1
Design of an Infrared Ship Signature Simulation
Software for General Emissivity Profiles
a Royal
Fabian D. Lapierrea , Jean-Paul Marcela , Marc Acheroya
Military Academy, Electrical Engineering Department (SIC), 30, Avenue de la Renaissance,
B-1000 Brussels, Belgium
Email: [email protected], Tel: +32/2/7376661, Fax: +32/2/7376472
Abstract— Design of modern war ships involves integrating
stealth technologies that aim at minimizing a vessel’s transmitted
and reflected energies. Increasing the ships’ survivability requires
the reduction of their signatures (radar, infrared, etc). The first
step towards the design of signature reduction techniques is to
design a signatures simulation software. In this paper, we propose
an infrared signature simulation software applied to military
ships. As opposed to existing softwares assuming the emissivity
to be a constant scalar value for reducing computational time,
the software proposed here integrates the dependence of the
emissivity upon the surface temperature, the wavelength, and
the elevation angle without requiring significant additional computational time. Example simulated signatures are presented and
the computational efficiency of our new algorithm is discussed.
I. I NTRODUCTION
Design of modern war ships nowadays involves integrating
stealth technologies. These technologies aim at minimizing a
vessel’s transmitted and reflected energies to deny an opponent
the opportunity to locate, identify, track, and attack it. There
are several kinds of energies and thus several kinds of signatures that have to be minimized. Reduction of these signatures
thus enhances the ships’ survavibility, because low signatures
make it more difficult to be detected, identified, and classified.
It also increases the effectiveness of own ship’s decoys and
(E)CM capabilities against incoming enemy targets.
Stealth technology makes full use of aggressive architecture, controlled reflection and absorption, colour variation,
machinery isolation, shielding, and electronic countermeasures
(jamming or false imaging) to mask a vessel’s existence. The
aim of any stealth technology is to decrease a given signature,
without increasing another one. The Visby corvette [1] is the
first warship fully equiped with stealth technology.
The project we are involved in focusses on the design of
infrared (IR) signature reduction techniques applied to military
ships, for which the conception begins with the design of a
software modelizing the IR signature of targets. Simulation
and modeling permit the scientific investigation of simulation
parameters that are not even available through experiment.
More and more emphasis is thus placed on simulation.
A lot of IR simulation softwares have been developed. An
example is SHIPIR developed by the Canadian Department
of National Defense and adopted by the U.S. Navy and by
NATO as the standard ship IR signature model. This model is
described and validated in [2], [3]. The major disadvantage of
such softwares is that they assume that the emissivity and the
reflectivity are constant to find the surface temperatures.
The software proposed in this paper assumes these parameters to have complex behaviour with respect to the surface
temperature, the direction of arrival of the IR wave, and
the wavelength. Moreover, the emissivity is assumed to be a
separable function of each of these variables. This paper describes the first version of the software. We temporarily neglect
the multiple reflections, the conduction, and the reflections
from the sea. Indeed, these effects will be included in further
versions of the software. This paper contains two original
contributions. First, we derive the heat transfer equation for
general parametric emissivity. IR literature typically presents
models for constant emissivity. Second, this model being an
integral equation for each point of the object’s surface, we
present a method to efficiently solve this equation, which is
the most important contribution of this paper.
In Sections II and III, we respectively present the general
architecture of the software and the meshing algorithm we use.
Section IV is devoted to the derivation of the heat transfer
equation for a particular facet and for a general parametric
emissivity. Section V describes the new method used to
efficiently solve the resulting integral equations to recover
the surface temperatures, while Section VI presents the way
we compute the atmospheric attenuation of the IR waves. In
Section VII, the method used to compute the voltage received
by each pixel of the camera is explored, and in Section IX,
some simulation results are shown and the complexity of the
proposed algorithm is analysed. Finally, Section IX concludes.
II. G ENERAL SYSTEM ARCHITECTURE
Obtaining the correct power levels at the input of the IR
camera, the correct voltages at the output of the pixel elements,
and finally the correct pixel intensities, are important and
delicate tasks. The basic image formation process is dominated
by the problem of computing the radiance leaving each facet.
Atmosphere attenuation is then taken into account by scaling
this “input” radiance appropriately. Below, we focus on getting
the correct power levels at each pixel. The analysis to follow
consists of four main phases: (1) meshing, (2) calculation
of facet radiance, (3) calculation of radiance attenuation by
atmosphere, and (4) calculation of image intensities.
The most complex part of the system is the computation
of the facets surface temperatures. The general processing
2
steps involved in this computation are depicted in Fig. 1 at
time tn . We assume that the surface temperatures at tn−1 are
available at tn . At initial time t0 , uniform surface temperatures are assumed. Four processing steps can be computed
in parallel: (1) the basic heat transfer equation including sun
and atmospheric effects, (2) the conduction process, (3) the
computation of the effect of multiple reflections that requires
a model of the sea surface, and (4) the computation of the
convection coefficients. Once all these processing steps are
completed, the output equations are fed into the processing
step dedicated to the computation of the surface temperatures
at tn . This paper is mainly devoted to the description of
the “sun and atmosphere” and in the “surface temperatures
computation” processing steps. We also quickly consider the
convection process.
Mesh
Sun and
atmosphere
Time tn
Conduction
Sea model
Convection
Surface
temperatures
computation
Multiple
reflections
Surface temperatures
at time tn−1
750 facets
Fig. 2.
Example coarse mesh for a simple ship geometry.
A. Conservation of energy (heat flux balance)
To find the surface temperatures, we apply the principle of
energy conservation to each facet. Initially, we deal with a
basic heat quantity which is the heat flux generically denoted
by Φ (in Watts). The law of energy conservation says that the
heat flux Φabs absorbed by the surface of the facet minus the
heat flux Φlost lost to the environment and minus the (spatial)
variation of internal energy U̇ , must be equal to the temporal
variation of the facet energy, i.e.,
∂T
= Φabs − Φlost − U̇ ,
(1)
∂t
where a is a constant. Below, we assume that the problem is
stationary. We thus have
a
Surface temperatures at time tn
Fig. 1.
facets.
Processing steps for the computation of the radiance leaving the
III. M ESHING
The first step towards the radiance computation at each
point of the object’s surface is the meshing. There are two
kinds of meshing algorithms [4], [5]. A structured mesh can
be recognized by all interior nodes of the mesh having an
equal number of adjacent elements, typically quad meshes.
Unstructured mesh generation, on the other hand, relaxes the
node valence requirement, allowing any number of elements
to meet at a single node, typically triangle meshes.
We decide to use an unstructured meshing algorithm with
triangular elements, since it is easier to generate. The mesh is
generated using GMSH. At this point, the surface of the object
is divided in M triangular facets. M typically goes from 104 to
106 . Computational time and memory management have then
to be taken into account. An example of a coarse mesh for a
simple ship geometry is shown in Fig. 2.
IV. D ERIVATION OF THE HEAT TRANSFER EQUATION
The facet radiance is the power per unit area of facet and per
unit of solid angle leaving the facet in the direction of interest,
typically that of the camera. The determination of the surface
temperature Ts of a facet is a key step in determining its
radiance. Once we have Ts , it is straightforward to compute the
radiance. Sections IV and V are dedicated to the computation
of the radiance leaving each facet.
(2)
Φabs = Φlost + U̇ .
As shown in Fig. 3, when a heat flux Φinc is incident on a
facet, a part of the flux Φabs is absorbed by the surface, another
part Φref l is reflected by the surface and does not contribute
to the computation of the facet’s surface temperature. The
remaining flux Φtr is transmitted through the facet. We assume
the materials to be opaque to IR radiations, i.e., Φtr = 0.
nm
Φinc
Φref l
Φlost
Φcond
Φabs
facet m
Φtr
Fig. 3. Heat flux involved in the heat transfer equation of a particular facet.
Φabs is composed of (1) the solar heat flux Φsol that comes
from the sun or by a diffusion process, (2) the heat flux Φsky
3
modeling the atmospheric radiation, which is basically temperature driven, (3) the heat flux Φmr coming from multiple
reflections on other facets, and (4) the heat flux Φsea coming
from the reflection of energy on the sea surface which is not
considered in this paper. We thus have
Φabs = Φsol + Φsky + Φmr .
(3)
Φlost is the heat flux lost to the environment and is mainly
composed of (1) the heat flux Φrad lost to the environment by
radiation, due to the non-zero emissivity of the material and
(2) the heat flux Φconv lost to the environment by convection.
We thus have
Φlost = Φrad + Φconv .
(4)
The variation of internal heat flux U̇ is due to the conduction
of heat into the neighboring facets from the facet of interest,
i.e., U̇ = Φcond .
Different notations are used for flux densities depending
upon whether the flux arrives on the object or departs from
it. An “arriving flux density” is an irradiance, denoted by E
(Watts/m2 ), whereas a “departing flux density” is an exitance,
denoted by M (Watts/m2 ). E and M can be placed “on the
same level” in an equation since they represent exactly the
same physical quantity: the only difference is the direction of
travel with respect to an object (“to” for E and “from” for
M ). Equation (2) thus becomes
Eabs = Mlost + Mcond ,
(5)
where
Eabs
Mlost
Mcond
= Esol + Esky + Emr
= Mrad + Mconv
(6)
(7)
= ”U̇ per unit area".
(8)
We can already point out that the radiance we are ultimately
interested in is closely related to Mrad . Below, we give a
mathematical expression for each of these terms.
B. The solar bean and diffuse irradiances
The solar extraterrestrial radiation that is not backscattered
to space when interacting with the earth’s atmosphere reaches
the ground in two different ways. The radiation, selectively
attenuated by the atmosphere, which is not scattered and
reaches the surface directly is beam (direct) irradiance Eb,sol .
The scattered radiation that reaches the ground is diffuse
irradiance Ed,sol . Below, we assume clear-sky radiation.
e
1) The beam irradiance Eb,sol : The radiance Msun
emitted
by 1 m2 of solar surface is typically modeled as that of a
inc
blackbody at T = 5762 K. The irradiance Eb,sol
incident on a
2
surface of 1m located on the earth’s atmosphere (normal to
the unit vector ns joining the earth and the sun) is given by
inc
e
Eb,sol
= D ∆ωs Msun
,
with Rs and Rav being the radius
where ∆ωs =
of the sun and the average distance between the sun and the
earth, respectively. D is a correction factor accounting for the
variation of Rav (±3%) [6].
2
Rs2 /Rav
n
The beam irradiance Eb,sol
(m) impinging on a facet m of
2
1 m located on the earth’s ground and normal to n s and with
surface temperature Ts,m is given by
Z ∞
n
Eb,sol
(m) = Vm D ∆ωs cos θs
c(λ) mb (λ, Tsol )
0
0λ (λ, θs , φs , Ts,m ) dλ,
(9)
where 0λ ( . ) is the directional-spectral emissivity, Vm is a
visibility factor which is equal to one if facet m is in direct
visibility of the sun and zero otherwise. The spectral coefficient
c(λ) accounts for the propagation through the atmosphere
and can be obtained from MODTRAN. mb (λ, Tsol ) is the
radiant exitance of a blackbody at the wavelength λ and at
sun temperature Tsol . It is given by
mb (λ, T ) =
2πC1
5
C
λ (e 2 /λT
− 1)
,
(10)
where C1 and C2 are constant values [7]. φs and θs are the
azimuth and zenith angles of the sun, respectively. They are
computed using the algorithm of [8].
Now, assume that the normal nm to the surface and ns are
not colinear. The irradiance is thus given by
n
n
(m) cos θn (m),
Eb,sol (m) = Eb,sol
(m) nm · ns = Eb,sol
(11)
where θn (m) is the angle between nm and ns . The computation of θn (m) is straightforward when φs and θs are known.
The computation of Vm is done using an octree structure [9]
for describing the object of interest. This structure allows to
compute the intersection of a ray with the object in O(log M ),
where M is the number of facets.
2) The diffuse irradiance Ed,sol : The estimate of the diffuse
n
irradiance Ed,sol
(m) on an horizontal surface m is made as
inc
a product of the normal extraterrestrial irradiance Eb,sol
, a
diffuse transmission function Tn dependent only on the Linke
turbidity factor TLK , and a diffuse solar altitude function Fd
dependent on the solar zenith angle θs and on TLK , i.e.,
n
inc
(m) = Eb,sol
Tn (TLK ) Fd (θs , TLK ).
Ed,sol
(12)
The interested reader should consult [10] for more details.
The model for estimating the clear-sky diffuse irradiance
Ed,sol (m) for an inclined surface m is described in [10] and
is not described here.
C. The atmospheric irradiance Esky
The atmospheric irradiance is typically modeled as a blackbody at temperature Tsky , which is estimated based on a model
presented in [11]. Future versions of the software will include
the spectral behavior of this irradiance. The atmospheric irradiance absorbed by facet m is thus given by
Z ∞
¯∆ω
(13)
Esky (m) =
λ (λ, Ts,m ) mb (λ, Tsky ) dλ
0
where the hemispherical spectral emissivity is given by
¯∆ω
λ (λ, Ts,m )
1
=
π
Z
∆ω
0λ (λ, θ, φ, Ts,m ) cos θ dω.,
(14)
4
where ∆ω is the solid angle for which the sky is in direct
visibility with the facet. It can be shown that
¯∆ω
λ (λ, Ts,m ) =
1
π
Z
2π
0
−
Z
M
X
π/2
0λ (λ, θ, φ, Ts,m ) cos θ sin θdθdφ
θs (φ)
F,n→m (λ, Ts,m ),
(15)
n=1
This part of the radiance lost by the surface to the environment corresponds to the radiation produced by the surface of
the material due to its own temperature Ts,m and its non-zero
emissivity. We have
Z ∞
Mrad (m) =
¯λ (λ, Ts,m ) mb (λ, Ts,m ) dλ,
(18)
0
where θs (φ) is function of φ and nm . This expression is not
given here for the sake of brevity. The second term takes into
account the fact that the surface illuminated by the sky can be
obscured by other surfaces. The surface of interest is facet m
with area Am and normal unit vector nm . Assume that facet
n, with area An and normal unit vector nn is obscuring m.
rnm is the distance between a point xm on m and a point xn
on n. θm is the angle between the vector xnm joining xm to
xn and nm . Similarly, θn is the angle between xnm and nn .
Finally, Vnm is the visibility factor from xm to xn . In that
case, it can be shown that
Z Z
1
Vnm 0λ (λ, θ, φ, Ts,m )
F,n→m (λ, Ts,m ) =
A n An Am
cos θn cos θm
dAn dAm .
(16)
2
rnm
If 0λ ( . ) is independent of θ and φ, F,n→m (λ, Ts,m )
reduces to the configuration factor between n and m [7].
The computation of the F,n→m (λ, Ts,m )’s is highly time
consuming. In the case where 0λ ( . ) is independent of θ and
φ, they can be computed analytically [12]. For the general
case, we must use, for example, the Monte Carlo Ray-Traced
(MCRT) method [13] to compute these factors.
D. The irradiance due to multiple reflections
If reflections are diffuse, we can use the (hierachical) radiosity method [14]. If the object presents specularly reflective
properties, we can use ray-tracing methods [15]. If the surface
properties are angle dependent and wavelength dependent, we
can use the MCRT method [7]. We choose to use the MCRT
technique due to its simplicity to deal with general parametric
emissivity and reflectivity.
This method emits a number of bundles as rays, each of
fixed energy. At each reflection of a ray with a facet, it is
checked if the bundle is absorbed by the intersected facet. If
not, the bundle propagation continues. The initial direction and
the reflection directions are randomly determined according to
the emissivity and reflectivity profiles with respect to θ and
φ, treated as 2D probability density functions. The irradiance
received by facet m due to multiple reflections is
M
X
Ak
Emr (m) =
Mrad(k)
Dm,n ,
Nk
n=1
E. The radiated energy flux
(17)
where Nk is the number of bundles emitted by facet k,
Mrad (k) is the radiance leaving k given by Eq. (18), and Dm,n
represents the irradiance received by facet m due to the heat
flux emitted by facet n. The determination of the Dm,n ’s is
done using the MCRT method.
where ¯λ (λ, Ts,m ) is given by Eq. (14) where ∆ω is the total
hemisphere (θ ∈ [0, π/2] and φ ∈ [0, 2π]).
F. The convected heat flux
The convected radiance Mconv (m) is given by
Mconv (m) = h̄(m)(Ts,m − Tamb ),
(19)
where the convective coefficient h̄(m) depends on fluid parameters such as wind speed and on the orientation of the facet
with respect to the wind. In this preliminary version of the
software, we assume a constant value for h̄(m). However, our
software is able to manage a h̄(m) function represented by a
polynomial function of Ts,m , the coefficients of the polynom
being dependent upon the facet number m, i.e.,
h̄(m) = h̄(Ts,m ) =
Nh
X
i
ai,m Ts,m
,
i=0
where Nh is the order of the polynom.
G. The conduction heat flux
The computation of Mcond (m) requires the numerical resolution of a 3D differential equation. We choose a finite-volume
method [16], [17], This method, which is simple to implement
for unstructured meshes, approximate Mcond (m) as a linear
combination of surface temperatures, i.e.,
Mcond (m) '
M
X
(20)
Tn δn,m ,
n=1,n6=m
where the proportionality coefficients δn,m are non-zero only
for facets that are in the direct neighborhood of m.
V. C ALCULATION
OF THE FACET RADIANCE
Inserting Eqs. (11), (12), (13), (17), (18), (19), and (20)
into Eq. (5), we obtain the global heat transfer equation, i.e.,
0 ' −h̄(Ts,m )(Ts,m − Tamb ) + Vm cos θn (m) cos θs D∆ωs
Z ∞
c(λ) mb (λ, Tsol ) 0λ (λ, θs , φs , Ts,m ) dλ + Ed,sol (m)
0
+
Z
0
∞
¯∆ω
λ (λ, Ts,m )mb (λ, Tsky ) dλ +
M
X
Tn α n
n=1
Z ∞
M
Ak X
¯λ (λ, Ts,k ) mb (λ, Ts,k ) dλ
Dm,n
Nk n=1
0
Z ∞
¯λ (λ, Ts,m ) mb (λ, Ts,m ) dλ.
−
+
0
(21)
5
This equation must be solved for each facet m. Since
there is a coupling between the equations for the various
facets, we must solve a system of integral equations. In this
first description of our software, we do not consider multiple
reflections and conduction. Future papers will describe these
complex tasks. Suppressing these terms decouples the system
of equations. We thus must solve M independent integral
equations. Each equation being an integral equation, it is time
consuming to solve it and this must be done for each facet.
Thus, we must find a way to simplify this equation so that it
can be solved in a reasonnable amount of time.
A. Emissivity approximation
We assume for simplicity that the emissivity is a separable
function of λ, θ, φ, and Ts,m . We thus have
0λ (λ, θ, φ, Ts,m ) = av λ (λ) θ (θ) φ (φ) T (Ts,m ),
(22)
where av is the average value of the emissivity. This assumption greatly simplifies Eq. (21) by enabling the computation
of various terms, such as the integration of the solar spectrum, offline. Below, we describe the additional assumptions
concerning the four factors of 0λ (λ, θ, φ, Ts,m ).
1) The angular dependence : θ (θ) and φ (φ) highly depend upon the material used (metal vs non-metal). Typical
emissivity profiles can be more adequately approximated using
a function of cos θ rather than a function of θ [7]. Moreover,
for metals, these profiles are typically smooth [7]. We thus
choose to approximate θ (θ) as a Taylor series expansion of
θ (cos θ) around cos θ0 , where θ0 is the angle of the facet’s
normal. Thus, θ0 = 0. We thus have
∞
X
(cos θ − cos θ0 )k dk θ (t) ,
θ (cos θ) = θ (cos θ0 )+
k!
dtk t=cos θ0
k=1
(23)
where θ (cos θ0 ) is the emissivity for normal incidence which
can be obtained from real data or from models. The derivative
coefficients are found by fitting Eq. (23) to real data or by
calculating the derivatives analytically from the model.
In most material, the dependence upon φ can be neglected.
In the following, we assume that
φ (φ) = 1.
(24)
The advantage of approximations given in Eqs. (23) and
(24) is that 0λ ( . ) can be easily integrated with respect to dω.
Eq. (14) can thus be computed analytically once ∆ω is known.
To speed up the computation, we use a pre-computed look-up
table and interpolation to compute Eq. (14) accurately. This
approach can be generalized to other models if it better fits to
real data. The constraint is that it must be possible to perform
the integral analytically to simplify the computation.
2) The temperature dependence : Simulations and real data
show that 0λ ( . ) also varies smoothly as a function of Ts,m [7].
Thus, as for θ (θ), we approximate T (Ts,m ) with a Taylor
series expansion around T0 = 273K for which the resistivity
of most materials are well known. We thus have
∞
X
(Ts,m − T0 )k dk T (Ts,m ) T (Ts,m ) = T (T0 )+
k
k!
dTs,m
T
k=1
,
s,m =T0
(25)
where the derivative coefficients are found by fitting Eq. (25)
to real data. This equation can also be simply rewritten as a
polynom of the surface temperature Ts,m , i.e.,
T (Ts,m ) =
∞
X
j
,
αj,p (T0 ) Ts,m
(26)
j=0
where the αj,p (T0 )’s are coefficients that are not a function
of Ts,m . We have different values for the αj,p ’s only for
each different material composing the surface of the object.
Hence, we typically have only a small number of different
sets of αj,p ’s. A set for a given material is denoted by p. This
approximation must be accurate only in the range of variation
of interest of Ts,m , i.e., typically from 250◦ K to 450◦ K.
3) The spectral dependence : The approximation typically
used in the literature is to divide the spectrum into Nb spectral
bands ∆λi and to consider a gray body with emissivity i inside each ∆λi [7], where ∆λi = [λi,min (p), λi,max (p)], with
λi,min (p) and λi,max (p) being the minimum and maximum
wavelengthes of spectral band i and for material p.
4) Approximation of the heat transfer equation: For spectral
band i, using Eqs. (22), (23), (24), and (26), we have
Ed,sol (m) =
µi,p
Z
λi,max (p)
mb (λ, Ts,m ) dλ − ζi,m
λi,min (p)
∞
X
!
(27)
j
+ h̄(Ts,m )(Ts,m − Tamb ),
αj,p (T0 ) Ts,m
j=0
where µ and ζi,m are constant values that do not depend
upon λ and Ts,m . The right-hand side of this equation must
be computed for each spectral band. The final equation is
obtained by summing all these equations. We clearly see that
this equation is again an integral equation in Ts,m . To find
the solution, we must find an approximation or an analytical
solution for the integral, since the computational time to solve
this equation is too high. In Section V-C, we discuss an
efficient way to solve this equation.
B. Spectral dependence approximation
We first define fi,p (Ts,m ) as
Z λi,max (p)
fi,p (Ts,m ) =
mb (λ, Ts,m ) dλ,
(28)
λi,min (p)
The classical approach to compute fi,p (Ts,m ) aims at approximating the integral by a series expansion. We have
Z λi,max (p)
fi,p (Ts,m ) =
mb (λ, Ts,m ) dλ
0
Z λi,min (p)
−
mb (λ, Ts,m ) dλ
0
4
F0−λi,max (p)Ts,m − F0−λi,min (p)Ts,m ,
= σTs,m
6
where σ is the Stefan-Boltzmann constant [7] and F0−λT is a
function of λT defined as
Z λT
mb (λ, T )
F0−λT =
d(λT ).
(29)
T5
0
A convenient expression for F0−λT is given by [18]
F0−λT
∞ 3ζ 2
6ζ
6
15 X e−nζ
3
ζ +
+ 2+ 3
,
= 4
π n=1
n
n
n
n
1) Generalization of the infinite series approximation:
Below, we present a generalization of Eq. (30) to cubic
polynoms in λ in each ∆λi . This implies that we approximate
λ (λ) by piecewise-cubic functions. This reduces considerably
the computational time by drastically reducing the number of
spectral bands. To simplify the notation, we intentionally omit
the dependence on p.
(30)
N
where ζ = C2 /(λT ). In practice, the series in Eq. (30)
converges very rapidly and the first three terms give good
results over most of the range of F0−λT . As T becomes large,
a larger number of terms is required. Eq. (28) thus becomes
1
0
fi,p (Ts,m ) '
3
X
»
e
−nζM
„
2
3ζM
λi
λN
«
where ζM
=
C2 /(λi,max (p)Ts,m ) and ζm
=
C2 /(λi,min (p)Ts,m ). The only unknown in Eq. (31) is
Ts,m . Equation (27) thus becomes
Ed,sol (m)
λ1
λ
6ζ
6
M
3
ζM
+
+ 2 + 3
n
n
n
n
Fig. 4. Division of the spectral interval from 0 to ∞ in subbands.
n=1
«–
„
−nζm
2
e
6ζ
6
3ζ
m
3
(31)
−
ζm
+ m + 2 + 3
Typically, we have real data on a limited spectral interval.
n
n
n
n
4
15σTs,m
π4
' (µi,p fi,p (Ts,m ) − ζi,m )
∞
X
j
αj,p (T0 ) Ts,m
j=0
+h̄(Ts,m )(Ts,m − Tamb ),
(32)
This equation is again highly non-linear with respect to
Ts,m . The non-linearity appears in two ways: (1) as a polynom
in Ts,m and (2) as an exponetial term e−γ/Ts,m . To solve this
equation, we must use a non-linear optimization algorithm,
such as the Brent algorithm [19]. This optimization algorithm
has to be applied to each facet. This can take a huge amount
of time! As a consequence, a more attractive approach must
be design. This is the subject of the next section.
C. Computationally-efficient spectral dependence approximation
The approximation of a spectral emissivity profile by a
piecewise-constant function has a great limitation: λ (λ) must
be constant in each ∆λi . Hence, to accurately represent λ (λ),
we need a great number of ∆λi . This implies a considerable
increase of the computational time needed to compute the
solution of Eq. (32).
To reduce the computational time, we must find a way to
reduce the number of ∆λi and an approximation of Eq. (32)
to find a solution in a reasonnable amount of time without
sacrifying the accuracy. Below, we present a computationally
efficient way of solving Eq. (32) for general emissivity profiles.
We proceed in two steps. First, we generalize Eq. (30) to nongray bodies in each ∆λi . Second, we propose an approximation of the series that has promizing performance.
The first step is thus to divide the spectral interval [0, +∞[ in
three bands as shown in Fig. 4. The first band starts at λ = 0
and ends at the first sample we have at λ1 . In this band, we
consider a gray body with emissivity 1 equal to the emissivity
of the first sample. The second band begins at λ1 and ends at
λN , which is the wavelength of the last sample we have. In
this band, the emissivity profile can be very complex. The
third band starts at λN and ends at +∞. In this last spectral
band, we consider a gray body with emissivity N equal to the
emissivity at λN .
The spectral band [λ1 , λN ] is then divided in N spectral
bands. In each band, λ (λ) is approximated by a polynom in
λ. Denoting I as
Z ∞
I=
λ (λ) mb (λ, Ts,m ) dλ,
0
we decompose I as a sum of three terms, i.e.,
I = I 1 + I2 + I3 ,
(33)
where
I1
= 1
I2
=
I3
= n
Z
λ1
mb (λ, Ts,m ) dλ
0
Nj
N
−1 Z λi+1 X
X
βi,j λj mb (λ, Ts,m ) dλ
λ
i
i=1
j=0
Z ∞
mb (λ, Ts,m ) dλ,
(34)
(35)
(36)
λN
where Nj is the order of the polynom used to represent λ (λ)
in each spectral band. I1 and I3 are computed using Eq. (30).
I2 must be investigated furthermore. We write Eq. (35) as a
sum of integrals, i.e.,
I2 =
N
−1
X
i=1
I2,i .
7
Below, we derive a mathematical expression for I2,i . First,
we perform the change of variable ξ = C2 /(λTs,m ). Using
Eq. (10), the integral I2,i thus becomes
I2,i =
Z ξi+1 3−j
Nj
ξ
2πC1 X 4−j j
T
C
β
dξ,
i,j
ξ −1
C24 j=0 s,m 2
e
ξi
(37)
Then, we replace the denominator by an infinite series using
the following well known result for geometrical series.
series, which can significantly increase the computational time.
We thus limit our result to third-order polynoms in λ.
Figure 5 shows plots of the error between the true integral
(computed with a very accurate numerical integration routine)
and the infinite series limited to three terms. Figure 5(a) shows
that the value of the error is very small (less than 0.1%) and
Fig. 5(b) shows the relative value of the error as the number
of terms increases. We clearly see that two terms are sufficient
for an accurate computation of the integral.
∞
X
1
=
e−kx , k ∈ N and |e−x | < 1.
1 − e−x
k=0
ξi
−4
x 10
1.4
∞
X
ξ 3−j
dξ =
ξ
e −1
k=1
Z
ξi+1
1.2
ξ 3−j e−kξ dξ.
Error
We thus have
Z ξi+1
ξi
We compute the integral using a result from [20], i.e.,
Z
n
eax X xn−m
(−1)m g(n, m),
xn eax dx =
a m=0 an
g(n, m) =
0.2
0
0.3
0.25
450
0.2
Dynamic
if m = 0
(n
−
q)
if
m>0
q=m−1
1
Q0
400
0.15
1.6
300
0.05
Temperature (K)
250
(a)
x 10
1.4
1.2
Relative error = 0.001
1
Error
3−j
3
∞
X
2πC1 X 4−j j
e−kξ X ξ 3−j−m
g(3 − j, m).
Ts,m C2 βi,j
4
C2 j=0
k m=0 km
350
0.1
−3
Pay attention that this result is true only if n ≥ 0. In our
case, x is replaced by ξ, a is replaced by −k, and n is replaced
by 3 − j. This implies that j ≤ 3. This is why our result is
limited to third-order polynomial. We thus have
I2,i =
0.6
0.4
where
1
0.8
0.8
0.6
k=1
As a last step, we regroup the sum over m and the sum
over j as a single sum over j. After some basic mathematical
operations and by noting that 2πC1 /C24 = 15σ/π 4 , we get
i+1
i
− I2,i
,
I2,i = I2,i
∞
3
4
X
15σTs,m
e−kC2 /(λi Ts,m ) X −j
Ts,m γi,j ,
π4
k
j=0
(38)
k=1
where the γi,j coefficients are given by
γi,j = C2j
1
k 3−j
j
X
0.2
0
(b)
1
2
3
4
5
6
7
8
9
10
Number of terms in the sum
where
i
I2,i
=
0.4
1
βi,p ( )j−p g(3 − p, 3 − j).
λi
p=0
Comparing Eq. (38) with Eq. (30), we see that the two
equations have the same general shape. We only must replace
the 1/nj ’s with the γi,j ’s . Eq. (30) can thus be generalized
to third-order polynomial approximation in each ∆λi .
Two remarks are in order. First, the computation of the terms
of the series requires a negligeable increase in the computational time since the γi,j ’s can be computed offline. Second,
Eq. (30) can be further generalized to high-order polynomials.
However, this requires the evaluation of a second infinite
Fig. 5. Plots of the error between the true integral (computed using a very
accurate method) and the series. (a) The 3D plot of the error for three terms
in the sum (the polynom coefficients are given by the coefficients of ∆λ 1 in
Fig. 6). The value in ordinate is the absolute error. Dynamic is the interval of
variation of λ (λ). (b) The relative error as the number of terms in the sum
increases.
The global heat transfer equation is thus again given by
Eq. (32) with the fi,p (Ts,m )’s replaced by Eq. (38) for each
∆λi . Outside this interval, i.e., for I1 and I2 , we use Eq. (31).
A remark concerning the choice of the number Nb of
spectral bands is in order. In our software, we automatically
determine Nb . The method used is the following. We first
perform a fit of a thrid-order polynom to the data and compute
the RMS error. If this error is smaller than a given threshold
T , we stop the process. If not, we divide the interval such
as to have the same RMS error in the two subintervals. The
extend ∆λi of each spectral band can thus be different. We
then perfom two fits in the two spectral bands independently
and we compute the RMS error in each band. If the global
8
error is less than T , we stop. If not, we continue to divide
the spectral interval. We continue this process until the error
is less than T . Figure 6 shows an example fit of a piecewisecubic function to data points: two subbands are enough to
accurately represent λ (λ). Observe that we have not imposed
a continuity condition at the boundaries of the subbands.
linearly with n. Approximating Eq. (31) with a polynom is thus
meaningfull. This is confirmed by Fig. 7 that shows the relative
error between Eq. (39) and the approximation of Mrad(m)
described in Section V-C. We see that a value of Np < 10
leads to a negligeable error. In our simulations, the order of
the polynom used to represent T (Ts,m ) is 6 and the order of
the polynom in Eq. (31) is 3. Thus, it is quite logical to have
a negligeable error for values of Np greater than 9.
1.00
Emissivity λ (λ)
∆λ1
∆λ2
0.95
0.05
0.90
Relative
Error
Fit
Data points
0.85
Junction between
two subbands
3.62e-7
0.6
0.80
0.75
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Emissivity
dynamic
0.01
Example fit of a piecewise-cubic function for given data points.
2) Polynomial approximation: The global heat transfer
equation obtained with the previous approximation can lead to
a huge decrease of the computational time by greatly reducing
Nb . However, the equation for a particular ∆λi remains nonlinear with respect to λ, mainly due to the exponential term.
Hence, a non-linear optimization algorithm must be used.
To find a faster way for obtaining the surface temperatures,
remind that the behavior of Eq. (32) is only relevant in
a range of temperatures that typically goes from 250◦K to
450◦K. Simulations shows that the exponential function has
a very smooth behaviour in this range. Hence, this can be
approximated by a much simpler function, such as a polynom
in Ts,m .
The strategy used is thus the following. We choose the order
of the polynom and then we fit Mrad (m) in Eq. (32) (for all
spectral bands) with that polynom and we use the resulting
equation to find the surface temperatures. We thus replace
!
Z λi,max (p)
Nb
X
Mrad (m) =
µi,p
mb (λ, Ts,m ) dλ
λi,min (p)
i=1
∞
X
j
αj,p (T0 ) Ts,m
j=0
by
Mrad (m) '
Np
X
k
bj,p Ts,m
,
Polynom
order
x1e-5
Wavelength λ (m)
Fig. 6.
4
(39)
k=0
where Np is the order of the polynom. The bj,p ’s can be
computed offline since there is a set of bj,p ’s only for each
different material.
Investigation of the analytical expression of the nth-order
derivative of Eq. (31) shows that this derivative decreases
15
Fig. 7. Error between the approximation using a polynom of a given order
and the series used to approximate the integral of the blackbody. The first
axis is the order of the polynom and the second axis is the dynamic of the
T (Ts,m ) profile.
3) Solving the global heat transfer equation: Inserting
Eq. (39) into Eq. (32) and summing over all spectral bands,
the resulting equation is a polynomial equation in Ts,m , with
coefficients that varies from one facet to another. This equation
can then be solved using the iterative method of Laguerre [21].
This technique is very fast and allows us to quickly compute
the surface temperatures for all facets.
VI. ATTENUATION OF RADIANCE BY ATMOSPHERE
Once the thermal energy leaves the facet, it undergoes attenuation through the atmosphere before reaching the camera.
The attenuation varies with λ. The propagation through the
atmosphere is handled via scaling by the atmospheric attenuation coefficient. The radiance Lout (θ) reaching the optical
system for the viewing direction is related to the radiance
Lin (θ) leaving each surface facet via Lout (θ) = τat Lin (θ),
where τat is the atmosphere transmittance coefficient. In our
simulator, we compute τat using MODTRAN.
VII. C ALCULATION OF IMAGE INTENSITIES
Going from Lout (θ) to the voltage at each detector element
is a delicate task. This voltage is given by [22]
πAd
Lout (θ),
Vd = kRD (λ)τopt
4F
where k is a constant, RD (λ) is the detector responsivity, τopt
is the transmission factor through the optical system, Ad is the
area of the detector element (collecting the incident power),
and F is the aperture number. Second, this voltage is mapped
to the scale of image intensities typically from 0 to 255.
9
VIII. S IMULATION
B. Computational complexity
EXAMPLES
We first show that our simulator is able to compute the
surface temperatures of simple objects. Then, we study the
computational complexity of our algorithm and compare it to
simulations. Finally, we show a typical IR image generated on
the basis on the pre-computed surface temperatures.
A. Surface temperatures computation
We first consider simulations for the computation of the
surface temperatures. Two objects are considered: a sphere and
an elementary ship geometry.
The motivation for considering a sphere is to show that our
algorithm is able to compute the surface temperatures for all
facet orientations for a given sun position. It also allows us
to show the robustness of the algorithm with respest to the
orientation of the facet with respect to the sun. This is also
a good sanity test for the robustness of the computation and
the use of the octree structure. Figure 8 shows the surface
temperatures for clear sky conditions (a) in the winter and (b)
in the summer. We clearly see the effect of the direct beam of
the sun and of the sky radiance.
Simulation 1: winter
Simulation 2: summer
Date : 06 december, 11h30
Clear sky, 5 deg. C
Date : 21 june, 15h30
Clear sky, 22 deg. C
z
-10283deg
surface temperatures xxx
346
y
50409deg
x
Z
Y
X
4301deg
surface temperatures xxx
360
62420deg
The complexity of the algorithm is composed of four distinct
parts. (1) the computation of the octree, (2) the building of
the equations for surface temperatures computation, (3) the
computation of the solution of the equations and, (4) the
computation of the IR image. We focus our attention to the
points (2), (3), and (4). Indeed, as opposed to the computation
of the octree which is done once for a given mesh and a given
geometry, the computation of the surface temperatures and of
the IR images is done at each time step.
The computation of the equations is of complexity
O(M log(M )). Indeed, for a particular facet, we need to
compute the visibility factor Vm in Eq. (11) between the
sun and facet m. Using an octree structure allows to have
a complexity of O(log M ). The global complexity is thus
O(M log(M )). Figure 10(a) shows the result of a fit of a curve
a + bM log(M ), where a and b are parameters, on simulation
data points. We see that these points are well approximated by
this curve.
The computation of the solution of the equations is O(M ),
since there is no coupling between the equations. Figure 10(b)
shows the result of a fit of a curve a + bM , where a and b
are parameters, on simulation data points. The complexity also
depends on the order of the polynom in Eq. (39). Simulations
show that the computational time increases linearly with the
order of the polynom.
We use classical ray-tracing to compute the IR image
based on the surface temperatures. The complexity is then
O(Nr log(M )), where Nr is the number of ray launched and
thus the number of pixels. The log(M ) factor is due to the
computation of the visibility factor of each ray. The complexity
of the computation of the IR image is thus mainly constant
with M . This is what is observed in practice. The complexity
is dominated by the number of launched rays.
Z
Y
X
Fig. 8. Example surface temperatures computation for a sphere (a) in the
winter and (b) in the summer.
Time (sec)
500
We choose a simple ship as the second test object because
the final aim of the project is the simulation of the IR signature
of a ship. The second reason is to ensure that the octree
structure is able to correctly manage the shadow’s effect.
Figure 9 shows that this is indeed the case.
400
300
100
0
0
Time (sec)
83000 facets
Fig. 9. Example surface temperatures computation for a simple ship geometry.
Fit
Data points
200
45
40
35
30
25
20
15
10
5
0
0
(a)
1
2
3
4
5
6
x1e4
Fit
Data points
(b)
1
2
3
4
Number of facets
5
6
x1e4
Fig. 10. Plot of the complexity of the algorithm for (a) equations building
and (b) solving these equations.
10
C. Infrared image computation
The last step is the computation of the IR image. Figure 11
shows a typical IR image for the simple ship geometry. We
clearly see the effect of the sun.
300
1.4
250
1.2
200
1.0
0.8
150
0.6
100
0.4
50
0
0
0.2
50
100
150
200
250
300
0.0
Fig. 11. Example IR images generated based upon the surface temperatures
for a simple ship geometry.
IX. C ONCLUSION
In this paper, we have presented a software able (a) to
compute the surface temperatures of an object exposed to sun
and atmospheric radiations and (b) to use these temperatures to
predict the infrared signature of the object. The particularity of
this software is that it can manage complex emissivity profiles
in a reasonnable amount of time using accurate approximations
for the heat transfer equations. Moreover, simulations show
that the complexity of this software is promizing.
ACKNOWLEDGMENT
This project was financed by the Belgian Ministry of
Defense. I also acknowledge Idesbald Van den Bosch for
constructive comments.
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