Three-dimensional urban acoustic simulations and scale-model measurements over real-life topography Open Access

Finite-difference time-domain (FDTD) simulations can be applied to the prediction of outdoor noise propagation over complex real-life topography where an accurate prediction by engineering models is difficult. The limiting factors in performing such simulations have been the computer resources required to accommodate the vast computational domain size and the absence of a geometry reproduction technique of the topography. Although the frequency range must still be low, such simulations are becoming feasible following recent advances of computer performances and a topography reproduction technique using digital geographic datasets. Validation of the simulation results are of interest. Albert and Liu¹ compared the results of two-dimensional simulations over a full-scale artificial village with experimental measurements up to approximately the 200 Hz range. Heimann² performed a low-frequency three-dimensional simulation of approximately a 10 Hz frequency range over a natural terrain. The results were validated with measured point-to-point level differences. Oshima et al.³ performed three-dimensional simulations over a dense urban topography of approximately the 200 Hz range. Validation of the FDTD implementation was performed by comparisons with theoretical free-field solutions.

However, no study to date has performed an extensive validation of three-dimensional simulations over a complex topography such as a dense urban terrain through comparisons with actual measurements. The main problems with real-life measurements are that a live urban area contains too many uncontrollable background noise sources and that sound propagation is affected by complex meteorological conditions.⁴ In contrast, a scale-model experiment allows precise control over the source and propagation conditions.

In this study, as a preliminary work toward an extensive validation, the geometry of a real-life urban area in Kanagawa, Japan is reconstructed by digital geographic datasets. A 1:100 scale model of the geometry is created by a combination of numerically controlled machining of the terrain part and meticulous fabrication of the building parts. Experiments are carried out to measure shielding and diffractions by the buildings and the terrain. In parallel with the experiments, FDTD simulations are performed under the same geometry and source-receiver settings. The results are compared to illustrate the preliminary validity of the FDTD simulations.

The subject of the study is a 380 m × 380 m area in Kanagawa, Japan, as shown in Fig. 1(a). The upward direction of the figure is the north direction. The sources (S1 and S2) and the receivers (RH1, RH2, RV2, and Ref1) drawn in the figure are explained in Sec. III A. The geographic coordinates of the southwest and northeast corners of the area are $(x, y)=(−19 600,−57 405)$ and $(−19220,−57025)$ [m] in zone IX of the rectangular plane of the Japanese Geodetic Datum 2000 (Ref. 5) or $(35°28′56″N, 139°37′2″E)$ and $(35°29′8″N, 139°37′17″E)$ in latitude and longitude, respectively. The area is chosen because it is a built-up urban area with a moderately undulating terrain. For reconstruction of the topography and the building shapes, the following three types of datasets are used: A digital surface model (DSM)⁶ with a random resolution of 2 m on average, a digital elevation model⁶ with a uniform resolution of 2 m, and a building outline dataset.⁷ For simplicity of the scale-model creation, small detached houses are not included in the reconstructed geometry.

The model used for the measurements is created on a scale of 1:100. The terrain part and the building parts of the model are created separately. The terrain part is created by numerically controlled machining. The building parts are created by meticulous fabrication. For simplicity of the model creation, the roof of each building is flat at the median height of the DSM points inside the building outline. Also for simplicity, courtyards present in some of the buildings are not reproduced but are included in the building volumes.

The material of the model is resin foam with a hardening treatment of the terrain surface and acrylic coating of the building surfaces so that the surfaces are regarded as acoustically rigid. The complete model is shown in Fig. 1(b) with the north direction facing toward the front. By comparison with Fig. 1(a), it is seen that the geometries of the terrain and the buildings are reproduced well.

The input geometry for the simulations is created in full scale from the same geographic datasets used for the scale model by applying the technique by Oshima et al.⁸ The geometry is shown in Fig. 1(c). Following the scale model, the roof of each building is flat at the median height of the DSM points inside the building outline. By comparison with Fig. 1(b), the shapes of the scale model and the polygonal data agree well.

Axes, sources, receivers, a reference point, and their coordinates in full scale are shown in Fig. 1(a). The coordinates are local ones that originate from the ground level of the southwest end of the subject area. The x, y, and z axes represent the eastward, northward, and upward directions, respectively. S1 and S2 are the sources, RH1 and RH2 are the horizontal receiver planes, and RV2 is a vertical receiver plane. RH1 is located at 1.5 m above ground level. The receiver planes consist of sets of uniformly spaced receiver points. The spacings and the number of receiver points in each receiver plane are shown in Table I.

The upper half of Table I shows the settings of the test cases common to both the measurements and the simulations. Cases 1 and 2 assume situations where noises from the building equipment located on the building roofs propagate to the subject area. Case 2 is for a visualization of the sound wave propagation by dense measurement by the receivers placed in 2 m spacing (2 cm on the model scale) on receiver planes RH2 and RV2. Nonetheless, the upper frequency of reproducible waves is limited to approximately 40 Hz in full scale (assuming four points per wavelength for clear visibility). The limited frequency range is still considered to be useful for a qualitative assessment of the topographical effects on sound propagation.² 

The measurements are taken in an anechoic room. A spark pulse generator is used as the sound source for case 1. With the recording sampling rate of 192 kHz, a set of sample waveforms are obtained for a single receiver point by a 1/4-inch microphone. Then, 25 to 32 mutually similar waveforms are chosen by correlative operations. Time synchronous averaging is performed on the chosen waveforms. To correct the variations in the acoustic powers of spark radiations, the amplitude of each waveform is normalized by the maximum amplitude of the acoustic pressure observed at reference point Ref1 during sampling of the waveform.

The source for case 2 is a sweep signal radiated from a dodecahedron loudspeaker with inscribed circle radius of 42 mm, which is suitably small for the scale-model experiment. The recording sample rate is 96 kHz. Time synchronous averaging of 256 measured waveforms for each receiver point at the shadow zone and 64 waveforms for the non-shadow zone is performed. The impulse response from the source waveform to the averaged waveform is calculated. An infinite impulse response (IIR) filter of the 4 kHz octave band is applied to the impulse response to avoid spatial aliasing of the visualized waves originating from the receiver spacing of 2 cm, as stated in Sec. III A.

In all cases, air absorption is corrected following ISO 9613-1:1993.⁹ All results are presented in full scale by rescaling the frequency to 1/100 following the law of similarity.

A FDTD method is coded in c++ and applied to the simulations. The simulations are conducted in full scale. The bottom half of Table I shows the simulation settings. The speeds of sound are calculated from the average temperatures of the corresponding scale-model measurements. The source for case 1 is an initial Gaussian pressure pulse of full-width half-maximum 1 m. In case 2, the waveform of the impulse response of the dodecahedron loudspeaker with an adjustment to full scale by stretching the time by a factor of 100 is given to the source grid point. Furthermore, a 40 Hz octave-band IIR filter is applied to the source waveform of case 2 to simulate the IIR filtering of the measured waveforms. The boundary conditions of the terrain and building surfaces are rigid in all cases. A perfectly matched layer¹⁰ with a thickness of 20 grid points is applied to the top and lateral boundaries of the computational domains of all cases. The simulation of case 1 is carried out in 512-way parallel on a Fujitsu FX10 supercomputer. The simulation of case 2 is carried out in 8-way parallel on an Apple Mac Pro computer.

Figure 2 shows the octave-band sound pressure levels (SPLs) from 125 to 500 Hz obtained by the measurement and the simulation of case 1. The SPLs are relative to those observed at the nearest receiver point from the source to the southwest. The overall distribution of SPLs between the measurement and the simulation agrees well. Specifically, the shapes of the shadow zones in the region of (x,y) = (80,130) − (130,250) and the high SPL region along the direct propagation path from S1 to the southwest direction show good agreement. The propagation path is caused by a canyon between the building at the center west and the hilly terrain at the south. Moreover, the gradual decrease of the SPLs with increasing octave bands observed at the south hill is commonly seen in the measurement and the simulation. However, the SPL at the shadow zones of the 500 Hz octave band is generally larger in the measurement than in the simulation. This is most likely because the air absorption correction in the measurement process amplifies not only signals but also background noise. At the highest frequency band of shadow zones where the signal-to-noise ratio is severe, the amplified background noise could be significant to affect the measured SPLs.

Figure 3 shows the scatter plots comparing the relative SPLs obtained by the measurement and the simulation from the 125 Hz to the 500 Hz octave bands. The root-mean-square errors (RMSEs) are indicated in the figure as well. As expected from Fig. 2, the results of the measurement and the simulation agree well in the 125 and 250 Hz bands with RMSEs of approximately 2 dB. However, the SPLs obtained by the simulation are lower in the 500 Hz octave band with a higher RMSE of approximately 4 dB, owing to the aforementioned reason.

Multimedia 1 is a movie showing the visual comparison of acoustic pressure wave propagation on receiver planes RH2 and RV2 from time 0 to 0.468 s obtained by synchronously mapping the waveforms of all receiver points onto the receiver planes. The left and right halves of the movie are visualizations of the measurement and the simulation results, respectively. The temporal scale of the movie is stretched from real time by a factor of 32. Radiations from the sources are seen not as short impulses but as long bursts because of the IIR filtering. Nonetheless, in both the measurement and the simulation, wave propagation phenomena such as reflections from the buildings and the ground, diffractions at the edge of the building, and interference patterns by the direct and reflected waves, are clearly visible. However, the reflection of the measurement wavefront that touches the building wall corner at 0.130 s does not appear until approximately 0.18 s. This is because its amplitude is smaller than succeeding wavefronts. Moreover, the blue-to-red color scaling is exaggerated for clear visibility of the wavefronts. This makes small variations caused by the interference of the reflected and succeeding incoming waves hard to be visually recognized. A similar observation also applies to the simulated wavefront.

As a preliminary study to validate the FDTD simulations of outdoor sound propagation problems over a real-life urban topography, 1:100 scale-model measurements and full-scale simulations over a 380 m × 380 m area in Kanagawa, Japan are carried out. The geometry of the 1:100 scale model and the full-scale simulations are created from three types of geographic datasets. The results obtained on the horizontal receiver plane of case 1 show that the experiments agree well except for the shadow zones of the 500 Hz band. The visualizations of propagations of the low-frequency acoustic waves of the 40 Hz band in case 2 show that the wave phenomena are clearly visible in the measurement and the simulation, including the reproduction of diffractions and reflections.

The overall results indicate the preliminary validity of using an FDTD simulation for a quantitative assessment of sound propagation over a real-life complex urban topography up to a medium frequency range of approximately 500 Hz. Also, an FDTD simulation allows a qualitative analysis of complex wave phenomena caused by a natural terrain and buildings. However, the study is conducted under the assumption that all surfaces are rigid. The effects of ground absorption could be further investigated. Moreover, the level differences of 30 to 35 dB at the shadow zones, as shown in Fig. 2, should be larger if the source is placed on the ground, due to more effective shielding by the buildings. The validity of the FDTD simulation in such a high-dynamic-range condition is another topic for investigation, because the evaluation of ground traffic noise, for example, is considered to be an important application field of the FDTD simulations.

This work was supported by JSPS Grant-in-Aid for Scientific Research (B) 23360255, Joint Usage/Research Center for Interdisciplinary Large-Scale Information Infrastructures and Center for Spatial Information Science, the University of Tokyo. Parts of the fundamental geospatial data by the Geospatial Information Authority of Japan were used.