Similarly to the image-source method, ray tracing assumes that sound energy travels around a scene in “rays”. The rays start at the sound source, and are all emitted in uniformly random directions at the same time, travelling at the speed of sound. When a ray hits a boundary, it loses some of its energy, depending on the properties of the boundary’s material. Then, the ray is reflected. When it intersects the receiver, the energy and time-delay of the ray is recorded. In these ways, the models are similar. However, there are some important differences between the two methods, explained below.

Image sources are deterministic, while ray tracing is stochastic. The image-source method finds exact specular reflections, which are fixed for given source, receiver, and boundary positions. Ray tracing is less accurate, aiming to compute a result which is correct, within a certain probability. A large number of rays are fired into the scene in random directions, and traced until they have undergone a certain number of reflections. Some of these rays may intersect with the receiver volume (see below), but some may not. Only rays that *do* intersect the receiver contribute to the final output. The proportion of rays which intersect the receiver, and therefore the measured energy at the receiver, can be found with greater accuracy simply by increasing the number of rays fired.

The random nature of ray tracing requires that the receiver must have a finite volume. The likelihood of any given random ray intersecting with a single point with no volume tends to zero (there is an infinite number of possible ray directions, only one of which passes through the point). If the probability of a ray-receiver intersection is to be non-zero, the receiver must have some volume. This is different to the image-source method, which traces reflections backwards from the receiver, allowing it to be represented as a point.

In ray tracing, each ray represents a finite portion of the initial source energy. The reduction of energy over a given distance is accounted for by the spreading-out of the rays. This can be illustrated very simply: Imagine a sphere placed very close to a point. Assuming rays are fired with a uniform random distribution from that point, a certain proportion of those rays will intersect with the sphere. If the sphere is moved further away, a smaller proportion of rays will hit it (see fig. 1).

The exact proportion of intersecting rays is equal to \(s/4r^2\) [1, p. 75], where \(s\) is the constant area covered by the receiver, and \(r\) is the distance between the source and receiver. That is, the proportion of rays intersecting the receiver is inversely proportional to the square of the distance between the source and receiver. The energy registered is proportional to the number of ray intersections recorded, therefore the ray model intrinsically accounts for the inverse-square law for energy, and the per-ray energy does not need to be scaled proportionally to the distance travelled. This differs to the image-source model, in which only valid specular reflections are recorded, and the inverse-square law must be applied directly.

The final major difference between ray tracing and the image-source method is to do with the way in which results are recorded. The image-source method finds exact specular reflections, each of which contributes an impulsive signal with specific frequency content at a precise time. This reflection data is precise and accurate, so it can be used to render an output signal at arbitrarily high sampling frequencies. Ray tracing, on the other hand, is inexact because it is based on stochastic methods. The accuracy of the output increases with the average number of rays detected per unit time. It is shown in [2, p. 191] that the mean number of intersections \(k\) per time period \(\Delta t\) is given by

\[k=\frac{N\pi r^2c\Delta t}{V}\](1)

where \(N\) is the number of rays, \(r\) is the radius of the receiver, \(c\) is the speed of sound, and \(V\) is the room volume.

For an output which covers the human hearing range, the sampling rate must be at least 40kHz, which corresponds to a sampling period of 25\(\mu\)s. Therefore, for a receiver radius of 0.1m (around the size of a human head), and assuming that one detection-per-sampling-period is adequate, the minimum number of rays is

\[N=\frac{kV}{\pi r^2c\Delta t} = \frac{V}{\pi \cdot 0.1^2 \cdot 340 \cdot 0.000025 } \approx 3745V\](2)

In actual simulations, especially of large and complex rooms, this number of rays is likely to produce results with large, inaccurate energy fluctuations. For higher accuracy, higher output sample rates, and smaller receivers the number of rays required becomes even greater. This sheer quantity of rays requires a vast number of independent calculations which will be prohibitively time-consuming, even on modern hardware.

If, on the other hand, audio-rate results are not required, then the number of necessary rays is much lower. Vorländer suggests a sampling period of the order of magnitude of milliseconds, which requires at least 40-times fewer rays [2, p. 186].

Now, the ray tracer can be thought to produce an *energy envelope* describing the decay tail of the impulse response. To produce the impulse response itself, this energy envelope is simply overlaid onto a noise-like signal. The process will be described in greater detail in the following Implementation section.

Here, Wayverb’s ray tracer will be described. Details of the boundary- and microphone-modelling processes are discussed separately, in the Boundary Modelling and Microphone Modelling sections respectively.

The simulation begins identically to the image-source process. A voxel-based acceleration structure is created, to speed up ray intersection tests.

Rays are fired in uniformly-distributed random directions from the source point. Each ray carries an equal quantity of energy (the method for determining the starting energy is described in the Hybrid section). If a ray intersects with the scene geometry, data is stored about that intersection: its position, the unique ID of the triangle which was intersected, and whether or not the receiver point is visible from this position. This data will be used later on, when calculating energy loss, and the directional distribution of received energy.

Next, a new ray direction is calculated using the *vector-based scattering* method, described by Christensen and Rindel [3]. A uniformly random vector is generated, within the hemisphere oriented in the same direction as the triangle normal. The ideal specular direction is also calculated, and the two vectors are combined by

\[\overrightarrow{R}_\text{outgoing}=s\overrightarrow{R}_\text{random} + (1-s)\overrightarrow{R}_\text{specular}\](3)

where \(s\) is the scattering coefficient, as described in the Theory chapter. Normally, the scattering coefficient would be defined per-band, but this would require running the ray tracer once per band, so that each frequency component can be scattered differently. Instead, the mean scattering coefficient is used, so that all bands can be traced in one pass. For eight frequency bands, this provides an eight-times speed-up, at the cost of inaccurate interpretation of the scattering coefficients. This is a reasonable trade-off, as scattering will also be modelled using the *diffuse rain* technique described in the Boundary Modelling section, which *is* able to account for different per-band scattering coefficients. The final output will therefore retain per-band scattering characteristics, but with much improved performance.

Having calculated a new ray direction, the energy carried in the ray is decreased, depending on the absorption coefficients of the intersected triangle. If the surface has an absorption coefficient of \(\alpha\) in a particular band, then the energy in that band is multiplied by \((1 - \alpha)\) to find the outgoing energy. This process is repeated, using the incoming energy and absorption coefficient for each band, to find outgoing energies in all bands. The new ray, with the computed outgoing energies and vector-scattered direction, is now traced.

The ray tracing process continues for a set number of reflections. Typically, each ray would be traced until the energy in all bands has fallen below a certain threshold, requiring an additional check per reflection per ray [2, p. 183]. Under such a scheme, some rays might reach this threshold faster than others, depending on the absorptions of intermediate materials. However, in Wayverb all rays are traced in parallel, so it is not feasible or necessary to allow rays to quit early. The time taken for a parallel computation will always be limited by the longest-running process. If some rays are “stopped” early, this does not improve the processing-speed of the continuing rays, so the simulation still takes the same time to complete. Instead, the maximum possible required depth is found before the simulation, and all rays are traced to this maximum depth.

To find the maximum required ray tracing depth, first the minimum absorption of all surfaces in the scene is found. The outgoing energy from a single reflection is equal to \(E_\text{incoming}(1-\alpha)\) where \(E_\text{incoming}\) is the incoming energy and \(\alpha\) is the surface absorption. It follows that the outgoing energy from a series of reflections is given by \(E_\text{incoming}(1-\alpha)^{n_\text{reflections}}\). Then, the maximum ray tracing depth is equal to the number of reflections from the minimally absorptive surface required to reduce the energy of a ray by 60dB:

\[(1-\alpha_\text{min})^{n_\text{reflections}} = 10^{-6} \therefore n_\text{reflections}=\left\lceil-\frac{6}{\log_{10}(1-\alpha_\text{min})}\right\rceil\](4)

The 60dB level decrease is somewhat arbitrary, but was chosen to correspond to the *RT60*, which is a common descriptor of recorded impulse responses. In a future version of the software, the level decrease might be set depending on the dynamic range of the output format. This would allow 16-bit renders (with around 48dB of dynamic range) to use fewer reflections, while 32-bit outputs with lower noise floors would require more reflections.

The output of the ray tracing process is a set of histograms, one per frequency band, plotting recorded energy per time step. This recorded energy may come from two different sources.

Firstly, if a ray intersects with the receiver volume, then the current per-band energy of that ray, which may have been attenuated by previous reflections, is added to the result histograms at the appropriate time step. The time of the energy contribution is given by the total distance travelled by the ray, divided by the speed of sound. This is the approach taken in typical acoustic ray tracers.

Secondly, each reflection point is considered to spawn a “secondary source” which emits scattered sound energy, depending on the scattering coefficients of the surface. If the receiver is visible from the reflection point, then an energy contribution is logged, at a time proportional to the distance travelled by the ray. Similarly to absorption coefficients, scattering coefficients are defined per frequency band, and these coefficients are used to weight the contributions logged in each band of the output histograms. This mimics the real world behaviour of rough surfaces, which cause some energy to be randomly diffused in non-specular directions during reflection of the wave-front. The exact level of this contribution is explained in the Geometric Implementation subsection of the Boundary Modelling page.

When ray tracing has completed, the result is a set of histograms which describe the energy decay envelope of each frequency band. These histograms will have the relatively low sampling rate, as explained above (Wayverb uses a sampling rate of 1kHz). As a result, these histograms are not directly suitable for auralisation. To produce audio-rate impulse responses, the “fine structure” of the decay tail must be synthesised and then the gain adjusted using the histogram envelopes. The process used in Wayverb to convert the histogram into an audio-rate impulse response is described in [4], and in greater depth in [1, p. 70], though an overview will be given here. Figure 2 outlines the process of estimating an audio-rate representation of low-sample-rate multi-band histograms.

First, a noise-like sequence of Dirac impulses is generated at audio-rate. This sequence is designed to mimic the density of reflections in an impulse response of a certain volume. Therefore it is modelled as a temporal Poisson process which starts sparse, and with increasing density of impulses over time. Specifically, the time between one impulse event and the next is given by

\[\Delta t_\text{event}(z) = \frac{\ln\frac{1}{z}}{\mu}\](5)

where \(z\) is a uniformly distributed random number \(0 < z \leq 1\). \(\mu\) here is the mean event occurrence, and is dependent upon the current simulation time \(t\), the enclosure volume \(V\) and the speed of sound \(c\):

\[\mu = \frac{4\pi c^3 t^2}{V}\](6)

It can be seen that the mean occurrence is proportional to the square of the current time, producing an increase in event density over time. The first event occurs at time \(t_0\):

\[t_0=\sqrt[3]{\frac{2V\ln 2}{4\pi c^3}}\](7)

The full-length noise signal is produced by repeatedly generating inter-event times \(\Delta t_\text{event}\), and adding Dirac impulses to a buffer, until the final event time is greater or equal to the time of the final histogram interval. Dirac deltas falling on the latter half of a sampling interval are taken to be negative-valued. The number of Dirac deltas per sample is limited to one, and the value of \(\mu\) is limited to a maximum of 10kHz, which has been shown to produce results absent of obvious artefacts [4].

The noise sequence is duplicated, once for each frequency band. Then, the noise sequence for each band is weighted according to that band’s histogram. This enveloping is not quite as simple as multiplying each noise sample with the histogram entry at the corresponding time. Instead, the enveloping process must conserve the energy level recorded over each time step.

For each time interval in the histogram, the corresponding range of samples in the noise sequence is found. If the output sample rate is \(f_s\) and the histogram time step is \(\Delta t\), then the noise sequence sample corresponding to histogram step \(h\) is \(\lfloor h \cdot f_s \cdot \Delta t \rfloor\). The corrected energy level for each histogram step is found by dividing the histogram energy value by the sum of squared noise samples for that step. This is converted to a corrected pressure level by \(P = \sqrt{Z_0 I}\), where \(I\) is the corrected energy level, and \(Z_0\) is the acoustic impedance of air. The weighting is now accomplished by multiplying each noise sequence sample corresponding to this histogram step by the corrected pressure level.

Now, we are left with a set of broadband signals, each with different envelopes. The output signal is found by bandpass filtering each of these signals, and then mixing them down.

The filter bank should have perfect reconstruction characteristics: a signal passed through all filters in parallel and then summed should have the same frequency response as the original signal. In the case where all the materials in a scene have the same coefficients in all bands, the input to each filter would be identical. Then, the expected output would be the same as the input to any band (though band-passed between the lower cutoff of the lowest band and the upper cutoff of the highest band). Perfect-reconstruction filters maintain the correct behaviour in this (unusual) case. It is especially important that the bandpass filters are zero-phase, so that Dirac events in all bands are in-phase, without group delay, after filtering. Finally, the filters should have slow roll-off and no resonance, so that if adjacent bands have very mismatched levels, there are no obvious filtering artefacts.

An efficient filtering method is to use a bank of infinite-impulse-response filters. These filters are fast, and have low memory requirements. They can also be made zero-phase when filtering is offline, by running the filter forwards then backwards over the input signal (though this causes filter roll-off to become twice as steep). This was the initial method used in Wayverb: the filter bank was constructed from second-order Linkwitz-Riley bandpass filters. This method had two main drawbacks: the roll-off is limited to a minimum of 24 dB/octave [5], which may cause an obvious discontinuity in level in the final output; and the forward-backward filtering method requires computing the initial filter conditions in order to maintain perfect-reconstruction, which is non-trivial to implement [6].

A better method, which allows for shallower filter roll-offs while retaining perfect-reconstruction capabilities is to filter each band directly in the frequency domain. The filtering of each signal is accomplished by computing the signal’s frequency-domain representation, attenuating bins outside the passband, and then transforming the altered spectrum back to the time domain. To ensure perfect reconstruction, and to avoid artificial-sounding discontinuities in the spectrum, the filter shape suggested in [7] is used. This paper suggests equations which describe the band-edge magnitudes:

\[ G_\text{lower}(\omega_\text{edge} + p) = \sin^2\left(\frac{\pi}{2}\phi_l(p)\right), \\ G_\text{upper}(\omega_\text{edge} + p) = \cos^2\left(\frac{\pi}{2}\phi_l(p)\right) \](8)

Here, \(G\) is a function of frequency, \(\omega_\text{edge}\) is the band-edge frequency, and \(p\) is the relative frequency of a nearby frequency bin. The equations are computed for a range of values \(p=P,\dots,P\) where \(P\) is the width of the crossover. The definition of \(\phi_l(p), l \geq 0\) is recursive:

\[ \phi_l(p)= \begin{cases} \frac{1}{2}(p / P + 1), & l = 0 \\ \sin(\frac{\pi}{2}\phi_{l-1}(p)), & \text{otherwise} \end{cases} \](9)

The variable \(l\) defines the steepness of the crossover, and is set to 0 in Wayverb, so that the transition between bands is as slow and smooth as possible. The absolute width of the crossover is denoted by \(P\), but it is more useful to specify the crossover width in terms of overlap \(0 \leq o \leq 1\). Assuming logarithmically-spaced frequency bands, spread over the range \(\omega_\text{lowest}, \dots, \omega_\text{highest}\) where the edge frequency of band \(i\) is defined as

\[\omega_{\text{edge}_i}=\omega_\text{lowest}\left(\frac{\omega_\text{highest}}{\omega_\text{lowest}}^\frac{i}{N_\text{bands}}\right)\](10)

the maximum width factor \(w\) is given by

\[w=\frac{x-1}{x+1}, x=\frac{\omega_\text{highest}}{\omega_\text{lowest}}^\frac{1}{N_\text{bands}}\](11)

For \(\omega_\text{lowest}=20\text{Hz}\), \(\omega_\text{highest}=20\text{kHz}\), and \(N_\text{bands}=8\), \(w \approx 0.4068\). Then, the band edge width \(P\) can be defined in terms of the overlap-amount \(o\), the frequency of this band edge \(\omega_\text{edge}\), and the maximum allowable width factor \(w\): \(P=\omega_\text{edge}ow\). Wayverb sets the overlap factor \(o=1\) to ensure wide, natural-sounding crossovers.

The final broadband signal is found by summing together the weighted, filtered noise sequences.

Like the image-source model, ray tracing assumes that energy is transported through the sound field in rays. However, ray tracing is stochastic, meaning that the overall accuracy of the simulation (including the maximum viable sampling rate) is governed by the number of rays used in the simulation. Tracing enough rays to produce audio-rate results is unnecessarily computationally expensive. It is much more efficient to trace at a reduced sampling rate, and to approximate the fine structure of the reverb tail during post-processing. This matches the goals of the Wayverb project, by increasing efficiency without negatively affecting the plausibility of the results.

In Wayverb, the ray tracer produces multi-band energy histograms, which are used to weight Poisson noise sequences. Perfect-reconstruction frequency-domain filtering is used to combine individual frequency bands into a single broadband IR. The final model is capable of modelling loss of sound energy due to distance, along with frequency-dependent boundary absorption and scattering.

[1] D. Schröder, *Physically based real-time auralization of interactive virtual environments*, vol. 11. Logos Verlag Berlin GmbH, 2011.

[2] M. Vorländer, *Auralization: Fundamentals of acoustics, modelling, simulation, algorithms and acoustic virtual reality*. Springer Science & Business Media, 2007.

[3] C. L. Christensen and J. H. Rindel, “A new scattering method that combines roughness and diffraction effects,” in *Forum Acousticum, Budapest, Hungary*, 2005.

[4] R. Heinz, “Binaural room simulation based on an image source model with addition of statistical methods to include the diffuse sound scattering of walls and to predict the reverberant tail,” *Applied Acoustics*, vol. 38, no. 2, pp. 145–159, 1993.

[5] S. H. Linkwitz, “Active Crossover Networks for Noncoincident Drivers,” *Journal of the Audio Engineering Society*, vol. 24, no. 1, pp. 2–8, Feb. 1976.

[6] F. Gustafsson, “Determining the initial states in forward-backward filtering,” 1994.

[7] J. Antoni, “Orthogonal-like fractional-octave-band filters,” *The Journal of the Acoustical Society of America*, vol. 127, no. 2, pp. 884–895, 2010.