In this section, we present background material on refractive media and how it affects light and sound propagation. The two most prominent refractive media in outdoor scenes: the atmosphere and the ocean, are often studied separately. They are in fact tightly connected by heat flow and general circulation of the water component (Dowling, 2013). We hereby focus our discussion on atmospheric properties, but we would like to point out that media properties and propagation in the ocean are analogous.
2.1.1 Refractive media properties
A non-linear media profile can be described by spatially-varying propagation speed c(x), or equivalently by index of refractionn(x) =c0/c(x), for each locationx, wherec0is the reference propagation speed. The refractive index or the propagation speed is in turn determined by a set of properties of the media, which will be discussed in details in the following subsections.
Properties affecting light refraction
Light propagation paths are governed by the spatial profile of refractive index, which can in turn be computed from atmospheric density and wavelength of the light. Starting from an atmospheric profile for a spatial locationx, density is computed from temperature and pressure using the Perfect Gas Law:
ρ(x) = P(x)M
RT(x), (2.1)
whereT is temperature,P is pressure,M andRare constants with typical values of 28.96×10−3kg/moland 8.3145J/mol·K respectively. The Cauchy’s formula (Born and Wolf, 1999) relates index of refraction with wavelength as: n(λ) =a·(1 +λb2) + 1, whereaand bare constants with typical values ofa= 2879×10
−5
andb= 567×10−5for air. The Gladstone-Dale Law (Dale and Gladstone, 1858) then representsn(λ,x) as a function of both densityρ(x) andn(λ): n(x, λ) =ρ(x)(n(λ)−1) + 1.
Properties affecting sound refraction
The atmospheric speed of sound is governed by the temperature as
c=pγRdTv, (2.2)
where γ = cp/cv is the ratio of the specific heats, Rd is the gas constant of dry air, Tv is the virtual
temperature considering humidity, and can typically be approximated by the absolute temperatureT when the humidity effects are ignored.
2.1.2 Common media profiles
In this section, we first introduce a few simple profiles with analytic ray solutions, two of which we have adopted as the foundation of our ray curve tracer. The known analytic solutions are presented here while their detailed derivations are included in an Appendix to this chapter for completeness of presentation. We then introduce models of general non-linear media that corresponds to physical reality.
Profiles with analytic ray solutions
In ray tracing for wave propagation, rays are defined as normal to the wavefront. The equation for ray trajectories is derived (also see Ch. 2.3.2) from the wave equation as:
d ds 1 c(x) dx ds =− 1 c(x)2∇c(x), (2.3a) d ds n(x)dx ds =∇n(x), (2.3b)
x={x, y, z}is the Cartesian coordinates andsis the arc-length along the ray.
The analytic ray trajectories are known for a set of profiles with constant media gradient, and we give the trajectories in a local coordinate system aligned with the gradient direction. If we place the origin of the coordinate system at the ray originx, and take the media gradient direction as thez-axis, the ray trajectory is a plane curve that lies in the plane formed by thez-axis and the initial ray directiond, i.e. theray plane. We then take the direction perpendicular to the z-axis as the r-axis within the ray plane. Figure 3.1 plots the analytic ray curves for the following profiles (see Appendix 2.4.1 for detailed derivations):
dandraxis, the ray trajectory inr-zcoordinates is derived from Equation (2.3a) to be: r(z) = p 1−ξ02 0c20− q 1−ξ02 0 (c0+αz) 2 ξ00α , (2.4)
which is a circular curve in the ray plane. n2-linear: n2(z) = n2
0+αz, α = k∇n2k, n0 is n at ray origin. We establish a similar coordinate system with origin at x, and take the direction of∇n2 as z-axis. Let ξ0
0 =n0cosθ0, where θ0 is the angle betweendandraxis, the ray trajectory is:
r(z) =2ξ 0 0 α q −ξ002+n2 0+αz− q −ξ002+n2 0 , (2.5)
which is a parabolic curve in the ray plane.
n-linear: The analytic ray curve for constant∇nwas used in (Cao et al., 2010), although unlike the previous two ray curves, it does not have an analytic solution for intersection tests with planar surfaces. There are also analytic solutions for the profiles that produce superior and inferior mirages (Khular et al., 1977). The two profiles are also described in (Cao et al., 2010), and we use their analytic solutions to validate our ray tracer:
Inferior mirage (V-IM), with the squared refractive index: n2(z) =µ2
0+µ21(1−exp(−βz)), Superior mirage (V-SM), with the squared refractive index: n2(z) = µ20+µ21exp(−βz), with
constants µ0= 1.000233, µ1= 0.4584, β= 2.303.
Realistic profiles
Within the surface layer close to the ground, a common wind profile based on the Monin-Obukhov similarity theory (Monin and Obukhov, 1954) computes the mean wind velocity as following a logarithmic law depending on the height. The same theory prescribes wind profiles for altitude beyond the surface layer with parameters representing stable and unstable atmospheric conditions (Panofsky and Dutton, 1984a).
A standard profile of atmospheric temperature and pressure is available with the 1976 USA Standard Atmosphere (USGPC, 1976). It can be de-standardized with the following model for localized heat sources: Hot spot (A-HS)is computed by Eq. 2.2 with combined temperature from (USGPC, 1976) and Eq.
2.6,
T =T0+ (Ts−T0)exp(−d/d0), (2.6)
where T0= 273K,Ts is the temperature at the hot spot,dis the distance to the hot spot, and d0 is the dropoff length.
The above profile requires detailed measured data for a particular location, time, and atmospheric con- dition. Alternatively, we can adopt a widely-used empirical models of the atmosphere (Salomons, 2001a) that gives the sound speed directly. The sound speed is modeled with a stratified component cstr and a
fluctuation component cf lu, so that c=cstr+cf lu. The stratified component follows a logarithmic profile
of the altitudez: cstr(z) =c0+bln z zg + 1 ), (2.7)
where c0 is the sound speed at the ground, and zg is the roughness length of the ground surface. Different
values of the parameterblead to different profiles:
Stratified profile, upward (A-LU) or downward (A-LD) refractive,computed by Eq. 2.7 with n0 = 1,c0= 340 m/s, andzg = 1 m. We take b= 1m/sfor A-LD andb=−1m/sfor A-LU.
The fluctuation component models the random atmospheric temperature and wind speed turbulence:
cf lu(x) =
X
i
G(~ki) cos(~ki·x+ϕi), (2.8)
where~kiis the wave vector describing thespatial frequency of the fluctuation,ϕi is a random angle∈[0,2π],
G(~ki) is a normalization factor, and we have:
Stratified-plus-fluctuation (A-LU+F, A-LD+F)A-LU or A-LD combined with Equation 2.8. For sound propagation, the wind profile plays a role that is as important as the temperature(L’Esp´erance et al., 1993; Lamancusa and Daroux, 1993a), and the wind profile is significantly modified above undulating terrains. For example, Jackson and Hunt (Jackson and Hunt, 1975) derived a closed-form wind profile for a hill of the shape: f(xL) =1+(1x
L2)
, wherexis the horizontal distance from the apex of the hill,Lis the radius of the base of the hill. According to the Monin-Obukhov similarity theory (Monin and Obukhov, 1954), the mean wind velocity follows the logarithmic law with heightz: u(z) = u∗
K ln z
zg, whereK is the von-Karmann
constant,zg is the aerodynamic roughness length, andu∗is the friction velocity (Businger et al., 1971; Oke,
velocityu(z), is given as: ∆u=u0(z=L) h L ln(zL 0) ln2(zl 0) (1−( x L) 2 1 + (xL)2ln( ∆z z0) −( 2(x/L) (1 + (x/L)2)2( ∆z−z0 l ) ln( ∆z z0 )), (2.9)
whereδzis the distance above the hill,lis the thickness of the hill’s influence region, in which the flow above the ground is perturbed, and we have:
Wind over hill (A-UW for upwind, A-DW for downwind) u(z) + ∆u is combined with temperature-induced sound speed profile based on the 1976 USA Standard Atmosphere (USGPC, 1976).