Convert references to footnotes in README

README.md

@@ -53,7 +53,7 @@ There are two models provided with the code, which you can start to run from the
 To start a project (a model) from scratch, follow the provided examples, but essentially you will need to:
 
 1. Build a Sketchup model, set source/receiver locations in CSV files, and export to a JSON file.
-2. Fit absorption/impedance data to the passive boundary impedance model used [BHBS16]. Only a simple routine is provided to fit to 11 octave-band absorption coefficients (16Hz to 16kHz).
+2. Fit absorption/impedance data to the passive boundary impedance model used [^BHBS16]. Only a simple routine is provided to fit to 11 octave-band absorption coefficients (16Hz to 16kHz).
 3. Fill out a setup script to configure the model and simulation.
 4. Run your setup script, which in turns runs a 'voxelization' routine and sets up input/output grid positions and signals.
 5. Simulate your model from the exported .h5 data files with either the CPU-based Python code (which includes energy + visualization), or the C/CPU-based engine, or the C/CUDA/GPU-based engine.
@@ -82,7 +82,7 @@ Once you have your .h5 files ready in some folder, you can run the Python FDTD e
 
 When the simulation has completed there will be a 'sim_outs.h5' file which has the raw signal read from grid locations. You can process these signals to get final output files, which do need some
 cleanup to remove, e.g., high frequencies with too much error (numerical dispersion) or to resample to a standard audio rate like 48kHz. There are also filters to apply air absorption to the output
-responses [Ham21a,Ham21b]. PFFDTD is only designed to output monoaural RIRs, but you can build arrays of outputs and feed those into frequency-domain microphone-array processing tools [(e.g.,)](http://research.spa.aalto.fi/projects/sparta_vsts/) for spatial audio, or encode Ambisonics directly in the time-domain [BPH19] (a similar approach can be used for directive sources [BAH19]).
+responses [^Ham21a] [^Ham21b]. PFFDTD is only designed to output monoaural RIRs, but you can build arrays of outputs and feed those into frequency-domain microphone-array processing tools [(e.g.,)](http://research.spa.aalto.fi/projects/sparta_vsts/) for spatial audio, or encode Ambisonics directly in the time-domain [^BPH19] (a similar approach can be used for directive sources [^BAH19]).
 
 ## Enhancements
 
@@ -91,29 +91,29 @@ mentioning below.
 
 ### Operation in single precision
 
-Instabilities can occur with FDTD simulations in finite-precision if you wait long enough, and that length will depend on the precision chosen and the particular simulation. You should not experience instabilities in double precision (indeed, the code conserves energy to machine precision). Single precision is generally faster and uses less memory, but operating in single precision can be give rise to instabilities (due to rounding errors) (see, e.g., [HBW14]).
+Instabilities can occur with FDTD simulations in finite-precision if you wait long enough, and that length will depend on the precision chosen and the particular simulation. You should not experience instabilities in double precision (indeed, the code conserves energy to machine precision). Single precision is generally faster and uses less memory, but operating in single precision can be give rise to instabilities (due to rounding errors) (see, e.g., [^HBW14]:).
 
-If you choose to use single precision, this code has a few safeguards to mitigate/delay long-time instabilities. First, the eigenvalues of the underlying finite-difference Laplacian operator are perturbed to prevent DC-mode instabilities. In view of a matrix operator, off-diagonal elements are rounded towards zero (RTZ), while diagonal elements are shifted (on the order of machine epsilon) such that the finite-difference Laplacian operator remains negative semi-definite (a condition for stability). These DC-mode safeguards are only active in the single-precision CUDA version (they aren't needed in double precision). Secondly, the input signal to the scheme (e.g., an impulse) must be differentiated (using 'diff_source' option for sim_setup.py) to remove any DC input. At post-processing, the output will be integrated with a combined integrator/high-pass Butterworth filter [HPB20], which integrates while suppressing any leftover problematic DC modes.
+If you choose to use single precision, this code has a few safeguards to mitigate/delay long-time instabilities. First, the eigenvalues of the underlying finite-difference Laplacian operator are perturbed to prevent DC-mode instabilities. In view of a matrix operator, off-diagonal elements are rounded towards zero (RTZ), while diagonal elements are shifted (on the order of machine epsilon) such that the finite-difference Laplacian operator remains negative semi-definite (a condition for stability). These DC-mode safeguards are only active in the single-precision CUDA version (they aren't needed in double precision). Secondly, the input signal to the scheme (e.g., an impulse) must be differentiated (using 'diff_source' option for sim_setup.py) to remove any DC input. At post-processing, the output will be integrated with a combined integrator/high-pass Butterworth filter [^HPB20], which integrates while suppressing any leftover problematic DC modes.
 
 ### Staircasing in FDTD
 
-Staircase effects in regular-grid FDTD schemes is a known issue [BHBS16]. In essence, surfaces areas of boundaries are incorrectly estimated, leading to an overestimation of decay times. A novel correction scheme is provided in this code, based on the idea of an effective surface area. This concept is simple – it amounts to a weighting factor on the boundary-surface based on the inner product between the target boundary surface and a voxel boundary surface. Surface area errors before and after correction are tabulated in the voxelizer, and for refined grids the corrections brings errors down from as high as 50% to generally below 1% (going to zero in the limit of small cells). This helps provide more consistent estimates of decay times, and makes the simulation more robust to changes in grid resolution or scene rotations, while keeping the boundary updates efficient to implement and carry out on GPUs.
+Staircase effects in regular-grid FDTD schemes is a known issue [^BHBS16]. In essence, surfaces areas of boundaries are incorrectly estimated, leading to an overestimation of decay times. A novel correction scheme is provided in this code, based on the idea of an effective surface area. This concept is simple – it amounts to a weighting factor on the boundary-surface based on the inner product between the target boundary surface and a voxel boundary surface. Surface area errors before and after correction are tabulated in the voxelizer, and for refined grids the corrections brings errors down from as high as 50% to generally below 1% (going to zero in the limit of small cells). This helps provide more consistent estimates of decay times, and makes the simulation more robust to changes in grid resolution or scene rotations, while keeping the boundary updates efficient to implement and carry out on GPUs.
 
 ### FCC scheme
 
-For efficiency the 13-point FCC scheme is recommended over the 7-point Cartesian scheme, as it is typically requires ~5x less memory than the Cartesian scheme for a 1%-2% levels of dispersion error [HW13,Ham16]. However, the FCC scheme is tricky to implement due to its setting on a non-Cartesian grid. One solution to this has been compress the FCC grid like an accordion so it fits on a Cartesian grid, but there's a better solution implemented in this code. Namely, the FCC subgrid is folded onto itself across one dimension such that the stencil operation is uniform throughout (the old solution has some branching involved). Only the C and C/CUDA versions have this. The Python engine version uses a straightforward, yet redundant, Cartesian grid (aka using the CCP scheme).
+For efficiency the 13-point FCC scheme is recommended over the 7-point Cartesian scheme, as it is typically requires ~5x less memory than the Cartesian scheme for a 1%-2% levels of dispersion error [^HW13] [^Ham16]. However, the FCC scheme is tricky to implement due to its setting on a non-Cartesian grid. One solution to this has been compress the FCC grid like an accordion so it fits on a Cartesian grid, but there's a better solution implemented in this code. Namely, the FCC subgrid is folded onto itself across one dimension such that the stencil operation is uniform throughout (the old solution has some branching involved). Only the C and C/CUDA versions have this. The Python engine version uses a straightforward, yet redundant, Cartesian grid (aka using the CCP scheme).
 
 ## Performance benchmarks
 
-See (TODO:) for some performance benchmark results using single-node Nvidia GPUs servers, with GPU architectures ranging from Kepler to Ampere. This software has been tested with up to 30b nodes on the FCC grid (~250GB). For even larger, multi-node (MPI-based) FDTD simulations, see [ParallelFDTD](https://github.com/AaltoRSE/ParallelFDTD) and [SCM18].
+See (TODO:) for some performance benchmark results using single-node Nvidia GPUs servers, with GPU architectures ranging from Kepler to Ampere. This software has been tested with up to 30b nodes on the FCC grid (~250GB). For even larger, multi-node (MPI-based) FDTD simulations, see [ParallelFDTD](https://github.com/AaltoRSE/ParallelFDTD) and [^SCM18].
 
 ## License
 
 This software is released under the MIT license. See the LICENSE file for details. The Sketchup models found in the `models` folder are released under different licenses; see the README files in their corresponding folders.
 
 ## Credits
 
-The development of this code was not funded by any body or institution, but some credit can be given to grants ERC-StG-2011-279068-NESS and ERC-PoC-2016-WRAM, which funded many of the underlying (published) simulation algorithms developed within the Acoustics & Audio Group at the University of Edinburgh (see, e.g., list of references below). Some of the performance benchmarks of this code were carried out on GPUs (K80 / GTX1080Ti) paid for by those grants. Also, the Sketchup models provided were initially created by Heather Lai [LH20] and Nathaniel Fletcher [HWFB16] .
+The development of this code was not funded by any body or institution, but some credit can be given to grants ERC-StG-2011-279068-NESS and ERC-PoC-2016-WRAM, which funded many of the underlying (published) simulation algorithms developed within the Acoustics & Audio Group at the University of Edinburgh (see, e.g., list of references below). Some of the performance benchmarks of this code were carried out on GPUs (K80 / GTX1080Ti) paid for by those grants. Also, the Sketchup models provided were initially created by Heather Lai [^LH20] and Nathaniel Fletcher [^HWFB16] .
 
 ## Citing this work
 
@@ -144,41 +144,54 @@ The above list is non-exhaustive. Use of the third-party software, libraries or
 
 ## Some background references
 
-[HW13] B. Hamilton and C. J. Webb. Room acoustics modelling using GPU-accelerated finite difference and
-finite volume methods on a face-centered cubic grid. In Proc. Digital Audio Effects (DAFx), pages
-336–343, Maynooth, Ireland, September 2013.
-
-[HBW14] B. Hamilton, S. Bilbao, and C. J. Webb. Revisiting implicit finite difference schemes for 3-D room
-acoustics simulations on GPU. In Proc. Digital Audio Effects (DAFx), pages 41–48, Erlangen, Germany,
-September 2014.
-
-[BHBS16] S. Bilbao, B. Hamilton, J. Botts, and L. Savioja. Finite volume time domain room acoustics simulation
-under general impedance boundary conditions. IEEE/ACM Trans. Audio, Speech, Lang. Process.,
-24(1):161–173, 2016.
-
-[Ham16] B. Hamilton. Finite Difference and Finite Volume Methods for Wave-based Modelling of Room
-Acoustics. Ph.D. thesis, University of Edinburgh, 2016.
-
-[HWFB16] B. Hamilton, C. J. Webb, N. D. Fletcher, and S. Bilbao. Finite difference room acoustics simulation with
-general impedance boundaries and viscothermal losses in air: Parallel implementation on multiple
-GPUs. In Proc. Int. Symp. Music & Room Acoust., La Plata, Argentina, September 2016.
-
-[SCM18] J. Saarelma, J. Califa, and R. Mehra. Challenges of distributed real-time finite-difference time-domain room acoustic simulation for auralization. In AES Int. Conf. Spatial Reproduction, July 2018.
-
-[BPH19] S. Bilbao, A. Politis, and B. Hamilton. Local time-domain spherical harmonic spatial encoding for
-wave-based acoustic simulation. IEEE Signal Process. Lett., 26(4):617–621, 2019.
-
-[BAH19] S. Bilbao, J. Ahrens, and B. Hamilton. Incorporating source directivity in wave-based virtual acoustics:
-Time-domain models and fitting to measured data. J. Acoust. Soc. Am., 146(4):2692–2703, 2019.
-
-[LH20] H. Lai and B. Hamilton. Computer modeling of barrel-vaulted sanctuary exhibiting flutter echo with
-comparison to measurements. Acoustics, 2(1):87–109, 2020.
-
-[HPB20] I. Henderson, A. Politis, and S. Bilbao. Filter design for real-time ambisonics encoding during
-wave-based acoustic simulations. In Proc. e-Forum Acusticum, Lyon, France, December 2020.
-
-[Ham21a] B. Hamilton. Adding air attenuation to simulated room impulse responses: A modal approach. In Proc.
-Int. Conf. Immersive & 3D Audio, Bologna, Italy, September 2021.
-
-[Ham21b] B. Hamilton. Air absorption filtering method based on approximate Green’s function for Stokes’
-equation. In Proc. Digital Audio Effects (DAFx), Vienna, Austria, September 2021.
+[^HW13]:
+    B. Hamilton and C. J. Webb. Room acoustics modelling using GPU-accelerated finite difference and
+    finite volume methods on a face-centered cubic grid. In Proc. Digital Audio Effects (DAFx), pages
+    336–343, Maynooth, Ireland, September 2013.
+
+[^HBW14]:
+    B. Hamilton, S. Bilbao, and C. J. Webb. Revisiting implicit finite difference schemes for 3-D room
+    acoustics simulations on GPU. In Proc. Digital Audio Effects (DAFx), pages 41–48, Erlangen, Germany,
+    September 2014.
+
+[^BHBS16]:
+    S. Bilbao, B. Hamilton, J. Botts, and L. Savioja. Finite volume time domain room acoustics simulation
+    under general impedance boundary conditions. IEEE/ACM Trans. Audio, Speech, Lang. Process.,
+    24(1):161–173, 2016.
+
+[^Ham16]:
+    B. Hamilton. Finite Difference and Finite Volume Methods for Wave-based Modelling of Room
+    Acoustics. Ph.D. thesis, University of Edinburgh, 2016.
+
+[^HWFB16]:
+    B. Hamilton, C. J. Webb, N. D. Fletcher, and S. Bilbao. Finite difference room acoustics simulation with
+    general impedance boundaries and viscothermal losses in air: Parallel implementation on multiple
+    GPUs. In Proc. Int. Symp. Music & Room Acoust., La Plata, Argentina, September 2016.
+
+[^BPH19]:
+    S. Bilbao, A. Politis, and B. Hamilton. Local time-domain spherical harmonic spatial encoding for
+    wave-based acoustic simulation. IEEE Signal Process. Lett., 26(4):617–621, 2019.
+
+[^BAH19]:
+    S. Bilbao, J. Ahrens, and B. Hamilton. Incorporating source directivity in wave-based virtual acoustics:
+    Time-domain models and fitting to measured data. J. Acoust. Soc. Am., 146(4):2692–2703, 2019.
+
+[^LH20]:
+    H. Lai and B. Hamilton. Computer modeling of barrel-vaulted sanctuary exhibiting flutter echo with
+    comparison to measurements. Acoustics, 2(1):87–109, 2020.
+
+[^HPB20]:
+    I. Henderson, A. Politis, and S. Bilbao. Filter design for real-time ambisonics encoding during
+    wave-based acoustic simulations. In Proc. e-Forum Acusticum, Lyon, France, December 2020.
+
+[^Ham21a]:
+    B. Hamilton. Adding air attenuation to simulated room impulse responses: A modal approach. In Proc.
+    Int. Conf. Immersive & 3D Audio, Bologna, Italy, September 2021.
+
+[^Ham21b]:
+    B. Hamilton. Air absorption filtering method based on approximate Green’s function for Stokes’
+    equation. In Proc. Digital Audio Effects (DAFx), Vienna, Austria, September 2021.
+
+[^SCM18]:
+    J. Saarelma, J. Califa, and R. Mehra. Challenges of distributed real-time finite-difference
+    time-domain room acoustic simulation for auralization. In AES Int. Conf. Spatial Reproduction, July 2018.