A numerical hybrid model for outdoor sound propagation in complex urban environments

The modeling of sound propagation around buildings in urban settings is a major topic of research due to its widespread practical applications. In the military, many localization techniques for enemy threats such as snipers rely on the sensing of the acoustic signals as well as an accurate knowledge of the multiple sound propagation paths.¹ Civil applications mainly include the propagation and reduction of road-traffic and airport community noise. Prediction of sound propagation around buildings is a challenging problem. Many effects must be accounted for, in particular, multiple reflections from surfaces and diffraction from edges. Numerical tools for outdoor sound propagation are well established and widely used in the acoustic community, e.g., ray-tracing, finite-element (FE), boundary-element (BE), transmission-line-matrix,² finite-difference time-domain (FDTD), geometrical-acoustics (GA),³ and equivalent-source (ES)⁴ methods. However, as the size and complexity of the domain increase, these tools become inefficient in terms of computational speed, memory, and accuracy over a wide frequency range (e.g., a computational time of up to 42 h with the ES method for two canyons, on a computer with a dual 2.66-GHz processor and 4 GB of RAM,⁴ or computational times up of to 12 and 365 h, with GA and BE methods, respectively, for predictions at 2751 receivers on a quad 3.42-GHz processor and 16 GB of RAM³). Attempts to overcome these limitations have been made through hybrid modeling such as the FDTD/Parabolic-Equation model,⁵ but three-dimensional (3D) computations remain expensive and impractical in terms of computational resources. Moreover, most of these models are limited to perfectly smooth building facades. Other techniques are well suited to modeling diffuse reflections, e.g., the radiosity-based models.^6,7 However, the efficiency of these models highly depends on the patch parameterization of the domain, meaning that computation on large 3D models might be discouraging. Only a few techniques^8,9 have been developed to model mixed specular and diffuse reflections, although the influence of scattering by the facades on the sound levels has been proven significant.^10,11 These techniques have been used on very specific urban configurations (e.g., single street canyon, street crossings) and are hardly applicable to large 3D environments with high-building-density configurations.

Recently, considerable efforts have been devoted to developing a diffusion model (DM) for efficient sound-propagation simulations. Initiated by Ollendorff¹² and further developed by Picaut et al. for rectangular enclosures¹³ and various street configurations,¹⁴ studies on the mathematical theory of diffusion have shown significant potential for efficient computation of sound levels in complex environments where diffusion conditions exist. However, several limitations of the DM have already been pointed out, such as the lack of direct sound field modeling and edge diffraction modeling. The DM has been proven accurate inside a single street but has shortcomings with intersections close to the source due to the inability for the DM to account for the destructive/constructive interferences between the direct, reflected, and diffracted waves.¹⁵ 

The objective of this letter is to describe a numerical hybrid model capable of addressing the limitations of the DM and improve the accuracy of the predictions. The proposed model combines a full-wave model with an energy-based model. In this work, FDTD modeling is selected as the full-wave approach for proof of concept purposes. Results from the hybrid simulations are validated against the FDTD simulations. The capabilities and limitations of the proposed approach are also discussed.

The proposed hybrid model is the combination of a full-wave model and an energy-based model. Full-wave models are accurate but require considerable computer capabilities and computational time to obtain results on large domains over a wide frequency range. Energy-based models such as those derived from the diffusion equation are attractive because they allow fast computation of the acoustic energy distribution in a given environment. However, they do not model diffracted waves and the destructive/constructive interactions among direct, reflected, and diffracted waves. Here, the virtues of each model are combined to develop a fast, yet reasonably accurate tool.

Figure 1 shows a generic schematic of a cluttered environment along with the main constituents of the proposed hybrid model. First, the numerical environment is divided into user-defined sub-domains with interface boundaries [see Fig. 1(a)]. One represents a reduced part of the computational domain, centered on the source, where the impact of the destructive/constructive wave interferences is significant and thus needs to be taken into account. The ability of the full-wave model to simulate accurately the sound field is used for predictions inside this first sub-domain, in the vicinity of the source. Typically, this solution is in the form of time or narrowband frequency-domain results. Another sub-domain describes the rest of the cluttered environment, where the diffracted sound field is less influential and diffusion conditions are met. The energy-based model is used to solve the sound field in this second sub-domain, typically in the form of steady-state energy over bands of frequencies, e.g., 1/3 octave bands. The two sub-domains have a common interface where the solutions have to match. The full-wave solution computed at the boundary of this sub-domain is then converted to steady-state energy values and used to compute the energy density along this interface, over a defined frequency range. These values are then used as input to the energy-based model, which is applied to the rest of the domain. Therefore, no conventional source input is defined in the energy-based model. The only input provided to the energy-based model is the value of the energy density along the interface between the two defined sub-domains (i.e., a boundary condition to the energy-based model). Thus, the obvious limitation of the proposed hybrid approach is that the full-wave solution of the reduced first sub-domain does not include the effect of the rest of the environment, i.e., no feedback from the second sub-domain into the first one. Reflections that may be coming back into the sub-domain defined around the source are not accounted for by the full-wave model. However, as shown in Sec. 3, it is reasonable to assume that the impact of such a flaw in the model is limited. A third sub-domain [see Fig. 1(a)] is defined at the limits of the computational domain, outside of the cluttered environment. There, the medium starts resembling a free-field environment and there are not enough reflections to create a local diffuse field. This part of the domain can potentially be solved by using a full-wave model. These regions have common interfaces with the energy-based solution. Therefore, the results from the energy-based solution are used to compute the acoustic power on the interface. Then, several fictitious point-like monopole sources with equivalent acoustic power are used as input to the full-wave model. Schematic representations of the two types of interface are given in Figs. 1(b) and 1(c). Note that in this study, the choice of subdomain sizes was ad hoc and made without any optimization. After running a few simulations with various subdomain sizes we noticed the following. The full-wave domains should be small enough that the model is still computationally efficient but large enough to encompass the building corners the closest to the source, where diffraction effects are the strongest.

A priori, any full-wave model (e.g., FE, ray-tracing, FDTD) and energy-based model (e.g., diffusion model, radiative transfer) could be combined. In the following, a description of the proposed hybrid model is given. The constituents of this model, namely FDTD and DM, have already been described in detail by Pasareanu et al.¹⁵ and will not be repeated here. Instead, we will focus on how these constituents may be combined to construct the hybrid model.

It is important to mention that there exist other approaches to account for diffraction effects and/or interactions between waves, such as the introduction of fictitious sources at diffracting edges¹⁶ or the addition of a general correction term¹⁷ (defined as positive or negative depending on zones where the diffraction has constructive or destructive effects). However, it seems that these techniques become increasingly complex in the case of a crossing or in the case of random dispositions of buildings. Also, the presence of singularities in these formulations and their increasing number with the number of edges, makes it difficult to obtain useful data. These techniques do not appear to be much more effective techniques, in terms of computational time and ease of implementation.

The numerical environment is divided into several sub-domains. The FDTD is used to simulate the sound propagation in the sub-domain defined around the source (unitary amplitude of the source). Then, the transient response obtained in that manner is converted to compute the steady-state root-mean-square (rms) pressure at each grid point in the sub-domain around the source, according to the following steps.

First, transfer functions were computed between the pressures in the domain and the source strength by taking the Fourier transforms of these quantities. Transfer functions, TF(f), are obtained at each spectral line of the narrowband frequency analysis. Then, the rms pressure can be computed from the magnitude square of the transfer function for any rms source strength, Q_rms(f), as follows:

where f is the frequency of each spectral line.

The narrowband results in Eq. (1) will show well-defined patterns due to both constructive and destructive interferences of the multiple waves present in the sound field, e.g., direct, reflected, and diffracted waves. These interference patterns are not present in the results from the energy-based model where the input is a steady-state sound source power. The energy-based model is actually applicable to frequency bands where many waves of different frequencies are present, thus generating a diffused field. For the proper comparison between the models used in this letter (FDTD, DM, and hybrid FDTD/DM), the narrowband FDTD results need to be collapsed by grouping the frequencies in wider bands. Thus, in the second step, the rms pressure in the 1/3 octave band with center frequency f_b is obtained by the incoherent addition of the rms pressure of the spectral lines within the band, such that

where f_l and f_u are the lower and upper frequency limits of the 1/3 octave band. The analysis is limited to six 1/3 octave bands, with center frequencies of 63, 80, 100, 125, 160, and 200 Hz, bounded at low frequency by the applicability range of the DM and at high frequency by the limited computational resources for the FDTD solution.

Then, the FDTD results obtained at the interface of the two sub-domains are used for solving the DM in the rest of the computational domain. The values of the energy density at each grid point on the interface are computed from the FDTD rms pressure as

and are then imposed as boundary conditions in the DM. In the absence of domain sources, the steady-state diffusion equation is simplified to

where D is the diffusion coefficient.^15,18 Boundary conditions are defined in a rather general fashion as

where $∂/∂n$ is the outward normal derivative to the boundary, and $h$ is the exchange coefficient. The analytical expression of the exchange coefficient has been the subject of several mathematical developments. In some recent work,¹⁹ the expression of an exchange coefficient valid for both low- and high-absorption coefficients, α, was given as

where c is the speed of sound in the medium. The absorption coefficient $α$ is taken equal to 1 at the limit of the computational domain (complete absorption), and equal to 0 for the modeling of the ground and building facades (rigid boundaries). A popular finite-difference scheme, the 7-point scheme,²⁰ was used to solve Eqs. (4) and (5). The solution is obtained in frequency bands such as 1/1 or 1/3 octave bands. Once the DM is solved, the energy density is known at each grid point of the sub-domain. Then, the rms pressure, and consequently the sound pressure level (SPL), are obtained from Eq. (3).

Once the limit of the cluttered environment is reached, the energy-based model is no longer valid. The medium starts resembling a free-field environment, and the full-wave model must be used again. In order to transition from the DM results to the FDTD propagation, the following steps are performed. First, the acoustic intensity is computed at the interface between the energy-based solution and the full-wave solution, as follows:

This allows the computation of the total acoustic power $W$ along the boundaries to be divided amongst several fictitious sources in the FDTD.

To be consistent with the DM formulation, we express the acoustic power in 1/3 octave bands, W_1/3. Last, the adequate source power is determined for each source and each frequency band based on the following expression:

In this study, the output data simulated with FDTD are the pressure time histories at each point of the domain, due to a Ricker pulse with a center frequency of 100 Hz.²¹ This pulse is implemented in the FDTD code as a source strength time history defined by

where A is the amplitude of the source, and f_c is the center frequency of the source. Note that due to the explicit nature of the discretization of the FDTD code used in this paper (interdependency of the grid spacing and the time increment values) and the resulting computational time restrictions, the results are valid only at low frequencies, from 50 to 250 Hz. The center frequency of 100 Hz for the Ricker pulse minimizes the frequency content above this upper frequency limit of 250 Hz.

The last two equations, Eqs. (8) and (9), can be combined to find the source amplitudes at the interfaces of the hybrid FDTD/DM model, such that

The FDTD, whose input is now defined by Eqs. (9) and (10), is used to obtain predictions in each defined sub-domains (red boxes in Fig. 2). The results for each fictitious source are added incoherently. Then, the rms pressures and the SPL are computed.

The DM was applied to the two-dimensional model shown in Fig. 2, which represents a simplified urban environment and consists of 16 regularly-spaced buildings with the same size and shape, forming a grid-type street plan. The streets aligned with the x- and y-directions have widths of 10 and 7 m, respectively. For the DM, the computational domain is discretized into 151 × 101 grid points with a grid resolution of 1 m. For the FDTD, the computational domain is discretized into 1201 × 801 grid points, i.e., with a resolution of 0.125 m. This conventional model is a natural extension of those proposed in the literature.¹⁴ It is assumed that the ground and building facades are perfectly reflective and that perfect absorption occurs at the boundaries of the acoustic domain. Also, an example of sub-domains is shown in Fig. 2 for a source position displayed by a blue circle.

Simulations were performed using a uniform diffusion coefficient, based on an average of the street half-width (i.e., L = 8.5 m) and assuming mostly specular reflections (i.e., s = 0.1). Although the FDTD models perfectly specular reflections (equivalent to s = 0), the aim of this letter is to determine if the DM can accurately model the sound field while reducing the computation time. Assuming s = 0 in the DM was not possible (since D would then tend to infinity), however it has been found that assuming a scattering coefficient close enough to 0 provided valid results.¹⁵ The hybrid model requires about 1 min to run on the described environment whereas the FDTD requires about 1.5 h.

The SPL distribution due to a steady source in the environment was computed using the FDTD, DM, and FDTD/DM hybrid models. Figure 3 presents the maps of absolute dB differences between (a) the DM and the FDTD, and between (b) the hybrid model and the FDTD, due to a source located at x_s = 126 m and y_s = 26 m, for the 1/3 octave band centered at 200 Hz. The results are presented on a scale from 0 to 10 dB. Qualitatively, the hybrid model seems to improve the predictions, especially in the top left quarter of the domain where the DM over-predicts the SPL. The prediction obtained near the source with the hybrid model differs slightly from those computed using full FDTD. However, these discrepancies are negligible, with a mean difference for each frequency band of about 1 dB. This confirms the assumption made earlier about the lack of feedback from the second sub-domain into the first, which does not significantly affect the predictions in this case. Outside the cluttered environment, the improvement in predictions is noticeable, though it is less significant for regions on the top of the maps.

Figure 4 reports the SPL predicted at three receiver positions for the six spectral bands considered. Note that, to minimize the effects of wave interferences, a spatial average SPL was computed for the FDTD, based on the 8 grid points surrounding the receiver (i.e., a 9-point average). The results confirm that the hybrid model improves the predictions with respect to DM alone, especially in the higher frequencies of the analysis. For the 200 Hz frequency band, the error is significantly reduced (from about 4 dB to less than 0.5 dB) for the receiver at (100, 48) as well as for that at (46, 76), for which the error decreases from 4 to 1 dB and that at (3, 77), for which the error drops from 6 dB to less than 2 dB. The improvement is not so dramatic at lower frequency, and even worsens in some cases. For receiver position (100, 48), the improvement is noticeable for the 100, 125, and 160 Hz frequency bands, but the prediction is worse for the 63 and 80 Hz frequency bands. For the two other receiver positions, the predictions are improved for all the frequency bands, except at 125 Hz for the receiver at (46, 76), and at 100 Hz for that at (3, 77).

In this letter, a new numerical hybrid sound propagation model has been proposed for cluttered urban environments to overcome the current limitations of the diffusion model, and to a greater extent, the limitations of any energy-based model that does not account for interferences and diffraction. This model combines a full-wave propagation model with a diffusion model. The full-wave propagation model is used for the computation of the SPL inside a sub-domain in the vicinity of the source, where the effects of interactions between direct, reflected, and diffracted waves are the most significant and cannot be modeled by the DM yet.

It has been shown that the hybrid model significantly improves the predictions, especially in the higher frequency bands. This fact was expected since the energy-based models are valid at mid to high frequencies. Considering the size of the buildings and the dimensions of the streets, it is reasonable to expect valid predictions for frequencies above 100 Hz. The results obtained here tend to support these claims. Also, it was demonstrated that the hybrid model has a significant potential to allow fast predictions of the sound field in large complex environments, since the computational time was reduced by a factor of 90 with respect to the full FDTD solution.

Several limitations of the proposed hybrid model should be discussed. First, the two models are not fully coupled (i.e., one-way coupling only). Therefore, the hybrid model does not account for the effects of waves that may be reflected back into the sub-domain. It seems reasonable to state that these effects are negligible in urban environments, where the sound field mostly propagates outwards. Second, it has been observed that the improvements of predictions outside the cluttered environment are limited for some regions. This could be the result of the incoherent sum of the contribution of each source. Interferences between waves may be more important in some regions, in which case incoherent sources are not the best approach. However, this limitation applies only if predictions are performed near the limits of the dense urban area. The proposed hybrid model can be limited to the inner urban area (cluttered parts). Last, the diffusion coefficient used in the DM is computed in an approximate fashion. Although the approach used in this letter seems to give proper results, additional work to determine an appropriate diffusion coefficient may be necessary.

The authors would like to acknowledge the financial support from the U.S. Army (Grant No. W913E5-10-C-0001).