## atmosphere/functions.glsl

This GLSL file contains the core functions that implement our atmosphere model. It provides functions to compute the transmittance, the single scattering and the second and higher orders of scattering, the ground irradiance, as well as functions to store these in textures and to read them back. It uses physical types and constants which are provided in two versions: a GLSL version and a C++ version. This allows this file to be compiled either with a GLSL compiler or with a C++ compiler (see the Introduction).

The functions provided in this file are organized as follows:

They use the following utility functions to avoid NaNs due to floating point values slightly outside their theoretical bounds:

Number ClampCosine(Number mu) {
return clamp(mu, Number(-1.0), Number(1.0));
}

Length ClampDistance(Length d) {
return max(d, 0.0 * m);
}

Length ClampRadius(IN(AtmosphereParameters) atmosphere, Length r) {
}

Length SafeSqrt(Area a) {
return sqrt(max(a, 0.0 * m2));
}


### Transmittance

As the light travels from a point $\bp$ to a point $\bq$ in the atmosphere, it is partially absorbed and scattered out of its initial direction because of the air molecules and the aerosol particles. Thus, the light arriving at $\bq$ is only a fraction of the light from $\bp$, and this fraction, which depends on wavelength, is called the transmittance. The following sections describe how we compute it, how we store it in a precomputed texture, and how we read it back.

#### Computation

For 3 aligned points $\bp$, $\bq$ and $\br$ inside the atmosphere, in this order, the transmittance between $\bp$ and $\br$ is the product of the transmittance between $\bp$ and $\bq$ and between $\bq$ and $\br$. In particular, the transmittance between $\bp$ and $\bq$ is the transmittance between $\bp$ and the nearest intersection $\bi$ of the half-line $[\bp,\bq)$ with the top or bottom atmosphere boundary, divided by the transmittance between $\bq$ and $\bi$ (or 0 if the segment $[\bp,\bq]$ intersects the ground):

Also, the transmittance between $\bp$ and $\bq$ and between $\bq$ and $\bp$ are the same. Thus, to compute the transmittance between arbitrary points, it is sufficient to know the transmittance between a point $\bp$ in the atmosphere, and points $\bi$ on the top atmosphere boundary. This transmittance depends on only two parameters, which can be taken as the radius $r=\Vert\bo\bp\Vert$ and the cosine of the "view zenith angle", $\mu=\bo\bp\cdot\bp\bi/\Vert\bo\bp\Vert\Vert\bp\bi\Vert$. To compute it, we first need to compute the length $\Vert\bp\bi\Vert$, and we need to know when the segment $[\bp,\bi]$ intersects the ground.

##### Distance to the top atmosphere boundary

A point at distance $d$ from $\bp$ along $[\bp,\bi)$ has coordinates $[d\sqrt{1-\mu^2}, r+d\mu]^\top$, whose squared norm is $d^2+2r\mu d+r^2$. Thus, by definition of $\bi$, we have $\Vert\bp\bi\Vert^2+2r\mu\Vert\bp\bi\Vert+r^2=r_{\mathrm{top}}^2$, from which we deduce the length $\Vert\bp\bi\Vert$:

Length DistanceToTopAtmosphereBoundary(IN(AtmosphereParameters) atmosphere,
Length r, Number mu) {
assert(mu >= -1.0 && mu <= 1.0);
Area discriminant = r * r * (mu * mu - 1.0) +
return ClampDistance(-r * mu + SafeSqrt(discriminant));
}


We will also need, in the other sections, the distance to the bottom atmosphere boundary, which can be computed in a similar way (this code assumes that $[\bp,\bi)$ intersects the ground):

Length DistanceToBottomAtmosphereBoundary(IN(AtmosphereParameters) atmosphere,
Length r, Number mu) {
assert(mu >= -1.0 && mu <= 1.0);
Area discriminant = r * r * (mu * mu - 1.0) +
return ClampDistance(-r * mu - SafeSqrt(discriminant));
}

##### Intersections with the ground

The segment $[\bp,\bi]$ intersects the ground when $d^2+2r\mu d+r^2=r_{\mathrm{bottom}}^2$ has a solution with $d \ge 0$. This requires the discriminant $r^2(\mu^2-1)+r_{\mathrm{bottom}}^2$ to be positive, from which we deduce the following function:

bool RayIntersectsGround(IN(AtmosphereParameters) atmosphere,
Length r, Number mu) {
assert(mu >= -1.0 && mu <= 1.0);
return mu < 0.0 && r * r * (mu * mu - 1.0) +
}

##### Transmittance to the top atmosphere boundary

We can now compute the transmittance between $\bp$ and $\bi$. From its definition and the Beer-Lambert law, this involves the integral of the number density of air molecules along the segment $[\bp,\bi]$, as well as the integral of the number density of aerosols and the integral of the number density of air molecules that absorb light (e.g. ozone) - along the same segment. These 3 integrals have the same form and, when the segment $[\bp,\bi]$ does not intersect the ground, they can be computed numerically with the help of the following auxilliary function (using the trapezoidal rule):

Number GetLayerDensity(IN(DensityProfileLayer) layer, Length altitude) {
Number density = layer.exp_term * exp(layer.exp_scale * altitude) +
layer.linear_term * altitude + layer.constant_term;
return clamp(density, Number(0.0), Number(1.0));
}

Number GetProfileDensity(IN(DensityProfile) profile, Length altitude) {
return altitude < profile.layers.width ?
GetLayerDensity(profile.layers, altitude) :
GetLayerDensity(profile.layers, altitude);
}

Length ComputeOpticalLengthToTopAtmosphereBoundary(
IN(AtmosphereParameters) atmosphere, IN(DensityProfile) profile,
Length r, Number mu) {
assert(mu >= -1.0 && mu <= 1.0);
// Number of intervals for the numerical integration.
const int SAMPLE_COUNT = 500;
// The integration step, i.e. the length of each integration interval.
Length dx =
DistanceToTopAtmosphereBoundary(atmosphere, r, mu) / Number(SAMPLE_COUNT);
// Integration loop.
Length result = 0.0 * m;
for (int i = 0; i <= SAMPLE_COUNT; ++i) {
Length d_i = Number(i) * dx;
// Distance between the current sample point and the planet center.
Length r_i = sqrt(d_i * d_i + 2.0 * r * mu * d_i + r * r);
// Number density at the current sample point (divided by the number density
// at the bottom of the atmosphere, yielding a dimensionless number).
Number y_i = GetProfileDensity(profile, r_i - atmosphere.bottom_radius);
// Sample weight (from the trapezoidal rule).
Number weight_i = i == 0 || i == SAMPLE_COUNT ? 0.5 : 1.0;
result += y_i * weight_i * dx;
}
return result;
}


With this function the transmittance between $\bp$ and $\bi$ is now easy to compute (we continue to assume that the segment does not intersect the ground):

DimensionlessSpectrum ComputeTransmittanceToTopAtmosphereBoundary(
IN(AtmosphereParameters) atmosphere, Length r, Number mu) {
assert(mu >= -1.0 && mu <= 1.0);
return exp(-(
atmosphere.rayleigh_scattering *
ComputeOpticalLengthToTopAtmosphereBoundary(
atmosphere, atmosphere.rayleigh_density, r, mu) +
atmosphere.mie_extinction *
ComputeOpticalLengthToTopAtmosphereBoundary(
atmosphere, atmosphere.mie_density, r, mu) +
atmosphere.absorption_extinction *
ComputeOpticalLengthToTopAtmosphereBoundary(
atmosphere, atmosphere.absorption_density, r, mu)));
}


#### Precomputation

The above function is quite costly to evaluate, and a lot of evaluations are needed to compute single and multiple scattering. Fortunately this function depends on only two parameters and is quite smooth, so we can precompute it in a small 2D texture to optimize its evaluation.

For this we need a mapping between the function parameters $(r,\mu)$ and the texture coordinates $(u,v)$, and vice-versa, because these parameters do not have the same units and range of values. And even if it was the case, storing a function $f$ from the $[0,1]$ interval in a texture of size $n$ would sample the function at $0.5/n$, $1.5/n$, ... $(n-0.5)/n$, because texture samples are at the center of texels. Therefore, this texture would only give us extrapolated function values at the domain boundaries ($0$ and $1$). To avoid this we need to store $f(0)$ at the center of texel 0 and $f(1)$ at the center of texel $n-1$. This can be done with the following mapping from values $x$ in $[0,1]$ to texture coordinates $u$ in $[0.5/n,1-0.5/n]$ - and its inverse:

Number GetTextureCoordFromUnitRange(Number x, int texture_size) {
return 0.5 / Number(texture_size) + x * (1.0 - 1.0 / Number(texture_size));
}

Number GetUnitRangeFromTextureCoord(Number u, int texture_size) {
return (u - 0.5 / Number(texture_size)) / (1.0 - 1.0 / Number(texture_size));
}


Using these functions, we can now define a mapping between $(r,\mu)$ and the texture coordinates $(u,v)$, and its inverse, which avoid any extrapolation during texture lookups. In the original implementation this mapping was using some ad-hoc constants chosen for the Earth atmosphere case. Here we use a generic mapping, working for any atmosphere, but still providing an increased sampling rate near the horizon. Our improved mapping is based on the parameterization described in our paper for the 4D textures: we use the same mapping for $r$, and a slightly improved mapping for $\mu$ (considering only the case where the view ray does not intersect the ground). More precisely, we map $\mu$ to a value $x_{\mu}$ between 0 and 1 by considering the distance $d$ to the top atmosphere boundary, compared to its minimum and maximum values $d_{\mathrm{min}}=r_{\mathrm{top}}-r$ and $d_{\mathrm{max}}=\rho+H$ (cf. the notations from the paper and the figure below):

With these definitions, the mapping from $(r,\mu)$ to the texture coordinates $(u,v)$ can be implemented as follows:

vec2 GetTransmittanceTextureUvFromRMu(IN(AtmosphereParameters) atmosphere,
Length r, Number mu) {
assert(mu >= -1.0 && mu <= 1.0);
// Distance to top atmosphere boundary for a horizontal ray at ground level.
// Distance to the horizon.
Length rho =
// Distance to the top atmosphere boundary for the ray (r,mu), and its minimum
// and maximum values over all mu - obtained for (r,1) and (r,mu_horizon).
Length d = DistanceToTopAtmosphereBoundary(atmosphere, r, mu);
Length d_min = atmosphere.top_radius - r;
Length d_max = rho + H;
Number x_mu = (d - d_min) / (d_max - d_min);
Number x_r = rho / H;
return vec2(GetTextureCoordFromUnitRange(x_mu, TRANSMITTANCE_TEXTURE_WIDTH),
GetTextureCoordFromUnitRange(x_r, TRANSMITTANCE_TEXTURE_HEIGHT));
}


and the inverse mapping follows immediately:

void GetRMuFromTransmittanceTextureUv(IN(AtmosphereParameters) atmosphere,
IN(vec2) uv, OUT(Length) r, OUT(Number) mu) {
assert(uv.x >= 0.0 && uv.x <= 1.0);
assert(uv.y >= 0.0 && uv.y <= 1.0);
Number x_mu = GetUnitRangeFromTextureCoord(uv.x, TRANSMITTANCE_TEXTURE_WIDTH);
Number x_r = GetUnitRangeFromTextureCoord(uv.y, TRANSMITTANCE_TEXTURE_HEIGHT);
// Distance to top atmosphere boundary for a horizontal ray at ground level.
// Distance to the horizon, from which we can compute r:
Length rho = H * x_r;
// Distance to the top atmosphere boundary for the ray (r,mu), and its minimum
// and maximum values over all mu - obtained for (r,1) and (r,mu_horizon) -
// from which we can recover mu:
Length d_min = atmosphere.top_radius - r;
Length d_max = rho + H;
Length d = d_min + x_mu * (d_max - d_min);
mu = d == 0.0 * m ? Number(1.0) : (H * H - rho * rho - d * d) / (2.0 * r * d);
mu = ClampCosine(mu);
}


It is now easy to define a fragment shader function to precompute a texel of the transmittance texture:

DimensionlessSpectrum ComputeTransmittanceToTopAtmosphereBoundaryTexture(
IN(AtmosphereParameters) atmosphere, IN(vec2) frag_coord) {
const vec2 TRANSMITTANCE_TEXTURE_SIZE =
vec2(TRANSMITTANCE_TEXTURE_WIDTH, TRANSMITTANCE_TEXTURE_HEIGHT);
Length r;
Number mu;
GetRMuFromTransmittanceTextureUv(
atmosphere, frag_coord / TRANSMITTANCE_TEXTURE_SIZE, r, mu);
return ComputeTransmittanceToTopAtmosphereBoundary(atmosphere, r, mu);
}


#### Lookup

With the help of the above precomputed texture, we can now get the transmittance between a point and the top atmosphere boundary with a single texture lookup (assuming there is no intersection with the ground):

DimensionlessSpectrum GetTransmittanceToTopAtmosphereBoundary(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
Length r, Number mu) {
vec2 uv = GetTransmittanceTextureUvFromRMu(atmosphere, r, mu);
return DimensionlessSpectrum(texture(transmittance_texture, uv));
}


Also, with $r_d=\Vert\bo\bq\Vert=\sqrt{d^2+2r\mu d+r^2}$ and $\mu_d= \bo\bq\cdot\bp\bi/\Vert\bo\bq\Vert\Vert\bp\bi\Vert=(r\mu+d)/r_d$ the values of $r$ and $\mu$ at $\bq$, we can get the transmittance between two arbitrary points $\bp$ and $\bq$ inside the atmosphere with only two texture lookups (recall that the transmittance between $\bp$ and $\bq$ is the transmittance between $\bp$ and the top atmosphere boundary, divided by the transmittance between $\bq$ and the top atmosphere boundary, or the reverse - we continue to assume that the segment between the two points does not intersect the ground):

DimensionlessSpectrum GetTransmittance(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
Length r, Number mu, Length d, bool ray_r_mu_intersects_ground) {
assert(mu >= -1.0 && mu <= 1.0);
assert(d >= 0.0 * m);

Length r_d = ClampRadius(atmosphere, sqrt(d * d + 2.0 * r * mu * d + r * r));
Number mu_d = ClampCosine((r * mu + d) / r_d);

if (ray_r_mu_intersects_ground) {
return min(
GetTransmittanceToTopAtmosphereBoundary(
atmosphere, transmittance_texture, r_d, -mu_d) /
GetTransmittanceToTopAtmosphereBoundary(
atmosphere, transmittance_texture, r, -mu),
DimensionlessSpectrum(1.0));
} else {
return min(
GetTransmittanceToTopAtmosphereBoundary(
atmosphere, transmittance_texture, r, mu) /
GetTransmittanceToTopAtmosphereBoundary(
atmosphere, transmittance_texture, r_d, mu_d),
DimensionlessSpectrum(1.0));
}
}


where ray_r_mu_intersects_ground should be true iif the ray defined by $r$ and $\mu$ intersects the ground. We don't compute it here with RayIntersectsGround because the result could be wrong for rays very close to the horizon, due to the finite precision and rounding errors of floating point operations. And also because the caller generally has more robust ways to know whether a ray intersects the ground or not (see below).

Finally, we will also need the transmittance between a point in the atmosphere and the Sun. The Sun is not a point light source, so this is an integral of the transmittance over the Sun disc. Here we consider that the transmittance is constant over this disc, except below the horizon, where the transmittance is 0. As a consequence, the transmittance to the Sun can be computed with GetTransmittanceToTopAtmosphereBoundary, times the fraction of the Sun disc which is above the horizon.

This fraction varies from 0 when the Sun zenith angle $\theta_s$ is larger than the horizon zenith angle $\theta_h$ plus the Sun angular radius $\alpha_s$, to 1 when $\theta_s$ is smaller than $\theta_h-\alpha_s$. Equivalently, it varies from 0 when $\mu_s=\cos\theta_s$ is smaller than $\cos(\theta_h+\alpha_s)\approx\cos\theta_h-\alpha_s\sin\theta_h$ to 1 when $\mu_s$ is larger than $\cos(\theta_h-\alpha_s)\approx\cos\theta_h+\alpha_s\sin\theta_h$. In between, the visible Sun disc fraction varies approximately like a smoothstep (this can be verified by plotting the area of circular segment as a function of its sagitta). Therefore, since $\sin\theta_h=r_{\mathrm{bottom}}/r$, we can approximate the transmittance to the Sun with the following function:

DimensionlessSpectrum GetTransmittanceToSun(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
Length r, Number mu_s) {
Number sin_theta_h = atmosphere.bottom_radius / r;
Number cos_theta_h = -sqrt(max(1.0 - sin_theta_h * sin_theta_h, 0.0));
return GetTransmittanceToTopAtmosphereBoundary(
atmosphere, transmittance_texture, r, mu_s) *
mu_s - cos_theta_h);
}


### Single scattering

The single scattered radiance is the light arriving from the Sun at some point after exactly one scattering event inside the atmosphere (which can be due to air molecules or aerosol particles; we exclude reflections from the ground, computed separately). The following sections describe how we compute it, how we store it in a precomputed texture, and how we read it back.

#### Computation

Consider the Sun light scattered at a point $\bq$ by air molecules before arriving at another point $\bp$ (for aerosols, replace "Rayleigh" with "Mie" below):

The radiance arriving at $\bp$ is the product of:

• the solar irradiance at the top of the atmosphere,
• the transmittance between the Sun and $\bq$ (i.e. the fraction of the Sun light at the top of the atmosphere that reaches $\bq$),
• the Rayleigh scattering coefficient at $\bq$ (i.e. the fraction of the light arriving at $\bq$ which is scattered, in any direction),
• the Rayleigh phase function (i.e. the fraction of the scattered light at $\bq$ which is actually scattered towards $\bp$),
• the transmittance between $\bq$ and $\bp$ (i.e. the fraction of the light scattered at $\bq$ towards $\bp$ that reaches $\bp$).

Thus, by noting $\bw_s$ the unit direction vector towards the Sun, and with the following definitions:

• $r=\Vert\bo\bp\Vert$,
• $d=\Vert\bp\bq\Vert$,
• $\mu=(\bo\bp\cdot\bp\bq)/rd$,
• $\mu_s=(\bo\bp\cdot\bw_s)/r$,
• $\nu=(\bp\bq\cdot\bw_s)/d$
the values of $r$ and $\mu_s$ for $\bq$ are
• $r_d=\Vert\bo\bq\Vert=\sqrt{d^2+2r\mu d +r^2}$,
• $\mu_{s,d}=(\bo\bq\cdot\bw_s)/r_d=((\bo\bp+\bp\bq)\cdot\bw_s)/r_d= (r\mu_s + d\nu)/r_d$
and the Rayleigh and Mie single scattering components can be computed as follows (note that we omit the solar irradiance and the phase function terms, as well as the scattering coefficients at the bottom of the atmosphere - we add them later on for efficiency reasons):
void ComputeSingleScatteringIntegrand(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
Length r, Number mu, Number mu_s, Number nu, Length d,
bool ray_r_mu_intersects_ground,
OUT(DimensionlessSpectrum) rayleigh, OUT(DimensionlessSpectrum) mie) {
Length r_d = ClampRadius(atmosphere, sqrt(d * d + 2.0 * r * mu * d + r * r));
Number mu_s_d = ClampCosine((r * mu_s + d * nu) / r_d);
DimensionlessSpectrum transmittance =
GetTransmittance(
atmosphere, transmittance_texture, r, mu, d,
ray_r_mu_intersects_ground) *
GetTransmittanceToSun(
atmosphere, transmittance_texture, r_d, mu_s_d);
rayleigh = transmittance * GetProfileDensity(
mie = transmittance * GetProfileDensity(
}


Consider now the Sun light arriving at $\bp$ from a given direction $\bw$, after exactly one scattering event. The scattering event can occur at any point $\bq$ between $\bp$ and the intersection $\bi$ of the half-line $[\bp,\bw)$ with the nearest atmosphere boundary. Thus, the single scattered radiance at $\bp$ from direction $\bw$ is the integral of the single scattered radiance from $\bq$ to $\bp$ for all points $\bq$ between $\bp$ and $\bi$. To compute it, we first need the length $\Vert\bp\bi\Vert$:

Length DistanceToNearestAtmosphereBoundary(IN(AtmosphereParameters) atmosphere,
Length r, Number mu, bool ray_r_mu_intersects_ground) {
if (ray_r_mu_intersects_ground) {
return DistanceToBottomAtmosphereBoundary(atmosphere, r, mu);
} else {
return DistanceToTopAtmosphereBoundary(atmosphere, r, mu);
}
}


The single scattering integral can then be computed as follows (using the trapezoidal rule):

void ComputeSingleScattering(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
Length r, Number mu, Number mu_s, Number nu,
bool ray_r_mu_intersects_ground,
assert(mu >= -1.0 && mu <= 1.0);
assert(mu_s >= -1.0 && mu_s <= 1.0);
assert(nu >= -1.0 && nu <= 1.0);

// Number of intervals for the numerical integration.
const int SAMPLE_COUNT = 50;
// The integration step, i.e. the length of each integration interval.
Length dx =
DistanceToNearestAtmosphereBoundary(atmosphere, r, mu,
ray_r_mu_intersects_ground) / Number(SAMPLE_COUNT);
// Integration loop.
DimensionlessSpectrum rayleigh_sum = DimensionlessSpectrum(0.0);
DimensionlessSpectrum mie_sum = DimensionlessSpectrum(0.0);
for (int i = 0; i <= SAMPLE_COUNT; ++i) {
Length d_i = Number(i) * dx;
// The Rayleigh and Mie single scattering at the current sample point.
DimensionlessSpectrum rayleigh_i;
DimensionlessSpectrum mie_i;
ComputeSingleScatteringIntegrand(atmosphere, transmittance_texture,
r, mu, mu_s, nu, d_i, ray_r_mu_intersects_ground, rayleigh_i, mie_i);
// Sample weight (from the trapezoidal rule).
Number weight_i = (i == 0 || i == SAMPLE_COUNT) ? 0.5 : 1.0;
rayleigh_sum += rayleigh_i * weight_i;
mie_sum += mie_i * weight_i;
}
rayleigh = rayleigh_sum * dx * atmosphere.solar_irradiance *
atmosphere.rayleigh_scattering;
mie = mie_sum * dx * atmosphere.solar_irradiance * atmosphere.mie_scattering;
}


Note that we added the solar irradiance and the scattering coefficient terms that we omitted in ComputeSingleScatteringIntegrand, but not the phase function terms - they are added at render time for better angular precision. We provide them here for completeness:

InverseSolidAngle RayleighPhaseFunction(Number nu) {
InverseSolidAngle k = 3.0 / (16.0 * PI * sr);
return k * (1.0 + nu * nu);
}

InverseSolidAngle MiePhaseFunction(Number g, Number nu) {
InverseSolidAngle k = 3.0 / (8.0 * PI * sr) * (1.0 - g * g) / (2.0 + g * g);
return k * (1.0 + nu * nu) / pow(1.0 + g * g - 2.0 * g * nu, 1.5);
}


#### Precomputation

The ComputeSingleScattering function is quite costly to evaluate, and a lot of evaluations are needed to compute multiple scattering. We therefore want to precompute it in a texture, which requires a mapping from the 4 function parameters to texture coordinates. Assuming for now that we have 4D textures, we need to define a mapping from $(r,\mu,\mu_s,\nu)$ to texture coordinates $(u,v,w,z)$. The function below implements the mapping defined in our paper, with some small improvements (refer to the paper and to the above figures for the notations):

• the mapping for $\mu$ takes the minimal distance to the nearest atmosphere boundary into account, to map $\mu$ to the full $[0,1]$ interval (the original mapping was not covering the full $[0,1]$ interval).
• the mapping for $\mu_s$ is more generic than in the paper (the original mapping was using ad-hoc constants chosen for the Earth atmosphere case). It is based on the distance to the top atmosphere boundary (for the sun rays), as for the $\mu$ mapping, and uses only one ad-hoc (but configurable) parameter. Yet, as the original definition, it provides an increased sampling rate near the horizon.
vec4 GetScatteringTextureUvwzFromRMuMuSNu(IN(AtmosphereParameters) atmosphere,
Length r, Number mu, Number mu_s, Number nu,
bool ray_r_mu_intersects_ground) {
assert(mu >= -1.0 && mu <= 1.0);
assert(mu_s >= -1.0 && mu_s <= 1.0);
assert(nu >= -1.0 && nu <= 1.0);

// Distance to top atmosphere boundary for a horizontal ray at ground level.
// Distance to the horizon.
Length rho =
Number u_r = GetTextureCoordFromUnitRange(rho / H, SCATTERING_TEXTURE_R_SIZE);

// Discriminant of the quadratic equation for the intersections of the ray
// (r,mu) with the ground (see RayIntersectsGround).
Length r_mu = r * mu;
Area discriminant =
Number u_mu;
if (ray_r_mu_intersects_ground) {
// Distance to the ground for the ray (r,mu), and its minimum and maximum
// values over all mu - obtained for (r,-1) and (r,mu_horizon).
Length d = -r_mu - SafeSqrt(discriminant);
Length d_min = r - atmosphere.bottom_radius;
Length d_max = rho;
u_mu = 0.5 - 0.5 * GetTextureCoordFromUnitRange(d_max == d_min ? 0.0 :
(d - d_min) / (d_max - d_min), SCATTERING_TEXTURE_MU_SIZE / 2);
} else {
// Distance to the top atmosphere boundary for the ray (r,mu), and its
// minimum and maximum values over all mu - obtained for (r,1) and
// (r,mu_horizon).
Length d = -r_mu + SafeSqrt(discriminant + H * H);
Length d_min = atmosphere.top_radius - r;
Length d_max = rho + H;
u_mu = 0.5 + 0.5 * GetTextureCoordFromUnitRange(
(d - d_min) / (d_max - d_min), SCATTERING_TEXTURE_MU_SIZE / 2);
}

Length d = DistanceToTopAtmosphereBoundary(
Length d_max = H;
Number a = (d - d_min) / (d_max - d_min);
Length D = DistanceToTopAtmosphereBoundary(
Number A = (D - d_min) / (d_max - d_min);
// An ad-hoc function equal to 0 for mu_s = mu_s_min (because then d = D and
// thus a = A), equal to 1 for mu_s = 1 (because then d = d_min and thus
// a = 0), and with a large slope around mu_s = 0, to get more texture
// samples near the horizon.
Number u_mu_s = GetTextureCoordFromUnitRange(
max(1.0 - a / A, 0.0) / (1.0 + a), SCATTERING_TEXTURE_MU_S_SIZE);

Number u_nu = (nu + 1.0) / 2.0;
return vec4(u_nu, u_mu_s, u_mu, u_r);
}


The inverse mapping follows immediately:

void GetRMuMuSNuFromScatteringTextureUvwz(IN(AtmosphereParameters) atmosphere,
IN(vec4) uvwz, OUT(Length) r, OUT(Number) mu, OUT(Number) mu_s,
OUT(Number) nu, OUT(bool) ray_r_mu_intersects_ground) {
assert(uvwz.x >= 0.0 && uvwz.x <= 1.0);
assert(uvwz.y >= 0.0 && uvwz.y <= 1.0);
assert(uvwz.z >= 0.0 && uvwz.z <= 1.0);
assert(uvwz.w >= 0.0 && uvwz.w <= 1.0);

// Distance to top atmosphere boundary for a horizontal ray at ground level.
// Distance to the horizon.
Length rho =
H * GetUnitRangeFromTextureCoord(uvwz.w, SCATTERING_TEXTURE_R_SIZE);

if (uvwz.z < 0.5) {
// Distance to the ground for the ray (r,mu), and its minimum and maximum
// values over all mu - obtained for (r,-1) and (r,mu_horizon) - from which
// we can recover mu:
Length d_min = r - atmosphere.bottom_radius;
Length d_max = rho;
Length d = d_min + (d_max - d_min) * GetUnitRangeFromTextureCoord(
1.0 - 2.0 * uvwz.z, SCATTERING_TEXTURE_MU_SIZE / 2);
mu = d == 0.0 * m ? Number(-1.0) :
ClampCosine(-(rho * rho + d * d) / (2.0 * r * d));
ray_r_mu_intersects_ground = true;
} else {
// Distance to the top atmosphere boundary for the ray (r,mu), and its
// minimum and maximum values over all mu - obtained for (r,1) and
// (r,mu_horizon) - from which we can recover mu:
Length d_min = atmosphere.top_radius - r;
Length d_max = rho + H;
Length d = d_min + (d_max - d_min) * GetUnitRangeFromTextureCoord(
2.0 * uvwz.z - 1.0, SCATTERING_TEXTURE_MU_SIZE / 2);
mu = d == 0.0 * m ? Number(1.0) :
ClampCosine((H * H - rho * rho - d * d) / (2.0 * r * d));
ray_r_mu_intersects_ground = false;
}

Number x_mu_s =
GetUnitRangeFromTextureCoord(uvwz.y, SCATTERING_TEXTURE_MU_S_SIZE);
Length d_max = H;
Length D = DistanceToTopAtmosphereBoundary(
Number A = (D - d_min) / (d_max - d_min);
Number a = (A - x_mu_s * A) / (1.0 + x_mu_s * A);
Length d = d_min + min(a, A) * (d_max - d_min);
mu_s = d == 0.0 * m ? Number(1.0) :
ClampCosine((H * H - d * d) / (2.0 * atmosphere.bottom_radius * d));

nu = ClampCosine(uvwz.x * 2.0 - 1.0);
}


We assumed above that we have 4D textures, which is not the case in practice. We therefore need a further mapping, between 3D and 4D texture coordinates. The function below expands a 3D texel coordinate into a 4D texture coordinate, and then to $(r,\mu,\mu_s,\nu)$ parameters. It does so by "unpacking" two texel coordinates from the $x$ texel coordinate. Note also how we clamp the $\nu$ parameter at the end. This is because $\nu$ is not a fully independent variable: its range of values depends on $\mu$ and $\mu_s$ (this can be seen by computing $\mu$, $\mu_s$ and $\nu$ from the cartesian coordinates of the zenith, view and sun unit direction vectors), and the previous functions implicitely assume this (their assertions can break if this constraint is not respected).

void GetRMuMuSNuFromScatteringTextureFragCoord(
IN(AtmosphereParameters) atmosphere, IN(vec3) frag_coord,
OUT(Length) r, OUT(Number) mu, OUT(Number) mu_s, OUT(Number) nu,
OUT(bool) ray_r_mu_intersects_ground) {
const vec4 SCATTERING_TEXTURE_SIZE = vec4(
SCATTERING_TEXTURE_NU_SIZE - 1,
SCATTERING_TEXTURE_MU_S_SIZE,
SCATTERING_TEXTURE_MU_SIZE,
SCATTERING_TEXTURE_R_SIZE);
Number frag_coord_nu =
floor(frag_coord.x / Number(SCATTERING_TEXTURE_MU_S_SIZE));
Number frag_coord_mu_s =
mod(frag_coord.x, Number(SCATTERING_TEXTURE_MU_S_SIZE));
vec4 uvwz =
vec4(frag_coord_nu, frag_coord_mu_s, frag_coord.y, frag_coord.z) /
SCATTERING_TEXTURE_SIZE;
GetRMuMuSNuFromScatteringTextureUvwz(
atmosphere, uvwz, r, mu, mu_s, nu, ray_r_mu_intersects_ground);
// Clamp nu to its valid range of values, given mu and mu_s.
nu = clamp(nu, mu * mu_s - sqrt((1.0 - mu * mu) * (1.0 - mu_s * mu_s)),
mu * mu_s + sqrt((1.0 - mu * mu) * (1.0 - mu_s * mu_s)));
}


With this mapping, we can finally write a function to precompute a texel of the single scattering in a 3D texture:

void ComputeSingleScatteringTexture(IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture, IN(vec3) frag_coord,
Length r;
Number mu;
Number mu_s;
Number nu;
bool ray_r_mu_intersects_ground;
GetRMuMuSNuFromScatteringTextureFragCoord(atmosphere, frag_coord,
r, mu, mu_s, nu, ray_r_mu_intersects_ground);
ComputeSingleScattering(atmosphere, transmittance_texture,
r, mu, mu_s, nu, ray_r_mu_intersects_ground, rayleigh, mie);
}


#### Lookup

With the help of the above precomputed texture, we can now get the scattering between a point and the nearest atmosphere boundary with two texture lookups (we need two 3D texture lookups to emulate a single 4D texture lookup with quadrilinear interpolation; the 3D texture coordinates are computed using the inverse of the 3D-4D mapping defined in GetRMuMuSNuFromScatteringTextureFragCoord):

TEMPLATE(AbstractSpectrum)
AbstractSpectrum GetScattering(
IN(AtmosphereParameters) atmosphere,
IN(AbstractScatteringTexture TEMPLATE_ARGUMENT(AbstractSpectrum))
scattering_texture,
Length r, Number mu, Number mu_s, Number nu,
bool ray_r_mu_intersects_ground) {
vec4 uvwz = GetScatteringTextureUvwzFromRMuMuSNu(
atmosphere, r, mu, mu_s, nu, ray_r_mu_intersects_ground);
Number tex_coord_x = uvwz.x * Number(SCATTERING_TEXTURE_NU_SIZE - 1);
Number tex_x = floor(tex_coord_x);
Number lerp = tex_coord_x - tex_x;
vec3 uvw0 = vec3((tex_x + uvwz.y) / Number(SCATTERING_TEXTURE_NU_SIZE),
uvwz.z, uvwz.w);
vec3 uvw1 = vec3((tex_x + 1.0 + uvwz.y) / Number(SCATTERING_TEXTURE_NU_SIZE),
uvwz.z, uvwz.w);
return AbstractSpectrum(texture(scattering_texture, uvw0) * (1.0 - lerp) +
texture(scattering_texture, uvw1) * lerp);
}


Finally, we provide here a convenience lookup function which will be useful in the next section. This function returns either the single scattering, with the phase functions included, or the $n$-th order of scattering, with $n>1$. It assumes that, if scattering_order is strictly greater than 1, then multiple_scattering_texture corresponds to this scattering order, with both Rayleigh and Mie included, as well as all the phase function terms.

RadianceSpectrum GetScattering(
IN(AtmosphereParameters) atmosphere,
IN(ReducedScatteringTexture) single_rayleigh_scattering_texture,
IN(ReducedScatteringTexture) single_mie_scattering_texture,
IN(ScatteringTexture) multiple_scattering_texture,
Length r, Number mu, Number mu_s, Number nu,
bool ray_r_mu_intersects_ground,
int scattering_order) {
if (scattering_order == 1) {
atmosphere, single_rayleigh_scattering_texture, r, mu, mu_s, nu,
ray_r_mu_intersects_ground);
atmosphere, single_mie_scattering_texture, r, mu, mu_s, nu,
ray_r_mu_intersects_ground);
return rayleigh * RayleighPhaseFunction(nu) +
mie * MiePhaseFunction(atmosphere.mie_phase_function_g, nu);
} else {
return GetScattering(
atmosphere, multiple_scattering_texture, r, mu, mu_s, nu,
ray_r_mu_intersects_ground);
}
}


### Multiple scattering

The multiply scattered radiance is the light arriving from the Sun at some point in the atmosphere after two or more bounces (where a bounce is either a scattering event or a reflection from the ground). The following sections describe how we compute it, how we store it in a precomputed texture, and how we read it back.

Note that, as for single scattering, we exclude here the light paths whose last bounce is a reflection on the ground. The contribution from these paths is computed separately, at rendering time, in order to take the actual ground albedo into account (for intermediate reflections on the ground, which are precomputed, we use an average, uniform albedo).

#### Computation

Multiple scattering can be decomposed into the sum of double scattering, triple scattering, etc, where each term corresponds to the light arriving from the Sun at some point in the atmosphere after exactly 2, 3, etc bounces. Moreover, each term can be computed from the previous one. Indeed, the light arriving at some point $\bp$ from direction $\bw$ after $n$ bounces is an integral over all the possible points $\bq$ for the last bounce, which involves the light arriving at $\bq$ from any direction, after $n-1$ bounces.

This description shows that each scattering order requires a triple integral to be computed from the previous one (one integral over all the points $\bq$ on the segment from $\bp$ to the nearest atmosphere boundary in direction $\bw$, and a nested double integral over all directions at each point $\bq$). Therefore, if we wanted to compute each order "from scratch", we would need a triple integral for double scattering, a sextuple integral for triple scattering, etc. This would be clearly inefficient, because of all the redundant computations (the computations for order $n$ would basically redo all the computations for all previous orders, leading to quadratic complexity in the total number of orders). Instead, it is much more efficient to proceed as follows:

• precompute single scattering in a texture (as described above),
• for $n \ge 2$:
• precompute the $n$-th scattering in a texture, with a triple integral whose integrand uses lookups in the $(n-1)$-th scattering texture

This strategy avoids many redundant computations but does not eliminate all of them. Consider for instance the points $\bp$ and $\bp'$ in the figure below, and the computations which are necessary to compute the light arriving at these two points from direction $\bw$ after $n$ bounces. These computations involve, in particular, the evaluation of the radiance $L$ which is scattered at $\bq$ in direction $-\bw$, and coming from all directions after $n-1$ bounces:

Therefore, if we computed the n-th scattering with a triple integral as described above, we would compute $L$ redundantly (in fact, for all points $\bp$ between $\bq$ and the nearest atmosphere boundary in direction $-\bw$). To avoid this, and thus increase the efficiency of the multiple scattering computations, we refine the above algorithm as follows:

• precompute single scattering in a texture (as described above),
• for $n \ge 2$:
• for each point $\bq$ and direction $\bw$, precompute the light which is scattered at $\bq$ towards direction $-\bw$, coming from any direction after $n-1$ bounces (this involves only a double integral, whose integrand uses lookups in the $(n-1)$-th scattering texture),
• for each point $\bp$ and direction $\bw$, precompute the light coming from direction $\bw$ after $n$ bounces (this involves only a single integral, whose integrand uses lookups in the texture computed at the previous line)

To get a complete algorithm, we must now specify how we implement the two steps in the above loop. This is what we do in the rest of this section.

##### First step

The first step computes the radiance which is scattered at some point $\bq$ inside the atmosphere, towards some direction $-\bw$. Furthermore, we assume that this scattering event is the $n$-th bounce.

This radiance is the integral over all the possible incident directions $\bw_i$, of the product of

• the incident radiance $L_i$ arriving at $\bq$ from direction $\bw_i$ after $n-1$ bounces, which is the sum of:
• a term given by the precomputed scattering texture for the $(n-1)$-th order,
• if the ray $[\bq, \bw_i)$ intersects the ground at $\br$, the contribution from the light paths with $n-1$ bounces and whose last bounce is at $\br$, i.e. on the ground (these paths are excluded, by definition, from our precomputed textures, but we must take them into account here since the bounce on the ground is followed by a bounce at $\bq$). This contribution, in turn, is the product of:
• the transmittance between $\bq$ and $\br$,
• the (average) ground albedo,
• the Lambertian BRDF $1/\pi$,
• the irradiance received on the ground after $n-2$ bounces. We explain in the next section how we precompute it in a texture. For now, we assume that we can use the following function to retrieve this irradiance from a precomputed texture:
IrradianceSpectrum GetIrradiance(
IN(AtmosphereParameters) atmosphere,
Length r, Number mu_s);

• the scattering coefficient at $\bq$,
• the scattering phase function for the directions $\bw$ and $\bw_i$
This leads to the following implementation (where multiple_scattering_texture is supposed to contain the $(n-1)$-th order of scattering, if $n>2$, irradiance_texture is the irradiance received on the ground after $n-2$ bounces, and scattering_order is equal to $n$):
RadianceDensitySpectrum ComputeScatteringDensity(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
IN(ReducedScatteringTexture) single_rayleigh_scattering_texture,
IN(ReducedScatteringTexture) single_mie_scattering_texture,
IN(ScatteringTexture) multiple_scattering_texture,
Length r, Number mu, Number mu_s, Number nu, int scattering_order) {
assert(mu >= -1.0 && mu <= 1.0);
assert(mu_s >= -1.0 && mu_s <= 1.0);
assert(nu >= -1.0 && nu <= 1.0);
assert(scattering_order >= 2);

// Compute unit direction vectors for the zenith, the view direction omega and
// and the sun direction omega_s, such that the cosine of the view-zenith
// angle is mu, the cosine of the sun-zenith angle is mu_s, and the cosine of
// the view-sun angle is nu. The goal is to simplify computations below.
vec3 zenith_direction = vec3(0.0, 0.0, 1.0);
vec3 omega = vec3(sqrt(1.0 - mu * mu), 0.0, mu);
Number sun_dir_x = omega.x == 0.0 ? 0.0 : (nu - mu * mu_s) / omega.x;
Number sun_dir_y = sqrt(max(1.0 - sun_dir_x * sun_dir_x - mu_s * mu_s, 0.0));
vec3 omega_s = vec3(sun_dir_x, sun_dir_y, mu_s);

const int SAMPLE_COUNT = 16;
const Angle dphi = pi / Number(SAMPLE_COUNT);
const Angle dtheta = pi / Number(SAMPLE_COUNT);

// Nested loops for the integral over all the incident directions omega_i.
for (int l = 0; l < SAMPLE_COUNT; ++l) {
Angle theta = (Number(l) + 0.5) * dtheta;
Number cos_theta = cos(theta);
Number sin_theta = sin(theta);
bool ray_r_theta_intersects_ground =
RayIntersectsGround(atmosphere, r, cos_theta);

// The distance and transmittance to the ground only depend on theta, so we
// can compute them in the outer loop for efficiency.
Length distance_to_ground = 0.0 * m;
DimensionlessSpectrum transmittance_to_ground = DimensionlessSpectrum(0.0);
DimensionlessSpectrum ground_albedo = DimensionlessSpectrum(0.0);
if (ray_r_theta_intersects_ground) {
distance_to_ground =
DistanceToBottomAtmosphereBoundary(atmosphere, r, cos_theta);
transmittance_to_ground =
GetTransmittance(atmosphere, transmittance_texture, r, cos_theta,
distance_to_ground, true /* ray_intersects_ground */);
ground_albedo = atmosphere.ground_albedo;
}

for (int m = 0; m < 2 * SAMPLE_COUNT; ++m) {
Angle phi = (Number(m) + 0.5) * dphi;
vec3 omega_i =
vec3(cos(phi) * sin_theta, sin(phi) * sin_theta, cos_theta);
SolidAngle domega_i = (dtheta / rad) * (dphi / rad) * sin(theta) * sr;

// The radiance L_i arriving from direction omega_i after n-1 bounces is
// the sum of a term given by the precomputed scattering texture for the
// (n-1)-th order:
Number nu1 = dot(omega_s, omega_i);
single_rayleigh_scattering_texture, single_mie_scattering_texture,
multiple_scattering_texture, r, omega_i.z, mu_s, nu1,
ray_r_theta_intersects_ground, scattering_order - 1);

// and of the contribution from the light paths with n-1 bounces and whose
// last bounce is on the ground. This contribution is the product of the
// transmittance to the ground, the ground albedo, the ground BRDF, and
vec3 ground_normal =
normalize(zenith_direction * r + omega_i * distance_to_ground);
dot(ground_normal, omega_s));
ground_albedo * (1.0 / (PI * sr)) * ground_irradiance;

// The radiance finally scattered from direction omega_i towards direction
// -omega is the product of the incident radiance, the scattering
// coefficient, and the phase function for directions omega and omega_i
// (all this summed over all particle types, i.e. Rayleigh and Mie).
Number nu2 = dot(omega, omega_i);
Number rayleigh_density = GetProfileDensity(
Number mie_density = GetProfileDensity(
atmosphere.rayleigh_scattering * rayleigh_density *
RayleighPhaseFunction(nu2) +
atmosphere.mie_scattering * mie_density *
MiePhaseFunction(atmosphere.mie_phase_function_g, nu2)) *
domega_i;
}
}
return rayleigh_mie;
}

##### Second step

The second step to compute the $n$-th order of scattering is to compute for each point $\bp$ and direction $\bw$, the radiance coming from direction $\bw$ after $n$ bounces, using a texture precomputed with the previous function.

This radiance is the integral over all points $\bq$ between $\bp$ and the nearest atmosphere boundary in direction $\bw$ of the product of:

• a term given by a texture precomputed with the previous function, namely the radiance scattered at $\bq$ towards $\bp$, coming from any direction after $n-1$ bounces,
• the transmittance betweeen $\bp$ and $\bq$
Note that this excludes the light paths with $n$ bounces and whose last bounce is on the ground, on purpose. Indeed, we chose to exclude these paths from our precomputed textures so that we can compute them at render time instead, using the actual ground albedo.

The implementation for this second step is straightforward:

RadianceSpectrum ComputeMultipleScattering(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
IN(ScatteringDensityTexture) scattering_density_texture,
Length r, Number mu, Number mu_s, Number nu,
bool ray_r_mu_intersects_ground) {
assert(mu >= -1.0 && mu <= 1.0);
assert(mu_s >= -1.0 && mu_s <= 1.0);
assert(nu >= -1.0 && nu <= 1.0);

// Number of intervals for the numerical integration.
const int SAMPLE_COUNT = 50;
// The integration step, i.e. the length of each integration interval.
Length dx =
DistanceToNearestAtmosphereBoundary(
atmosphere, r, mu, ray_r_mu_intersects_ground) /
Number(SAMPLE_COUNT);
// Integration loop.
for (int i = 0; i <= SAMPLE_COUNT; ++i) {
Length d_i = Number(i) * dx;

// The r, mu and mu_s parameters at the current integration point (see the
// single scattering section for a detailed explanation).
Length r_i =
ClampRadius(atmosphere, sqrt(d_i * d_i + 2.0 * r * mu * d_i + r * r));
Number mu_i = ClampCosine((r * mu + d_i) / r_i);
Number mu_s_i = ClampCosine((r * mu_s + d_i * nu) / r_i);

// The Rayleigh and Mie multiple scattering at the current sample point.
GetScattering(
atmosphere, scattering_density_texture, r_i, mu_i, mu_s_i, nu,
ray_r_mu_intersects_ground) *
GetTransmittance(
atmosphere, transmittance_texture, r, mu, d_i,
ray_r_mu_intersects_ground) *
dx;
// Sample weight (from the trapezoidal rule).
Number weight_i = (i == 0 || i == SAMPLE_COUNT) ? 0.5 : 1.0;
rayleigh_mie_sum += rayleigh_mie_i * weight_i;
}
return rayleigh_mie_sum;
}


#### Precomputation

As explained in the overall algorithm to compute multiple scattering, we need to precompute each order of scattering in a texture to save computations while computing the next order. And, in order to store a function in a texture, we need a mapping from the function parameters to texture coordinates. Fortunately, all the orders of scattering depend on the same $(r,\mu,\mu_s,\nu)$ parameters as single scattering, so we can simple reuse the mappings defined for single scattering. This immediately leads to the following simple functions to precompute a texel of the textures for the first and second steps of each iteration over the number of bounces:

RadianceDensitySpectrum ComputeScatteringDensityTexture(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
IN(ReducedScatteringTexture) single_rayleigh_scattering_texture,
IN(ReducedScatteringTexture) single_mie_scattering_texture,
IN(ScatteringTexture) multiple_scattering_texture,
IN(vec3) frag_coord, int scattering_order) {
Length r;
Number mu;
Number mu_s;
Number nu;
bool ray_r_mu_intersects_ground;
GetRMuMuSNuFromScatteringTextureFragCoord(atmosphere, frag_coord,
r, mu, mu_s, nu, ray_r_mu_intersects_ground);
return ComputeScatteringDensity(atmosphere, transmittance_texture,
single_rayleigh_scattering_texture, single_mie_scattering_texture,
multiple_scattering_texture, irradiance_texture, r, mu, mu_s, nu,
scattering_order);
}

IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
IN(ScatteringDensityTexture) scattering_density_texture,
IN(vec3) frag_coord, OUT(Number) nu) {
Length r;
Number mu;
Number mu_s;
bool ray_r_mu_intersects_ground;
GetRMuMuSNuFromScatteringTextureFragCoord(atmosphere, frag_coord,
r, mu, mu_s, nu, ray_r_mu_intersects_ground);
return ComputeMultipleScattering(atmosphere, transmittance_texture,
scattering_density_texture, r, mu, mu_s, nu,
ray_r_mu_intersects_ground);
}


#### Lookup

Likewise, we can simply reuse the lookup function GetScattering implemented for single scattering to read a value from the precomputed textures for multiple scattering. In fact, this is what we did above in the ComputeScatteringDensity and ComputeMultipleScattering functions.

The ground irradiance is the Sun light received on the ground after $n \ge 0$ bounces (where a bounce is either a scattering event or a reflection on the ground). We need this for two purposes:

• while precomputing the $n$-th order of scattering, with $n \ge 2$, in order to compute the contribution of light paths whose $(n-1)$-th bounce is on the ground (which requires the ground irradiance after $n-2$ bounces - see the Multiple scattering section),
• at rendering time, to compute the contribution of light paths whose last bounce is on the ground (these paths are excluded, by definition, from our precomputed scattering textures)

In the first case we only need the ground irradiance for horizontal surfaces at the bottom of the atmosphere (during precomputations we assume a perfectly spherical ground with a uniform albedo). In the second case, however, we need the ground irradiance for any altitude and any surface normal, and we want to precompute it for efficiency. In fact, as described in our paper we precompute it only for horizontal surfaces, at any altitude (which requires only 2D textures, instead of 4D textures for the general case), and we use approximations for non-horizontal surfaces.

The following sections describe how we compute the ground irradiance, how we store it in a precomputed texture, and how we read it back.

#### Computation

The ground irradiance computation is different for the direct irradiance, i.e. the light received directly from the Sun, without any intermediate bounce, and for the indirect irradiance (at least one bounce). We start here with the direct irradiance.

The irradiance is the integral over an hemisphere of the incident radiance, times a cosine factor. For the direct ground irradiance, the incident radiance is the Sun radiance at the top of the atmosphere, times the transmittance through the atmosphere. And, since the Sun solid angle is small, we can approximate the transmittance with a constant, i.e. we can move it outside the irradiance integral, which can be performed over (the visible fraction of) the Sun disc rather than the hemisphere. Then the integral becomes equivalent to the ambient occlusion due to a sphere, also called a view factor, which is given in Radiative view factors (page 10). For a small solid angle, these complex equations can be simplified as follows:

IrradianceSpectrum ComputeDirectIrradiance(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
Length r, Number mu_s) {
assert(mu_s >= -1.0 && mu_s <= 1.0);

// Approximate average of the cosine factor mu_s over the visible fraction of
// the Sun disc.
Number average_cosine_factor =
mu_s < -alpha_s ? 0.0 : (mu_s > alpha_s ? mu_s :
(mu_s + alpha_s) * (mu_s + alpha_s) / (4.0 * alpha_s));

GetTransmittanceToTopAtmosphereBoundary(
atmosphere, transmittance_texture, r, mu_s) * average_cosine_factor;

}


For the indirect ground irradiance the integral over the hemisphere must be computed numerically. More precisely we need to compute the integral over all the directions $\bw$ of the hemisphere, of the product of:

• the radiance arriving from direction $\bw$ after $n$ bounces,
• the cosine factor, i.e. $\omega_z$
This leads to the following implementation (where multiple_scattering_texture is supposed to contain the $n$-th order of scattering, if $n>1$, and scattering_order is equal to $n$):
IrradianceSpectrum ComputeIndirectIrradiance(
IN(AtmosphereParameters) atmosphere,
IN(ReducedScatteringTexture) single_rayleigh_scattering_texture,
IN(ReducedScatteringTexture) single_mie_scattering_texture,
IN(ScatteringTexture) multiple_scattering_texture,
Length r, Number mu_s, int scattering_order) {
assert(mu_s >= -1.0 && mu_s <= 1.0);
assert(scattering_order >= 1);

const int SAMPLE_COUNT = 32;
const Angle dphi = pi / Number(SAMPLE_COUNT);
const Angle dtheta = pi / Number(SAMPLE_COUNT);

vec3 omega_s = vec3(sqrt(1.0 - mu_s * mu_s), 0.0, mu_s);
for (int j = 0; j < SAMPLE_COUNT / 2; ++j) {
Angle theta = (Number(j) + 0.5) * dtheta;
for (int i = 0; i < 2 * SAMPLE_COUNT; ++i) {
Angle phi = (Number(i) + 0.5) * dphi;
vec3 omega =
vec3(cos(phi) * sin(theta), sin(phi) * sin(theta), cos(theta));
SolidAngle domega = (dtheta / rad) * (dphi / rad) * sin(theta) * sr;

Number nu = dot(omega, omega_s);
result += GetScattering(atmosphere, single_rayleigh_scattering_texture,
single_mie_scattering_texture, multiple_scattering_texture,
r, omega.z, mu_s, nu, false /* ray_r_theta_intersects_ground */,
scattering_order) *
omega.z * domega;
}
}
return result;
}


#### Precomputation

In order to precompute the ground irradiance in a texture we need a mapping from the ground irradiance parameters to texture coordinates. Since we precompute the ground irradiance only for horizontal surfaces, this irradiance depends only on $r$ and $\mu_s$, so we need a mapping from $(r,\mu_s)$ to $(u,v)$ texture coordinates. The simplest, affine mapping is sufficient here, because the ground irradiance function is very smooth:

vec2 GetIrradianceTextureUvFromRMuS(IN(AtmosphereParameters) atmosphere,
Length r, Number mu_s) {
assert(mu_s >= -1.0 && mu_s <= 1.0);
Number x_r = (r - atmosphere.bottom_radius) /
Number x_mu_s = mu_s * 0.5 + 0.5;
}


The inverse mapping follows immediately:

void GetRMuSFromIrradianceTextureUv(IN(AtmosphereParameters) atmosphere,
IN(vec2) uv, OUT(Length) r, OUT(Number) mu_s) {
assert(uv.x >= 0.0 && uv.x <= 1.0);
assert(uv.y >= 0.0 && uv.y <= 1.0);
mu_s = ClampCosine(2.0 * x_mu_s - 1.0);
}


It is now easy to define a fragment shader function to precompute a texel of the ground irradiance texture, for the direct irradiance:

const vec2 IRRADIANCE_TEXTURE_SIZE =

IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
IN(vec2) frag_coord) {
Length r;
Number mu_s;
atmosphere, frag_coord / IRRADIANCE_TEXTURE_SIZE, r, mu_s);
}


and the indirect one:

IrradianceSpectrum ComputeIndirectIrradianceTexture(
IN(AtmosphereParameters) atmosphere,
IN(ReducedScatteringTexture) single_rayleigh_scattering_texture,
IN(ReducedScatteringTexture) single_mie_scattering_texture,
IN(ScatteringTexture) multiple_scattering_texture,
IN(vec2) frag_coord, int scattering_order) {
Length r;
Number mu_s;
atmosphere, frag_coord / IRRADIANCE_TEXTURE_SIZE, r, mu_s);
single_rayleigh_scattering_texture, single_mie_scattering_texture,
multiple_scattering_texture, r, mu_s, scattering_order);
}


#### Lookup

Thanks to these precomputed textures, we can now get the ground irradiance with a single texture lookup:

IrradianceSpectrum GetIrradiance(
IN(AtmosphereParameters) atmosphere,
Length r, Number mu_s) {
vec2 uv = GetIrradianceTextureUvFromRMuS(atmosphere, r, mu_s);
}


### Rendering

Here we assume that the transmittance, scattering and irradiance textures have been precomputed, and we provide functions using them to compute the sky color, the aerial perspective, and the ground radiance.

More precisely, we assume that the single Rayleigh scattering, without its phase function term, plus the multiple scattering terms (divided by the Rayleigh phase function for dimensional homogeneity) are stored in a scattering_texture. We also assume that the single Mie scattering is stored, without its phase function term:

• either separately, in a single_mie_scattering_texture (this option was not provided our original implementation),
• or, if the COMBINED_SCATTERING_TEXTURES preprocessor macro is defined, in the scattering_texture. In this case, which is only available with a GLSL compiler, Rayleigh and multiple scattering are stored in the RGB channels, and the red component of the single Mie scattering is stored in the alpha channel).

In the second case, the green and blue components of the single Mie scattering are extrapolated as described in our paper, with the following function:

#ifdef COMBINED_SCATTERING_TEXTURES
vec3 GetExtrapolatedSingleMieScattering(
IN(AtmosphereParameters) atmosphere, IN(vec4) scattering) {
// Algebraically this can never be negative, but rounding errors can produce
// that effect for sufficiently short view rays.
if (scattering.r <= 0.0) {
return vec3(0.0);
}
return scattering.rgb * scattering.a / scattering.r *
(atmosphere.rayleigh_scattering.r / atmosphere.mie_scattering.r) *
(atmosphere.mie_scattering / atmosphere.rayleigh_scattering);
}
#endif


We can then retrieve all the scattering components (Rayleigh + multiple scattering on one side, and single Mie scattering on the other side) with the following function, based on GetScattering (we duplicate some code here, instead of using two calls to GetScattering, to make sure that the texture coordinates computation is shared between the lookups in scattering_texture and single_mie_scattering_texture):

IrradianceSpectrum GetCombinedScattering(
IN(AtmosphereParameters) atmosphere,
IN(ReducedScatteringTexture) scattering_texture,
IN(ReducedScatteringTexture) single_mie_scattering_texture,
Length r, Number mu, Number mu_s, Number nu,
bool ray_r_mu_intersects_ground,
vec4 uvwz = GetScatteringTextureUvwzFromRMuMuSNu(
atmosphere, r, mu, mu_s, nu, ray_r_mu_intersects_ground);
Number tex_coord_x = uvwz.x * Number(SCATTERING_TEXTURE_NU_SIZE - 1);
Number tex_x = floor(tex_coord_x);
Number lerp = tex_coord_x - tex_x;
vec3 uvw0 = vec3((tex_x + uvwz.y) / Number(SCATTERING_TEXTURE_NU_SIZE),
uvwz.z, uvwz.w);
vec3 uvw1 = vec3((tex_x + 1.0 + uvwz.y) / Number(SCATTERING_TEXTURE_NU_SIZE),
uvwz.z, uvwz.w);
#ifdef COMBINED_SCATTERING_TEXTURES
vec4 combined_scattering =
texture(scattering_texture, uvw0) * (1.0 - lerp) +
texture(scattering_texture, uvw1) * lerp;
single_mie_scattering =
GetExtrapolatedSingleMieScattering(atmosphere, combined_scattering);
#else
texture(scattering_texture, uvw0) * (1.0 - lerp) +
texture(scattering_texture, uvw1) * lerp);
texture(single_mie_scattering_texture, uvw0) * (1.0 - lerp) +
texture(single_mie_scattering_texture, uvw1) * lerp);
#endif
return scattering;
}


#### Sky

To render the sky we simply need to display the sky radiance, which we can get with a lookup in the precomputed scattering texture(s), multiplied by the phase function terms that were omitted during precomputation. We can also return the transmittance of the atmosphere (which we can get with a single lookup in the precomputed transmittance texture), which is needed to correctly render the objects in space (such as the Sun and the Moon). This leads to the following function, where most of the computations are used to correctly handle the case of viewers outside the atmosphere, and the case of light shafts:

RadianceSpectrum GetSkyRadiance(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
IN(ReducedScatteringTexture) scattering_texture,
IN(ReducedScatteringTexture) single_mie_scattering_texture,
Position camera, IN(Direction) view_ray, Length shadow_length,
IN(Direction) sun_direction, OUT(DimensionlessSpectrum) transmittance) {
// Compute the distance to the top atmosphere boundary along the view ray,
// assuming the viewer is in space (or NaN if the view ray does not intersect
// the atmosphere).
Length r = length(camera);
Length rmu = dot(camera, view_ray);
Length distance_to_top_atmosphere_boundary = -rmu -
// If the viewer is in space and the view ray intersects the atmosphere, move
// the viewer to the top atmosphere boundary (along the view ray):
if (distance_to_top_atmosphere_boundary > 0.0 * m) {
camera = camera + view_ray * distance_to_top_atmosphere_boundary;
rmu += distance_to_top_atmosphere_boundary;
} else if (r > atmosphere.top_radius) {
// If the view ray does not intersect the atmosphere, simply return 0.
transmittance = DimensionlessSpectrum(1.0);
}
// Compute the r, mu, mu_s and nu parameters needed for the texture lookups.
Number mu = rmu / r;
Number mu_s = dot(camera, sun_direction) / r;
Number nu = dot(view_ray, sun_direction);
bool ray_r_mu_intersects_ground = RayIntersectsGround(atmosphere, r, mu);

transmittance = ray_r_mu_intersects_ground ? DimensionlessSpectrum(0.0) :
GetTransmittanceToTopAtmosphereBoundary(
atmosphere, transmittance_texture, r, mu);
if (shadow_length == 0.0 * m) {
scattering = GetCombinedScattering(
atmosphere, scattering_texture, single_mie_scattering_texture,
r, mu, mu_s, nu, ray_r_mu_intersects_ground,
single_mie_scattering);
} else {
// Case of light shafts (shadow_length is the total length noted l in our
// paper): we omit the scattering between the camera and the point at
// distance l, by implementing Eq. (18) of the paper (shadow_transmittance
// is the T(x,x_s) term, scattering is the S|x_s=x+lv term).
Length r_p =
ClampRadius(atmosphere, sqrt(d * d + 2.0 * r * mu * d + r * r));
Number mu_p = (r * mu + d) / r_p;
Number mu_s_p = (r * mu_s + d * nu) / r_p;

scattering = GetCombinedScattering(
atmosphere, scattering_texture, single_mie_scattering_texture,
r_p, mu_p, mu_s_p, nu, ray_r_mu_intersects_ground,
single_mie_scattering);
GetTransmittance(atmosphere, transmittance_texture,
}
return scattering * RayleighPhaseFunction(nu) + single_mie_scattering *
MiePhaseFunction(atmosphere.mie_phase_function_g, nu);
}


#### Aerial perspective

To render the aerial perspective we need the transmittance and the scattering between two points (i.e. between the viewer and a point on the ground, which can at an arbibrary altitude). We already have a function to compute the transmittance between two points (using 2 lookups in a texture which only contains the transmittance to the top of the atmosphere), but we don't have one for the scattering between 2 points. Hopefully, the scattering between 2 points can be computed from two lookups in a texture which contains the scattering to the nearest atmosphere boundary, as for the transmittance (except that here the two lookup results must be subtracted, instead of divided). This is what we implement in the following function (the initial computations are used to correctly handle the case of viewers outside the atmosphere):

RadianceSpectrum GetSkyRadianceToPoint(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
IN(ReducedScatteringTexture) scattering_texture,
IN(ReducedScatteringTexture) single_mie_scattering_texture,
Position camera, IN(Position) point, Length shadow_length,
IN(Direction) sun_direction, OUT(DimensionlessSpectrum) transmittance) {
// Compute the distance to the top atmosphere boundary along the view ray,
// assuming the viewer is in space (or NaN if the view ray does not intersect
// the atmosphere).
Direction view_ray = normalize(point - camera);
Length r = length(camera);
Length rmu = dot(camera, view_ray);
Length distance_to_top_atmosphere_boundary = -rmu -
// If the viewer is in space and the view ray intersects the atmosphere, move
// the viewer to the top atmosphere boundary (along the view ray):
if (distance_to_top_atmosphere_boundary > 0.0 * m) {
camera = camera + view_ray * distance_to_top_atmosphere_boundary;
rmu += distance_to_top_atmosphere_boundary;
}

// Compute the r, mu, mu_s and nu parameters for the first texture lookup.
Number mu = rmu / r;
Number mu_s = dot(camera, sun_direction) / r;
Number nu = dot(view_ray, sun_direction);
Length d = length(point - camera);
bool ray_r_mu_intersects_ground = RayIntersectsGround(atmosphere, r, mu);

transmittance = GetTransmittance(atmosphere, transmittance_texture,
r, mu, d, ray_r_mu_intersects_ground);

atmosphere, scattering_texture, single_mie_scattering_texture,
r, mu, mu_s, nu, ray_r_mu_intersects_ground,
single_mie_scattering);

// Compute the r, mu, mu_s and nu parameters for the second texture lookup.
// If shadow_length is not 0 (case of light shafts), we want to ignore the
// scattering along the last shadow_length meters of the view ray, which we
// do by subtracting shadow_length from d (this way scattering_p is equal to
// the S|x_s=x_0-lv term in Eq. (17) of our paper).
d = max(d - shadow_length, 0.0 * m);
Length r_p = ClampRadius(atmosphere, sqrt(d * d + 2.0 * r * mu * d + r * r));
Number mu_p = (r * mu + d) / r_p;
Number mu_s_p = (r * mu_s + d * nu) / r_p;

atmosphere, scattering_texture, single_mie_scattering_texture,
r_p, mu_p, mu_s_p, nu, ray_r_mu_intersects_ground,
single_mie_scattering_p);

// Combine the lookup results to get the scattering between camera and point.
if (shadow_length > 0.0 * m) {
// This is the T(x,x_s) term in Eq. (17) of our paper, for light shafts.
r, mu, d, ray_r_mu_intersects_ground);
}
scattering = scattering - shadow_transmittance * scattering_p;
single_mie_scattering =
#ifdef COMBINED_SCATTERING_TEXTURES
single_mie_scattering = GetExtrapolatedSingleMieScattering(
atmosphere, vec4(scattering, single_mie_scattering.r));
#endif

// Hack to avoid rendering artifacts when the sun is below the horizon.
single_mie_scattering = single_mie_scattering *
smoothstep(Number(0.0), Number(0.01), mu_s);

return scattering * RayleighPhaseFunction(nu) + single_mie_scattering *
MiePhaseFunction(atmosphere.mie_phase_function_g, nu);
}


#### Ground

To render the ground we need the irradiance received on the ground after 0 or more bounce(s) in the atmosphere or on the ground. The direct irradiance can be computed with a lookup in the transmittance texture, via GetTransmittanceToSun, while the indirect irradiance is given by a lookup in the precomputed irradiance texture (this texture only contains the irradiance for horizontal surfaces; we use the approximation defined in our paper for the other cases). The function below returns the direct and indirect irradiances separately:

IrradianceSpectrum GetSunAndSkyIrradiance(
IN(AtmosphereParameters) atmosphere,
IN(TransmittanceTexture) transmittance_texture,
IN(Position) point, IN(Direction) normal, IN(Direction) sun_direction,
Length r = length(point);
Number mu_s = dot(point, sun_direction) / r;

// Indirect irradiance (approximated if the surface is not horizontal).