Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 1
Thermal Infrared Identification of Buried Landmines
Thành Trung Nguyen1 , Dinh Nho Hào1 , Paula Lopez2 , Frank Cremer3 and Hichem Sahli1,3
1
Vrije Universiteit Brussel, ETRO Department, Pleinlaan 2, B-1050 Brussel, Belgium
Universidad de Santiago de Compostela, Departamento de Electrónica y Computación,
E-15782, Santiago de Compostela, Spain
3
Interuniversity Micro Electronics Centre, ETRO Department, Pleinlaan 2, B-1050 Brussel,
Belgium
2
ABSTRACT
This paper deals with a three-dimensional thermal model for landmine detection problems and an inverse problem
for reconstructing the physical parameters of buried objects. Moreover, solutions are given for the estimation of
the soil thermal diffusivity and meteorological parameters, needed for solving the inverse problem. The paper
describes the main fundamental principles of thermal modelling for buried object identification and illustrates
the results on data acquired from a real minefield, together with qualitative and quantitative results illustrating
the validity of the model.
Keywords: Thermal model, infrared technique, heat equations, inverse problems
1. INTRODUCTION
During the last decade, research in landmines detection received a growing interest in finding structured and
reliable solutions for solving the problem of the detection and subsequent removal of mines and unexploded
ordonnance. Several ’new’ technologies, based on different physical principles, e.g. electromagnetic detection,
vapor/builk detection, and optoelectronic imaging, have been investigated. Among the used technologies, dynamic thermal infrared (IR) technique seems to be effective to detect non-metallic landmines for which metal
detectors cannot be applied14.
Dynamic thermal IR technique has been used since the 80s for nondestructive evaluation3 with the wellknown paper of K. Watson25 for geologic applications of IR images. The use of thermal IR technique is based on
the thermal radiation contrast of objects, with respect to their background. All objects at temperatures greater
than absolute zero emit electromagnatic radiations at all wavelengths in which the radiations corresponding to
the wavelengths from 3 µm to 100 µm is referred to be the thermal IR radiations. The magnitude of spectral
radiations of an object depends on its temperature. The total energy radiated by a blackbody is given by
Stefan-Boltzmann’s formula9, 13, 26. According to Wien’s displacement Law 9, almost all objects on the earth emit
electromagnatic radiation mainly in the range of longwave infrared (8 µm to 14 µm).
The difference of thermal characteristics, i.e., heat capacity c, thermal conductivity k and thermal diffusivity
α, between buried objects and the background is the base of using IR technique for detecting landmines. Indeed,
the presence of a buried object affects the heat conduction inside the soil during natural heating conditions.
Consequently, the temperature of the soil surface, above the buried object, may be different from that of the
surrounding area. This temperature contrast is measured by an IR imaging system. In the case of surface
laid landmines, the thermal radiation, measured by the IR camera, may be different for the landmines and the
surrounding. This high contrast is due to the difference in radiant characteristics (emissivity, absorptivity and
Email addresses: {ntthanh, dnhohao, hsahli, fcremer}@etro.vub.ac.be, paula@dec.usc.es
Copyright 2005 Society of Photo-Optical instrumentaion Engineers
This paper was published in Proc. of SPIE 5794 and is made available as an electronic preprint with the permission of
SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution
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purposes, or modification of the content of the paper are prohibited.
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 2
reflectivity) between the landmines and the background soil. From the measured images, one can easily detect
the presence of abnormal objects.
However, to detect and identify buried objects, a thorough analysis of the thermal behaviour of the surveyed
area, by considering the thermal parameters of both background and buried objects, is needed. This is achieved
in two steps.
• The first step, referred to as forward thermal model, aims at modelling the temporal behaviour of the soil
temperature, under natural heating conditions, with the presence of buried landmines. This is based on
the solution of an initial boundary value problem for a three-dimensional heat transfer equation. Although
heat transfer processes in solids have been analysed thoroughly6, 20, only few thermal models have been
proposed for landmine detection problems. Among the existing models, I. K. Sendur and B. A. Baertlein21
proposed a three-dimensional thermal model for homogeneous soil containing a buried anti-tank mine
modelled as TNT object. The authors considered smooth and Gaussian distributed rough surfaces, and
proposed a radiometric model for thermal signatures released from the soil surface and from other sources.
The considered radiometric model, instead of using apparent temperature, allows approximating the real
temperature of the surface from measured IR images. The same radiometric model has been also studied
by P. Pregowski et al.19. A three-dimensional model, for homogeneous soil containing buried landmines and
considering both the explosive (TNT) and the casing of the landmine, has been introduced by K. Khanafer
and K. Vafai11. In their paper, the authors analysed the effect of the casing on the temporal behaviour of
the soil temperature distribution taking into account rough soil surfaces and diurnal cycle.
These models helped understanding the effect of buried landmines on the soil temperature. However, their
validity, by comparing the simulated and experimental results, have not been studied. This has been done
in our early works14, 16, where we proposed and validated, using indoor and outdoor experiments, a 3D
thermal model for homogeneous soil and buried landmines characterised by their TNT and considering flat
soil surface. A 2D thermal model has been also investigated by S. Sjokvist et al.22–24.
• The second step, referred to as inverse problem setting for landmine detection, consists of using these
thermal models for the detection and identification of the buried objects, i.e., estimate the position, (c1 , c2 ),
the sectional radius r, the height h, the depth d, and the thermal diffusivity αt of the buried object (see
figure 1). Here, we are faced with an ill-posed inverse problem of determining the location (position, depth
and height), size and the physical properties of buried objects from diurnal thermal observations on the
soil surface. Although inverse problems in heat conduction have been intensively studied1, 2, 4, 5, 8, 12, to our
knowledge, only our previous work, P. Lopez et al.14, 15, applied them for landmine detection. In14, 15 the
depth (of burial) and the thermal diffusivity, of a buried object have been estimated, when its location in
the soil volume is known. Moreover, in the later work, we considered known the soil characteristics and
the incoming heat flux parameters.
To apply thermal modelling for detecting landmines in a real minefield7, several parameters, such as specific
heat, thermal conductivity, thermal diffusivity, emissivity, solar absorptivity, etc. have to be estimated. Moreover,
one should take into account the relationship between the real soil surface temperature and the ’intensity’ of the
acquired IR images. This relationship is not straightforward, indeed, the measured thermal ’intensity’ is affected
by the solar radiation, the emission of the air, as well as the soil roughness. Finally, the ill-posedness of the
inverse problem may cause large errors, in the estimated parameters, with small errors in the input data. All
these aspects are considered in this paper.
The paper is organised as follows. In Section 2, we first state mathematically the full process of detecting
landmines using thermal modelling. This model requires, among other parameters, the soil thermal diffusivity,
the soil thermal emissivity, the solar absorption, the sky absorption, and the convective heat transfer coefficients.
These parameters depend on the local soil characteristics and the meteorological (weather) conditions during the
thermal observations. From measured soil temperatures at different locations and depths, in Section 3, we present
a method of estimating the soil thermal diffusivity and the derivation of the soil temperature (in the vertical
direction) proportionally to the heat flux on the soil surface. The other parameters such as solar absorptivity,
the soil emissivity, etc. are estimated as described in Section 4 using weather data. The model given by Lopez14
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 3
is used for the estimation of the thermal diffusivity of the buried objects and their depths of burial. In Section 5,
we present some results of the inverse problems for the estimation of the physical parameters of soil and buried
objects, using data acquired in a real minefield within the United Nations Buffer Zone in Cyprus. The operated
stand-off thermal IR survey system and the anomaly detection algorithm are described in Cremer et al.7
2. MATHEMATICAL FORMULATION
In this section, we describe the mathematical equations governing the heat transfer process inside the soil and the
buried objects, as well as the necessary boundary conditions. Moreover, the formulation of the inverse problems
for buried object detection is given.
2.1. Governing equation
Consider a rectangular parallelepiped domain Ω within the three-dimensional soil volume containing a possibly
buried target (see figure 1). We denote by Γ1 and Γ2 the top and bottom surfaces of the domain, respectively.
Γ1 , being the soil surface and the only portion of the volume accessible for measurements. We assume that the
soil and the target are isotropic solids with specific heat c(x, y, z) (J/(kg K)), density ρ(x, y, z) (kg/m3 ) and
thermal conductivity k(x, y, z) (W/(m K)). We also assume that the soil moisture content variation is negligible.
Then, the temperature distribution, T (x, y, z, t), (x, y, z, t) ∈ Ω × [0, te ], of the considered volume (the soil and
the possibly buried object) under natural heating conditions, satisfies the following partial differential equation
µ
¶
µ
¶
µ
¶
∂T
∂
∂T
∂
∂T
∂
∂T
cρ
=
k
+
k
+
k
.
(1)
∂t
∂x
∂x
∂y
∂y
∂z
∂z
In the case of homogeneous or piecewise homogeneous solid, equation (1) has the form
∂T
= α(x, y, z)∆T,
∂t
(2)
where α = k/(cρ) (m2 /s) is the thermal diffusivity of the solid.
2.2. Initial and boundary conditions
In order to solve (1) or (2), one should know the initial distribution of temperature (T (x, y, z, t = 0)), as well as
the boundary conditions, e.g., the incoming or outgoing heat flows on the boundary, ∂Ω, of the domain. These
conditions are described as follows
1. The initial condition expresses the soil temperature at the initial time instant t = 0
T (x, y, z, t = 0) = g(x, y, z), ∀(x, y, z) ∈ Ω.
(3)
2. The sufficient deep depth condition, assumes that the soil temperature at sufficient deep depth, z0 , does
not depend on the diurnal heat transfer process.
T (x, y, z0 , t) = T∞ , ∀(x, y, z0 , t) ∈ Γ2 × [0, te ].
(4)
From our experimental observations, z0 is set to 50 cm (see section 5).
3. The surface heat flux, establishes the incoming heat flux, qnet , through the portion of the soil volume
accessible for measurements, i.e. the air-soil interface (soil surface) Γ1
−k
∂T
= qnet for (x, y, z) ∈ Γ1 ,
∂z
The estimation of qnet will be considered in detail in Section 4.
(5)
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 4
y
Γ
1
c2
d
r
x
c
1
Ω
1
h
Γ
2
z
Figure 1. Model of the soil domain containing a buried object
4. The domain is assumed to insulate the outgoing as well as the incoming heat flows from surrounding areas
of the soil, i.e., there is no heat flowing in the horizontal direction on the boundary of Ω. This hypothesis
is reasonable if the soil is homogeneous and the considered domain is large enough so that we can neglect
the effect of the buried objects to the boundary of the domain. In this case, the heat balance takes place
on the surrounding boundary of the domain during the analysis period. This condition is given by
∂T
= 0 for (x, y, z) ∈ ∂Ω \ (Γ1 ∪ Γ2 ),
∂~n
(6)
where ~n is the outward unit normal vector of ∂Ω.
Equation (1) or (2) with the initial and boundary conditions (3)–(6) define the forward thermal model for
landmine detection.
2.3. Inverse problems for buried landmines detection
As mentioned before, the aim of the inverse problem setting for landmine detection is to detect (locate) the
presence of an abnormal object and characterise it as being mine or not, i.e. estimate it’s physical thermal
characteristics. In this problem, we aim at reconstructing the internal characteristics of the soil based on
measured temperatures on the surface. This problem is stated as follows.
Let αs being the thermal diffusivity of the bare soil, and suppose the existence of a buried object with
thermal diffusivity αt occupying a sub-domain Ω1 of Ω. Moreover, the sub-domain is assumed to be a cylinder
with sectional center (c1 , c2 ) in the horizontal plane, height h, sectional radius r and is located at depth d (see
figure 1).
The soil temperature distribution T (x, y, z, t) is described by the boundary value problem (2)–(6) with piecewise constant thermal diffusivity
(
αt in Ω1
α=
(7)
αs in Ω \ Ω1
The aim of the inverse problem is to reconstruct the parameters αt , r, h, d, c1 , c2 from measured soil surface
temperatures, T (x, y, z = 0, t) = θ(x, y, t), at t = t1 , t2 , . . . , tn . This problem can be written in the form of an
optimization problem:
tn Z
1X
2
|T (x, y, 0, t) − θ(x, y, t)| dxdy,
(8)
min
αt ,r,h,d,c1 ,c2 2
t=t
1
Γ1
where T (x, y, z, t) is the solution of the forward model (2)–(6) with piecewise constant thermal diffusivity α. In
section 5 we will consider numerical methods for solving the forward problem (2)–(6) and inverse problem (8).
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 5
3. SOIL THERMAL DIFFUSIVITY ESTIMATION
In order to model the heat transfer process in a given soil, it is necessary to know its specific heat, density,
thermal conductivity and thermal diffusivity characteristics. Although the characteristics of different materials,
including soil types, have been given in the literature, they can not be used directly. Indeed, soil characteristics
are dependent on the location and moisture content of the soil.
In this section, we propose a simple method to estimate the soil diffusivity from soil temperature profiles
measured by thermocouples placed at different depths. We consider a domain of homogeneous soil without
buried objects. In this domain, the three-dimensional equation (1), with the given temperature on the domain
boundary, can be simplified to a one-dimensional problem describing the soil diurnal temperature distribution
in depth:
 ∂T
∂2T

 ∂t = α ∂x2 , a ≤ x ≤ b, 0 ≤ t ≤ te ,

T (x, 0) = g(x), a ≤ x ≤ b,
(9)
T (a, t) = fa (t), 0 ≤ t ≤ te ,



T (b, t) = fb (t), 0 ≤ t ≤ te ,
where α is a constant (the soil is considered homogeneous), and T = T (x, t; α) is the soil temperature at depth
x and time t. The functions fa (t) and fb (t) are temperature profiles at given depths a and b, respectively, while
g(x) is the initial distribution of the soil temperature in depth.
Consistent estimates of the coefficient α is obtained by least squares minimization, i.e., find the coefficient α
such that the difference between the simulated solution of (9) and the measured data is minimal:
1
min
α>0 2
Zte
[T (c, t; α) − fc (t)]2 dt,
(10)
0
where T (x, t; α) is the solution of the Dirichlet problem (9). For which, (i) fa and fb are approximated from
the measured data, (ii) the initial condition g is approximated by interpolating the measured temperatures
at different depths at time t = 0, and (iii) c, (a < c < b), a depth for which soil temperature profile fc is
measured. For solving the inverse problem (10), we refer the reader to the paper of Kravaris and Seinfeld12
and the references therein for different numerical methods. For the forward model (9) we used the well-known
Crank-Nicolson scheme17 for heat equations.
Figure 2 illustrates the obtained results using data acquired during minefield trials7. Figure 2 (left) shows the
measured soil temperature profiles at different depths. Figure 2 (right) shows the error evolution when solving
equation (9); the minimum is obtained with α = 5.3 × 10−7 (m2 /s). Figure 2 (middle) shows a good agreement
between the measured and the simulated (using the estimated soil thermal diffusivity) soil temperature.
Measured soil temperature and simulation at diffusivity = 5.3e−007(m2/s)
Soil temperature at different depths
20
2
measured at 3cm
simulated at 3cm
measured at 8cm
simulated at 8cm
measured at 18cm
simulated at 18cm
14
12
10
5
1.8
1.6
1.4
10
Error (0C)
Soil temperature (oC)
15
Soil temperature (0C)
Evolution of error of soil diffusivity estimation
16
Surface
3cm
8cm
18cm
48cm
8
1.2
1
0.8
6
0.6
4
0
0.4
2
−5
0
5
10
15
20
Time (hour)
25
30
35
40
0
0.2
0
5
10
15
20
25
Time (hour)
30
35
40
0
1
2
3
4
5
6
Soil diffusivity (m2/s)
7
8
9
10
−7
x 10
Figure 2. Soil thermal diffusivity estimation; Left: Measured soil temperature profile; Middle: Measured soil temperatures
versus simulated soil temperatures; Right: Error Evolution
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 6
4. BOUNDARY PARAMETERS ESTIMATION
As mentioned in Section 2.2, the boundary conditions (3)-(5) are estimated using soil temperature data at
different depths and locations (see Figure 4 (left)), and meteorological data, namely, air temperature, solar
irradiance, sky irradiance, and wind speed (Figure 3), measured near the suspected minefield during the thermal
IR image acquisition7.
Air temperature
Solar and sky irradiance on the earth surface
288
4.5
Solar irradiance
Sky irradiance
600
286
4
3.5
Irradiance (W/m2)
282
280
Wind speed (m/s)
500
284
Air temperature (C)
Wind speed
700
400
300
200
3
2.5
2
1.5
278
100
276
0
274
−100
8
10
12
14
16
18
Time (hour)
20
22
24
1
0.5
8
10
12
14
16
18
Time (hour)
20
22
24
0
8
10
12
14
16
18
Time (hour)
20
22
24
Figure 3. Measured meteorological data by the weather station; Left: Air temperature; Middle: Solar and sky irradiances;
Right: Wind speed
Assuming that the soil is homogeneous, the initial condition (3) could be approximated from the measured
soil temperatures. Figure 4 (left), shows the approximated initial soil temperature for both linear and fourth
order polynomial interpolations of the data of Figure 2 (left).
The sufficient deep depth condition (4), can be estimated from the measured soil temperature at a depth
where the temperature is invariant for a diurnal cycle, and it is not effected by the presence of buried objects.
This condition is reasonable as we are dealing with the detection of shallowly buried objects. From Figure 2
(left) one can notice that the soil temperature at the depth of 48 cm is almost constant over time.
The surface heat flux (5) can be approximated as follows11, 16, 21, 25
qnet (x, y, t) = qsun (x, y, t) + qsky (x, y, t) + qconv (x, y, t) − qemis (x, y, t),
(11)
where
• qsun = ²sun Esun is the solar irradiance absorbed by the soil; ²sun is the solar absorption coefficient or solar
absorptivity of the soil, and Esun (W/m2 ) the solar irradiance on the earth surface. In practice, Esun is
either measured (see Figure 3(middle)) or approximated using equation (12)25
Esun (t) = S0 (1 − C)H(t),
(12)
with S0 = 1353(W/m2 ) the solar constant, C is a factor that accounts for the reduction in the solar
irradiation due to cloud cover, and H(t) a local insolation function. In our approach, we use the measured
solar irradiance.
• qsky = ²sky Esky is the sky irradiance absorbed by the soil; ²sky is the sky absorption coefficient or sky
absorptivity of the soil, and Esky (W/m2 ) the sky irradiance on the earth surface. Esky can be formulated
by Stefan-Boltzmann’s law9, 21
4
Esky = σTsky
,
(13)
with Tsky the sky temperature, and σ the Stefan-Boltzman constant (σ = 5.67 × 10−8 W/(m2 K4 )). The
sky temperature can be approximated by21
√
Tsky = Tair (0.61 + 0.05 ω)0.25 ,
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 7
with ω the water vapour pressure in the atmosphere, expressed in mmHg, and Tair the air temperature.
Combining the two above equations, we get:
√
4
4
Esky = σTair
(0.61 + 0.05 ω) ≈ 0.61σTair
.
(14)
The sky irradiance can also be measured as shown in Figure 3 (middle), from which, one can notice the
correlation between the solar and sky irradiances.
• qconv (W/m2 ) is the heat transfer, by convection, between the soil and the air. It refers to the transport of
heat between the surface and the atmosphere by motion of the air. In 1701, Isaac Newton considered the
convective process and suggested that the cooling would be such that13
dTs
∼ (Ts − Tair ).
dt
So the net heat flux by convection can be written as,
qconv = h(Tair − Tsoil ),
where h is the convective heat transfer coefficient; it depends on the wind speed. In practice, Kahle10
approximated it by
qconv (x, y, t) = ρa cpa Cd (W (t) + 2)(Tair (t) − Tsoil (x, y, t)),
(15)
with ρa the density of the air, cpa the specific heat of the air, Cd the wind drag coefficient (normally chosen
to be 0.002) and W (t) the wind speed.
• qemis (W/m2 ) is the thermal emittance of the soil. It corresponds to the thermal radiation emitted by the
soil:
4
qemis = ²soil σTsoil
,
(16)
where ²soil is the thermal emissivity of the soil, and Tsoil the temperature at the soil surface. From
Kirchoff’s Law9, the thermal emissivity of the soil is equal to the sky absorptivity ²sky .
In summary, equation (11) can be rewritten as:
qnet (x, y, t) = ²sun Esun (t) + ²conv (W (t) + 2)(Tair (t) − Tsoil (x, y, t))
4
+ ²sky Esky (t) − ²soil σ Tsoil
(x, y, t)
(17)
with, Tair (t), Esun (t), Esky (t), and W (t) measured, with the help of a weather station, as illustrated in Figure
3.
To estimate the the boundary condition (5), using equation (17), one should estimate the unknown coefficients
²sun , ²sky , ²conv and ²soil . This is done by solving the following minimization problem
min
²sun ,²sky ,²conv ,²soil
1
2
Zte
|−
0
−
∂T
²sun
²conv
(t) −
Esun (t) −
(W (t) + 2)(Tair (t) − Tsoil (x, y, t))
∂z
k
k
(18)
²soil
²sky
4
Esky (t) +
σ Tsoil
(x, y, t)|2 dt.
k
k
where − ∂T
∂z (t) is an approximation of the heat flux in the bare soil area, calculated as the derivative of the
solution of the problem (9) using the estimated soil diffusivity (equation (10)). Note that, from equation (5),
− ∂T
∂z (t) = qnet (t)/k, k being the unknown soil conductivity, so in (18), we estimate ²sun /k, ²sky /k, ²conv /k
and ²soil /k instead of the coefficients ²sun , ²sky , ²conv and ²soil . Figure 4 (right) shows the approximation
of the surface heat flux. Solving the above minimization problem, we obtain ²sun /k = 0.4574 (m K/W),
²sky /k = 0.5416 (m K/W), ²soil /k = 0.6725 (m K/W), ²conv /k = 1.8634 (m K/W).
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 8
Approximation of initial condition
Incoming heat flux on the soil surface
18
150
4th poly. inter.
Linear inter.
Data
16
From simulation
With estimated parameters
100
12
Heat flux (W/m2)
Soil temperature (0C)
14
10
8
6
50
0
−50
4
−100
2
0
0
0.1
0.2
0.3
0.4
−150
0.5
8
10
12
14
Depth (m)
16
18
Time (hour)
20
22
24
Figure 4. Initial and boundary conditions estimation; Left: initial temperature condition; Right: Approximated surface
heat flux
Detected anomaly
An IR image at 0:5
An IR image at 9:58
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
35
35
35
40
40
5
10
15
20
25
30
35
40
5
10
15
20
25
30
35
5
10
15
20
25
30
35
Figure 5. IR images of a suspected target; Left: Detected anomaly; Middle: Image acquired at 0:05 , Right: Image
acquired at 9:58
5. NUMERICAL RESULTS AND COMPARISON TO EXPERIMENTS
In this section, we present some numerical results of the forward model (2)–(6) and the inverse problem (8),
using data acquired in a real minefield within the United Nations Buffer Zone in Cyprus. The operated stand-off
thermal IR survey system, the acquired data, and the image processing steps, are described in Cremer et al.7.
However, to help the reader understanding the full process of diurnal thermal IR landmines detection using
the developed system, we outline here after the main system components and data processing approaches. The
system allows (i) a stand-off observation of hazardous area, (ii) acquiring thermal IR data during a diurnal cycle,
(iii) measuring meteorological data (solar irradiance, sky irradiance, air temperature, etc.), and soil temperatures
at different depths, (iv) estimating the soil thermal diffusivity as described in Section 3, and the meteorological
coefficients, ²sun , ²sky , ²conv and ²soil , as described in Section 4, (v) temporally calibrating and co-registering the
IR images, (vi) detecting anomalies, i.e. locating and searching for targets which are thermally and spectrally
distinct from their surroundings; Figure 5 shows two IR images of a suspected target and the results of the
anomaly detection procedure, and (vii) finally estimating the depth of burial and thermal diffusivity of the
detected anomaly.
In the following we illustrate the forward model (2)–(6) and the inverse problem (8), using the data of Figure
2(left), Figure 3 and Figure 5, the estimated soil diffusivity α = 5.3 × 10−7 (m2 /s) (see Section 3), the initial
condition as approximated in Figure 4 (left), and the meteorological coefficients ²sun /k = 0.4574 (m K/W),
²sky /k = 0.5416 (m K/W), ²soil /k = 0.6725 (m K/W), ²conv /k = 1.8634 (m K/W), as estimated in Section 4.
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 9
The forward thermal model (2)–(6), has been solved using a simple explicit finite difference scheme as described in Ozisik18, page 420. The numerical result of the simulation for the NR22C1, anti-personnel plastic,
mine buried at 1 cm deep is plotted in Figure 6. In this simulation we assumed the mine as a TNT object
with thermal diffusivity of 1.139 × 10−7 (m2 /s). Figure 6 (left) depicts the simulated soil surface temperatures
above the mine and in the surrounding area, and the measured soil surface temperature. The figure shows a
good agreement between the simulation and the measured soil surface temperature. Figure 6 (right) shows the
thermal contrast between the soil temperature above the mine and the surrounding area. As it can be seen,
during sun shine, the soil above the mine was hotter than the surrounding area and it became cooler during
the night. Moreover, the highest thermal contrast took place around noon. This result can be explained from
the fact that, when the sun was shining, there was an incoming heat flux through the soil surface. In this case,
the mine blocked the heat flow from the surface due to its small thermal conductivity or thermal diffusivity.
Consequently, the soil above the mine became hotter than the surrounding area. During the night, there was
an outgoing heat flow from the bottom of the soil volume. The mine blocked the heat flow resulting in cooler
temperature above the mine than the surrounding area. The same results have been obtained in the previous
works of Sendur and Baertlein21, Khanafer and Vafai11 and Lopez et al.16.
Evolution of soil temperature
Temperature contrast between mine and bare soil on the surface
20
2
Simulation at bare soil area
Simulation on the top of mine
Measured soil temperature
1.5
Temperature contrast (oC)
Temperature (oC)
15
10
5
1
0.5
0
−0.5
0
−1
−5
5
10
15
20
Time (hour)
25
30
35
−1.5
5
10
15
20
Time (hour)
25
30
35
Figure 6. Forward model; Left: Simulated and measured soil temperatures on the surface; Right: Thermal contrast
The quality of the acquired images has been analysed before solving the inverse problem. As it can be seen
from Figure 5 (middle), Figure 5 (right), and the size of the detected anomaly of Figure 5 (left), the temporal
co-registration is not perfect. This explains the irregular in shape of the measured contrast of Figure 7 (right).
However, from Figure 7 (left), depicting the measured soil surface temperature versus the apparent thermal IR
temperature, one can notice that the two temperatures follow the same behaviour with a small shift. Figure 7
(right) shows the simulated thermal contrast, considering that the suspected target is a NR22C1 mine buried at
3 cm, versus the measured thermal contrast. As it can be seen, during the night, with reasonable co-registrated
images, the measured and simulated thermal contrasts are similar in shape.
For solving the inverse problem, we used an early work14, 15 to estimate the depth of burial and the thermal
diffusivity of the detected object of Figure 5. The inputs to the inverse problem are the acquired thermal IR
images, and the spatial position (center) and radius of the object, estimated from the image of Figure 5 (left).
Figure 8 illustrates the error evolution of the thermal diffusivity and depth of burial estimation, respectively. As
it can be seen, the depth is estimated at 6 cm and thermal diffusivity 2 × 10−7 (m2 /s).
6. CONCLUSIONS
In this paper, we presented the full process for landmine detection using thermal modelling. The difficulty
in estimating physical and meteorological parameters of the soil has been solved. The forward model, using
these estimated parameters, showed a good agreement with the measured data. The inverse problem has been
Preprint Proc. SPIE Vol. 5794, Det. and Rem. Techn. for Mines and Minelike Targets X, Orlando FL, USA, Mar. 2005 10
Measured real and apparent temperature
Thermal behaviour of the soil temperature measured by an IR camera
20
Thermal contrast on the soil surface
20
2
Real temperature
Apparent temperature
Above the mine
Bare soil
Simulated
Measured
1.5
15
15
10
5
Thermal contrast (0C)
Soil temperature (0C)
Soil temperature (0C)
1
10
5
0.5
0
−0.5
0
0
−1
−5
5
10
15
20
Time (hour)
1
6
11
−5
5
10
15
20
Time (hour)
1
6
−1.5
5
11
10
15
20
Time (hour)
1
6
11
Figure 7. Image quality analysis; left: measured soil surface temperature versus the apparent thermal IR temperature,
Middle: Apparent thermal IR behaviour; Right: Thermal IR contrast versus simulated contrast
Evolution of the eror of thermal diffusivity estimation
Evolution of the eror of depth estimation
0.62
1.05
1
0.61
0.95
0.9
0.59
Error (0C)
Error (0C)
0.6
0.58
0.85
0.8
0.75
0.7
0.57
0.65
0.56
0.6
0.55
0
5
10
Iteration
15
20
0.55
0
5
10
15
Depth (cm)
Figure 8. Evolution of error of depth and thermal diffusivity estimation; Left: Error of diffusivity estimation; Right:
Error of depth of burial estimation
solved using a the model presented in early works14–16, where the location and radius of the buried object are
considered known using several anomaly detection procedures7, 15, 16 we developed. Future work consists in
developing efficient numerical methods for solving the forward problem and considering the full inverse problem
i.e. augmenting the estimation for both the physical parameters and the location of the buried objects.
ACKNOWLEDGEMENTS
This work has been partly funded by the European Commision, under the CLEARFAST project IST-2000-25173.
The authors would like to thank the other project partners, namely, TAMAM, RLS, and BACTEC. Furthermore
we would like to thank the UN (UMAS, UNDP), the MACC and Armour Group for their support and for giving
us the opportunity to perform the field trial in Cyprus.
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