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 U˙ , must be equal to the temporal variation of the facet energy, i.e., ∂T = Φabs − Φlost − U˙ , (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.
Φabs = Φlost + U˙ .
(2)
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 U˙ is due to the conduction of heat into the neighboring facets from the facet of interest, i.e., U˙ = 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 = ”U˙ per unit area".
(6) (7) (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) = h(m)(T s,m − Tamb ),
(19)
¯ where the convective coefficient h(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 h(m). However, our ¯ software is able to manage a h(m) function represented by a polynomial function of Ts,m , the coefficients of the polynom being dependent upon the facet number m, i.e., ¯ ¯ s,m ) = h(m) = h(T
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 ' −h(T )(T − Tamb ) + Vm cos θn (m) cos θs D∆ωs Z ∞ s,m s,m 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 ¯ s,m )(Ts,m − Tamb ), + h(T 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
¯ s,m )(Ts,m − Tamb ), +h(T
(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. R EFERENCES [1] J. Nilsson and A. Kockums, “Stealth phylosophy, real solutions from sweden.” [2] D. A. Vaitekunas, K. Alexan, O. E. Lawrence, and F. Reid, “SHIPIR/NTCS: a naval ship infrared signature countermeasure and threat engagement simulator,” in Infrared Technology and Applications XXII, B. F. Andresen and M. S. Scholl, Eds., vol. 2744, no. 1. SPIE, 1996, pp. 411–424. [3] D. A. Vaitekunas, “Validation of shipir (v3.2): methodology and results,” in Targets and Backgrounds: Characterization and Representation XII, W. R. Watkins, D. Clement, and W. R. Reynolds, Eds., vol. 6239. SPIE, 2005. [4] J. Thompson, B. Soni, and N. Weatherill, Handbook of Grid Generation. CRC Press, 1999. [5] J. Thompson, “A reflection on grid generation in the 90s: Trends, needs and influences,” in Proceedings of the 5th International Conference on Numerical Grid Generation in Computational Field Simulations, April 1996, pp. 1029–1110. [6] J. Duffie and W. Beckman, Solar Energy Thermal Processes. John Wiley & Sons, 1974. [7] R. Siegel and J. Howell, Thermal Radiation Heat Transfer, Third Edition. Hemisphere Publishing Corporation, 1992.
[8] I. Reda and A. Andreas, “Solar position algorithm for solar radiation applications,” National Renewable Energy Laboratory, Tech. Rep. TP560-34302, 2003. [9] A. Glassner, “Space subdivision for fast raytracing,” vol. 10. IEEE Computer Graphics & Applications 4, 1984, pp. 15–22. [10] J. Hofierka and M. Suri, “The solar radiation model for open source gis: Implementation and applications,” in Proceedings of the Open Source GIS - GRASS users conference, 11-13 September 2002. [11] G. Walton, “Thermal analysis research program,” U.S. Government, 1983. [12] P. Schroeder and P. Hanrahan, “A closed form expression for the form factor between two plygons,” Department of Computer Science, University of Princeton, Tech. Rep. TR-404-93, 1993. [13] J. Maham, Radiation Heat Transfer: A Statistical Approach. John Wiley & Sons, INC, 2002. [14] M. Cohen and J. Wallace, Radiosity and Realistic Image Synthesis. Academic Press, 1993. [15] A. Glassner, J. Arvo, R. Cook, R. Haines, P. Hanrahan, P. Heckbert, and D. Kirk, An Introduction to Ray-Tracing. Academic Press, 1989. [16] J. Hyman, R. Knapp, and J. Scovel, “High order finite volume approximation of differential operators on nonuniform grids,” vol. 60, 1992. [17] T. Barth, “Aspects of unstructured grids and finite-volume solvers for the euler and navier-stokes equations,” Von Karman Institute for Fluid Dynamics, Tech. Rep. Lecture Series 1994-05, March 21-25 1994. [18] S. Chang and K. Rhee, “Blackbody radiation functions,” International Communications Heat Mass Transfer, vol. 11, pp. 451–455, 1984. [19] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C. Cambridge University Press, 1992. [20] M. Spiegel, Formules et Tables de Mathémathiques. McGraw-Hill, 1992. [21] F. Litt, “Analyse numérique,” Montefiore Institute, University of Liège, Belgium, 2001. [22] G. Holst, Common Sense Approach to Thermal Imaging. SPIE Optical Eng. Press, 2000.
