complex implementation and is not considered in the presented work.

The next set of artefacts that can be seen in the results is due to invalidation of the conditions for the diffraction model. For example, in the building corner scenario (see Sect. 4.4), the conditions for the HOD model are not met anymore as the source moves towards the corner of the building. In this work, two approaches are used to reduce the artefact. A fading approach, where the problematic path is faded in and out, and a clamping approach, where the diffraction coefficients are clamped to a maximum absolute value of one. Both approaches work well for the presented scenarios, however they are not physically correct solutions and should be improved. Possible solutions are approximating the double edge into a single edge when the distance between two edges becomes smaller than $\frac{\lambda }{4}$. This however would require cross-fading between the HOD and single edge approximations. Another limitation of the presented method is that only infinite edges are considered. This also resulted in artefacts, especially in the additional scenarios in the supplementary material. For this, the utilized diffraction model can be extended to consider finite length edges as is done in [60] or through vertex diffraction [61]. Another approach to tackle these issues could be to consider different diffraction models like BTMS [50] or the incremental theory of diffraction (ITD) [62] and their derivatives/extensions. Both models can inherently consider finite length edges. Thanks to the modular implementation of the presented method and the open interfaces, these models could be integrated with minimal effort. However, it has to be noted that these would probably increase the computational complexity significantly.

The presented method only considers specular reflections and diffraction of comparably low orders. Thus, the scattered sound components of the sound field are not considered. These however are an important part of the sound field as for example discussed in a recent publication by Vos et al. [63]. For these propagation effects, other simulation methods like ray-tracing [15] or radiosity [64, 65] could be used and combined with the presented method. Connected to the consideration of the stochastic and diffuse sound components is the consideration of surface impedance. For simplification purposes, in the presented work, only rigid surfaces with a reflection factor of 1 were considered. The method and implementation can however consider arbitrary reflection factors and impedances. Yet, for urban environments where distances are usually large and incident angles for reflections are small, the implemented plane-wave reflection method could be extended to consider spherical reflection factors as described in Ingard [66]. Furthermore, when impedances are considered, impedance jumps cannot be auralized without artefacts with this method. The presented model could be extended to consider impedance jumps, though a physically correct implementation is a complex addition and was considered out of scope for this work [67].

The presented method is dependent on the IEM [45] for finding the propagation paths. Due to the required limitations of the path orders, the IEM can find combinations of paths that can lead to artefacts. As discussed in Stienen [40], a fading approach for these scenarios could be used to reduce the artefacts. This however is not yet implemented in the presented work. In addition to the order limitation, the IEM can also occasionally fail to find all paths for a given scenario. This can lead to artefacts, especially if for example a propagation path is missing for a single audio block. In the testing of the presented method, this issue was only encountered once. A possible solution could be an interpolation in the propagation parameter dataset of the missing paths based on the previous and next blocks.

Even though the IEM was validated for a thin screen scenario in Erraji et al. [45], the results are not yet validated for more complex urban scenarios and in conjunction with the presented propagation modeling. As such, a comprehensive evaluation and comparison of the complete method with reference recordings and numerical simulations is still necessary to validate the absolute accuracy of the method.

In general, only minor artefacts were observed in the spectrograms. An informal listening session with the authors and four additional listeners indicated that, for the pink-noise excitation, these artefacts were perceived as barely audible. For the triangle-wave excitation, some of the discussed artefacts were audible. Primarily, the artefacts in the building corner scenario (Sect. 4.4) were noticeable. The first-sample artefact, present in multiple scenarios, could be detected, but was generally considered less noticeable. Still, these artefacts did not noticeably degrade the perceived quality of the auralization. Although it is important to discuss such artefacts, it must be emphasized that they are expected to be of little practical relevance in more typical listening situations, for example in complex urban soundscapes, where ambient noise and contextual cues provide masking and source signals are rarely as tonal as the triangle-wave signal. The pink-noise result, where the artefacts were barely audible, further supports this expectation. Nevertheless, the present assessment remains limited: it relies solely on visual inspection of the spectrograms together with a small-scale informal listening session, and should not be interpreted as a general statement on audibility or plausibility. A comprehensive evaluation – ideally a controlled listening test with an adequate sample size and reference recordings – remains necessary to quantify the perceptual impact of the artefacts under representative acoustic conditions. This could be in form of forced-choice-tests or rating-tests either comparing to reference recordings or for the detection of artefacts. The detection of artefacts could also be tested with the presence of background noise, different source signals or more complex scenarios.

5 Summary and conclusion

In this work, an auralization approach for arbitrarily built urban environments is presented. The approach is a continuation on a path-based geometric acoustic approach extended from Stienen and Schäfer et al. [40–42]. The method is able to calculate the propagation paths and the corresponding propagation parameters for each path separately based on the geometry and the source and receiver positions/movements. The propagation parameters are then applied to the source signal via a block- and path-based VDL and convolution. In addition to considering the propagation phenomena of specular reflection, air absorption, propagation delay and spreading loss, the approach also considers diffraction. For this, the diffraction model from Kim et al. [38] was extended to consider heterogeneous diffraction paths.

An in-depth evaluation of the presented method is done with multiple scenarios ranging from simple test cases to more complex urban-like environments. In the supplementary material, additional scenarios are included that further evaluate the method though do not provide additional insights. The results indicate that the presented method is capable of auralizing the scenarios with few to no artefacts. The artefacts that are present can be mitigated by refining the method further. A full discussion of the artefacts, limitations and possible improvements is given in Section 4.5.

As was also discussed in Section 4.5, the presented method is yet to be evaluated and validated in a listening test. However, the results of the presented method are promising and show that the method is able to auralize urban environments.

From this, we can conclude that the presented auralization method works well and the ability to auralize even complex scenarios without fading serves as a strong indicator that the underlying concepts are correct. While some artefacts still occur, they are relatively insignificant, and their causes are mostly understood and can be addressed in future work. For instance, extending the diffraction model to incorporate finite-length edges could be beneficial. Additionally, although block fading did not lead to significant improvements in the results of this work, it may still be worthwhile to explore this approach further in future iterations.

The issue of missing paths could also be addressed in future work. For example, an interpolation in the propagation parameter dataset of the missing paths based on the previous and next blocks could be implemented. Such an interpolation approach could also be used to reduce the computation time of the method by up-sampling the propagation parameter dataset [42].

Acknowledgments

The authors would like to thank Philipp Schäfer and Chalotorn Möhlmann for their valuable feedback and discussions during the development of the method. In addition, the authors would like to thank Chalotorn Möhlmann, Jithin Thilakan and Lukas Aspöck for proofreading the manuscript. Large language models have been used to assist in the writing and editing process of this manuscript.

Funding

The authors would like to express their gratitude to the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) for funding the project Auralization of Urban Environments: Benchmark and Evaluation under grant number 499742946, which made this contribution possible.

Conflicts of interest

The authors declare no conflict of interest.

Data availability statement

The data supporting the findings of this study can be reproduced using the openly available package pynamic-pigeon-auralization (https://git.rwth-aachen.de/ihta/auralization/pynamic-pigeon-auralization; see [56]). Supplementary material is available at https://doi.org/10.5281/zenodo.19112715. It includes auralization results for the scenarios presented in this paper and additional scenarios used to further evaluate the method, the corresponding plots, the scripts required to generate the results, and the software packages used.

Author contribution statement

Pascal Palenda: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft. Michael Vorländer: Conceptualization, Project administration, Funding acquisition, Supervision, Writing – review & editing.

Software availability statement

PIGEON [54], version 1.0.0

Software available from: https://git.rwth-aachen.de/ihta/ihta-package-registry/-/packages/12417

Source code available from: https://git.rwth-aachen.de/ita/ITAGeometricalAcoustics/-/tree/master/apps/pigeon

License: Apache License 2.0

PELICAN [52], version 1.0.0

Software available from: https://git.rwth-aachen.de/ihta/ihta-package-registry/-/packages/12416

Source code available from: https://git.rwth-aachen.de/ita/ITAPropagationModels/-/tree/master/apps/pelican

License: Apache License 2.0

DOVELET [55], version 0.3.1

Software available from: https://git.rwth-aachen.de/ihta/ihta-package-registry/-/packages/12429

Source code available from: https://git.rwth-aachen.de/ihta/auralization/dovelet

License: Apache License 2.0

pynamic-pigeon-auralization [56], version 0.1.0

Software available from: https://git.rwth-aachen.de/ihta/ihta-package-registry/-/packages/12431

Source code available from: https://git.rwth-aachen.de/ihta/auralization/pynamic-pigeon-auralization

License: Apache License 2.0

References

Appendix A

Diffraction coefficient equations

In the following, the equations for the diffraction coefficients used in this work are summarized. These equations are taken from Kim et al. [38], for more details the reader is referred to the original publication. Additionally, please refer to equations (12) and (15) for the definitions of A_Q and B_Q. Figure A.1 defines the angles used in the equations.

Figure A.1. Definition of the angles used in the diffraction coefficient equations. Adapted from Kim et al. [38].

$$\begin{array}{c}\hfill D_Q=V_Q(A_Q,B_Q,{\theta }_R-{\theta }_S)+V_Q(A_Q,B_Q,{\theta }_S+{\theta }_R)\end{array}$$(A.1)

$$\begin{array}{c}\hfill \begin{array}{cc}& V_Q(A_Q,B_Q,\mathrm{\Theta })=\hfill \\ \hfill & -\frac{e^{-i\pi /4}}{\sqrt{2\pi k}}\frac{1}{2\nu }cot\left(\frac{\pi \pm \mathrm{\Theta }}{2\nu }\right)F^{\ast }\left(B_Q{X}_Q^{\pm }(\mathrm{\Theta })\right)\hfill \end{array}\end{array}$$(A.2)

$$\begin{array}{c}\hfill F^{\ast }(x)=2i\sqrt{x}exp(ix){\int }_{\sqrt{x}}^{\infty }exp(-i{\tau }^2)d\tau \end{array}$$(A.3)

$$\begin{array}{c}\hfill X_Q^{\pm }(\theta )=2k{A}_Q{cos}^2\left(\frac{2N^{\pm }\nu \pi -\theta }{2}\right)\end{array}$$(A.4)

$$\begin{array}{c}\hfill N^{+}=\{\begin{array}{cc}0\hfill & \text{for}\theta \le \nu \pi -\pi \hfill \\ 1\hfill & \text{for}\theta >\nu \pi -\pi \hfill \end{array}\end{array}$$(A.5)

$$\begin{array}{c}\hfill N^{-}=\{\begin{array}{cc}-1\hfill & \text{for}\theta <\pi -\nu \pi \hfill \\ 0\hfill & \text{for}\pi -\nu \pi \le \theta \le \pi +\nu \pi \hfill \\ 1\hfill & \text{for}\theta >\pi +\nu \pi \hfill \end{array}\end{array}$$(A.6)

For the Fresnel integral F^*(x) the approximation from Kawai [39] is used:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F^{\ast }(x)& =exp(i\frac{\pi }{4}·(1-\sqrt{\frac{x}{x+1.4}}))\hfill \\ \hfill & \{\begin{array}{cc}\sqrt{\pi ·x}·\frac{1-\sqrt{x}}{0.7·\sqrt{x}+1.2}\hfill & \text{for}x<0.8\hfill \\ 1-\frac{0.8}{{(x+1.25)}^2}\hfill & \text{for}x\ge 0.8\hfill \end{array}\hfill \end{array}\end{array}$$(A.7)

All Tables

All Figures

	Figure 1. Schematic definition of convex and concave geometries. In convex geometries, not all face normals (red arrows) point towards each other, while in concave geometries, all face normals point towards each other.
In the text

	Figure 2. Schematic overview of the proposed auralization scheme. The process consists of three main steps: determining propagation paths, calculating propagation parameters, and rendering the audio. The process is repeated for each audio block to adapt to dynamic scenarios.
In the text

	Figure 3. Example for two HOD propagation paths over a wide barrier with a ground. Due to the reflection on the ground, a second path Path 2 exists. More paths exist for this scenario, but are omitted for clarity.
In the text

Figure 4. Considered geometry for the wedge scenario. The wedge has an exterior angle of 270°. The source (in blue) rotates around the edge and moves along the z-axis of a cylindrical coordinate system. The receiver (in red) is static. Both positions are described in detail on equations (17) and (18). The yellow line indicates the source trajectory. The origin of the coordinate system and a 1 m grid are also depicted.

In the text

	Figure 5. Results for the wedge scenario. Figure 5a and 5b show the spectrograms for the pink noise and triangle wave signals, respectively. Figure 5c shows a detailed view of the triangle wave signal to investigate the discussed artefact. Figure 5d shows the Doppler effect for the triangle wave signal.
In the text

	Figure 6. Geometry of the wide barrier scenario. The barrier is 6 m wide and 3 m high. The source (in blue) is on an elliptical trajectory around the barrier. The source trajectory, indicated by the yellow line, is also described in detail in equation (19). The static receiver (in red) position is described in equation (20). The origin of the coordinate system and a 1 m grid are also included.
In the text

	Figure 7. Time domain view of the triangle wave signal using a sequential first-order diffraction approximation. At around 0.45 s, where the source transitions from the double shadow zone to the single shadow zone, a significant change in amplitude can be seen.
In the text

Figure 8. Schematic illustration showing the cause of the error in the sequential HOD approach. The source (blue) rotates around the wide barrier as in Figure 6. Also indicated by the arrow denoted with t. The gray dashed lines show the receiver’s (red) shadow boundaries for the two edges of the barrier. These boundaries separate the space into multiple regions, here only two are denoted: (I) double shadow zone, (II) single shadow zone. The petrol dotted arrow indicates the regions where the propagation path A with first-order diffraction on the right edge of the barrier exists. Indicated by the violet dashed arrow are the regions where the propagation path B with second-order diffraction exists. The two paths also exist further to the right (indicated by the three dots), but are omitted for clarity.

In the text

	Figure 9. Auralization results for the wide barrier scenario using the HOD model.
In the text

Figure 10. Geometry of the interior corner scenario with an opening angle of 95°. The source (in blue) moves on a circular trajectory with a radius of 5 m around the corner. The source trajectory, indicated by the yellow line, is also described in detail in equation (21). The static receiver (in red) position is described in equation (22). The origin of the coordinate system and a 1 m grid are also included.

In the text

	Figure 11. Results for the interior corner scenario.
In the text

Figure 12. Considered geometry for the building corner scenario. The source (in blue) moves on a linear trajectory from the left edge to the right edge of the building corner. The source trajectory, indicated by the yellow line, is also described in detail in equation (23). The static receiver (in red) position is described in equation (24). The origin of the coordinate system and a 1 m grid are also included.

In the text

	Figure 13. Initial results for the building corner scenario. At around 2 s a strong artefact can be seen.
In the text

	Figure 14. Highlighted propagation path that causes the artefact as seen in Figure 13 for different points in time.
In the text

	Figure 15. Results for the building corner scenario with the fading applied to reduce artefacts due to model limitations.
In the text

	Figure 16. Results for the clamping approach with the triangle wave signal. Compared to the fading approach, at around 2 s an additional artefact can be seen. This is likely caused by the path over the building not being compensated correctly by another path.
In the text

Figure 17. Simplified illustration of the diffraction field around an edge via the angle θ as described in equation (4) [49]. From top to bottom the different sound field components are depicted: incident field E i , reflected field E r , diffraction field E d and total sound field E. The incident field E i and reflected field E r are added in a step like behavior. The diffraction field compensates for the discontinuities at the boundaries of the incident and reflected fields such that the sound field E is continuous. The same logic applies if instead of the angle θ, the incident and reflected field are considered to be included in the time domain.

In the text

	Figure A.1. Definition of the angles used in the diffraction coefficient equations. Adapted from Kim et al. [38].
In the text