Subsurface Imaging with
Ground Penetrating Radar
Carey M. Rappaport
CenSSIS
Dept. Elect. and Comp. Engineering
Northeastern University
April 2011
© Carey Rappaport 2011
Propagation Characteristics in Real
Soil
•Concepts of dielectric constant, electrical
conductivity
•Velocity, attenuation, dispersion, reflection
and refraction at interfaces
•Moisture and density dependence
•Nonmetallic target scattering in lossy media
•Rough surface effects
Wave and Helmholtz Equation:
Lossy Media (Soil, Water, Tissue)
The electric field for a wave traveling in linear, homogeneous,
non-dispersive, and lossy medium is given by:
 2E -   E/  t -   2E/  t2 = 0
 = conductivity (S/m), ranging from ~ 0 to 107
For time harmonic wave, the Helmholtz Equation remains:
2E + k2E=0
But the dispersion relation is modified by :
k =   [00 ’(1 - j tan)] =  - j 
With Loss Tangent defined by:
tan =  / ( ’0)
Electromagnetic Waves in Lossy
Media
k    j       '  0 1  j
Propagation (Wave) Number
Slightly lossy medium   '  0  1
Very lossy medium
    '  0
    /  ' 0 / 2
 ' / c
 0 / 2  '
  '  0  1      / 2
Velocity
v   / ,   2  ,
Impedance
=

Slightly lossy medium
Very lossy medium

 '  0





 '0
1
1   j '  0

 

1  j

 '0 
2 '  0 

skin depth  2 / 

(1 j)
2

1
  0
j tan  
'
Propagation in Soil is
Frequency Dependent
Frequency f (1 MHz – 10 GHz)
Dielectric constant ’ (1 – 25)
Electrical conductivity  (0.0001— 1)
Wave Number, k (meters-1)
E ( x, t )  e  j[  ( f ) j ( f )] x e j 2ft
 2 Ex
  2 Ex
2
z
Exact derivation of Wave Numbers
in Lossy Media
Starting from scalar Helmholtz Eqn.
 2U
2

k
U  0, U  Ex , or E y , or Ez
2
z
where the complex wave number is:

  2
2
2
  2  ' 1  j tan  
k    0 0   ' j
 0  c

Separate into real and imaginary components (k =  – j )
 2  2 
2
c
2
2 
2  2
c  0
'
Solve for the quadratic equations for  and 
2
 

  
  ' 

  1 
 
1  

c 2 
  '  0 



1
2
2
 

  
  ' 

  1 
 
1  

c 2 
  '  0 



1
2
Decibel Scale
The decibel (dB) is a logarithmic transformation of ratios of amplitudes or powers.
A power ratio R corresponds to r = 10log10R (dB). An amplitude ratio R
corresponds to 20log10R (dB).
1/10 power
1/2 power
 10log10(1/10)
 10log10(1/2)
1/10 amplitude
1/2 amplitude
= -10 dB.
= -3 dB.
 20log10(1/10) = -20 dB.
 20log10(1/2) = -6 dB.
An intensity attenuation by a factor exp(-a) is equivalent to -4.3a dB .
The decibel changes multiplication into addition
When a wave is transmitted through a cascade of two media resulting in intensity
reduction by factors R1 and R2, the overall reduction is a factor R = R1R2.
The change in dB units is r = r1+ r2.
If the rate of attenuation of a medium is a dB/m, a distance z (m), corresponds to
attenuation of az (dB).
Courtesy of B. Saleh, BU
Logarithms Without Calculators
•
•
•
•
•
•
•
•
Log 10 = 1.0
Log 1 = 0
Log 2 ~ 0.3
Log 5 = Log 10/2 = Log 10 – Log 2 = 0.7
Log 3 ~ Log 101/2 = ½ Log 10 = 0.5Log 4 = Log 22 = 2 Log 2 = 0.6
Log 6 = Log (2 X 3) = Log 2 + Log 3 = 0.8
Log 8 = Log 23 = 3 Log 2 = 0.9
Log10 e = 1/ Loge 10 = 1/2.302
Penetration Depth v. Frequency
for Various Dielectric Materials
Penetration
Depth d10
(19%)
(26%)
= Distance for
the power to
drop by a factor
of 10 (—10 dB)
Wavelengths for Various Dielectric
Materials
Wavelength:
= 2/
Fields for Different Soil Types
f =2.5 GHz
1
Dry Sand
0
-1
10
5
10
15
20
YPG
0
-1
10
5
10
15
20
5
10
15
20
0
-1
10
A.P. Hill
0
-1
10
5
10
15
20
5
10
Distance (cm)
15
20
0
-1
Saturated
Sand
0
Bosnian
(Alicia); 25%
moisture
Exercise: Microwave Penetration in Soil
Determine the loss in dB for a wave at 300
GHz penetrating 1.0 mm into uniform soil
and then reflecting back out for a) Yuma
and b) AP Hill Soil
Hint: Extrapolate the loss curves from
previous slide.
Extrapolated Penetration Depths
at 300 GHz (Terahertz range)
Return signal power (in dB) from a radar source incident on a
metallic target buried a depth D in lossy soil:
-20 D/d
d=Penetration Depth
Radar Return (dB)
(D = 1 mm)
Yuma PG
55.7 cm
-0.036
Dry Sand
4.57 cm
-0.44
Wet Sand
0.31 cm
-6.5
Bosnian soil
54.3 m
-368
A P Hill
40.0 m
-500
Soil Type
Wire on Flat Ground:
Bosnian Soil 26% Moisture
E-field parallel
to wire
H-field parallel
to wire
Difference
Wire on Rough Ground:
Bosnian Soil 26% Moisture
E-field parallel
to wire (Ez)
(Ez) no wire
Modeling Soil Media for
Electromagnetic Wave Propagation
• Type of models
• Simulated wave response
Summary of Dielectric Mixing Models
Source: Kansas Geological Survey, 2001
Category
Method
Types
Advantages
Disadvantages
References
Phenomenological
Relate frequency
dependent behavior
to characteristic
relaxation times.
Cole-Cole; Debye,
Lorentz
- Component
properties/geometry
relationships
unnecessary
- Dependent on frequencyspecific parameters.
Powers, 1997;
Ulaby 1986;
Wang, 1980.
Volumetric
Relate bulk
dielectric properties
of a mixture to the
dielectric properties
of its constituents.
ComplexRefractive
Index (CRI);
Arithmetic average;
Harmonic average;
LicheteneckerRother;
- Volumetric data
relatively easy to
obtain.
- Do not account for microgeometry of components,
-Do not account for
electrochemical interaction
between components.
Alharthi 1987;
Birchak 1974;
Knoll, 1996;
Lange, 1983;
Lichtenecker
1931; Roth 1990;
Wharton 1980.
Empirical
and Semiempirical
Mathematical
relationship
between dielectric
and other
measurable
properties.
Logarithmic;
Polynomial.
- Easy to develop
quantitative
relationships,
-Able to handle
complex materials in
models.
- No physical justification for the
relationship,
-Valid only for the specific data
used to develop the relation may
not be applicable to other data
sets.
Dobson 1985;
Olhoeft 1975;
Topp 1980; Wang
1980.
Effective
medium
Compute dielectric
properties by
successive
substitutions.
Bruggeman-HanaiSen (BHS)
- Accurate for known
geometries.
- Cumbersome to implement,
- Must choose number of inputs,
initial material, and order and
shape of replacement material.
Sen 1981; Ulaby
1986.
Fourier Transform
• Short pulse in time
transforms into
broadband frequency
signal

Y( f ) 
 y(t ) exp( j 2 f t ) dt


y (t )   Y ( f ) exp(  j 2 f t ) df

t
• Long pulse in time
transforms into narrow
frequency signal
1/t
t
f
Temporal Dispersion
• Pulses in time are composed of many frequencies (Fourier
relationship)
• Most real material has frequency-dependent dielectric
parameters
• If material has constant loss, it is strongly dispersive    0  ' j /  0 
• Each frequency component travels at a different velocity
and with a different decay rate
• Amplitude of each frequency component lessens by a
different amount with distance
• Because of dispersion, multiplication in frequency domain
becomes temporal convolution in the time domain
D( )   ( ) E ( )
D(t )   (t ) * E (t )
Dispersion of a Pulse
3 Fourier Components of Pulse at t0
Same components at t>t0
• Each component travels at a different
velocity (dispersion)
• Amplitude of component lessens in
time (loss)
Modeling Dispersion for Easy
Transformation to Time Domain
Standard (2nd Order) Debye Model: simple form for complex permittivity,
easily transformed to time domain differential equation
N
   0 '  0 v
p 1
For
 2   and  0 A2 /  2  
   0 '  0
Ap
1  j p
N=2
A1


1  j 1 j 0
Lorentz Model: 2nd order when N = 1
N
   0 '  0 
p 1
Ap
 2   2  2 j p
0
[Cole-Cole Model is more accurate, not easily converted to time domain]
   0 '  0
 's  '

1   j 
Conversion of Dispersion Models
to Time Domain
Replace  by D/E and multiply through by denominator
D j 0 1  j1   E j 0 1  j1  0 '  0  j 0 A1  1  j1  
Convert to time domain with
Debye
j 

t
 2 D D
2E 
  E 
1 2 
  0 ' 1 2   0 '  0 A1  1 
 E
t
t
t
 0  t  0





D  2   20  2 j p  E  2   20  2 j p  0 '  0 A1
Convert to time domain with
j 


t


2
2D
D

E
E
2
2

2



D



'

2



'



'

 A1 E
p
0

p
0

0

2
2
0
0
t
t
t
Lorentz t
Modeling Dispersion for Easy
Transformation to Time Domain
Z-Transform model keeps real permittivity constant, and matches
conductivity to measured values in terms of Z-1 [4 Zero Model]
’ = Constant,
  Z 1/ 2
b0  b1 Z 1  b2 Z 2  b3 Z 3
1  a1 Z 1
,Z = e jt
Since Z-1 transforms to unit time delay, application to FDTD is simple
D( Z )   Av E ( Z )
J (Z )   (Z ) E (Z )
D(t )   Av E (t )
t
3t
J (t  )  a1 J (t 
)
2
2
b0 E (t )  b1 E (t  t )  b2 E (t  2t )  b3 E (t  3t )
Dielectric Constant and Conductivity for
Puerto Rican Clay Loam (1.2 g/cc)
-1
9
10
Rappaport
Debye
data
8
-2
7
5%
10
10%
6
2.5%

’
10%
5%
-3
5
10
4
3
2.5%
-4
0
500
Frequency (MHz)
1000
10
0
500
Frequency (MHz)
1000
Real and Imaginary Wave Number for
Puerto Rican Clay Loam (1.2g/cc)
60
0
Rappaport
Debye
data
50
10%
-0.5
2.5%
5%
- (1/m)
-1
(1/m)
40

5%
-1.5
30
2.5%
20
-2
10
-2.5
0
7
7.5
8
8.5
9
Log Frequency
9.5
-3
7
10%
7.5
8
8.5
9
Log Frequency
9.5
Wave Propagation Variation as a
Function of Clay Loam Moisture
Rough Surface Test Geometry
40
Transmitter
30
Receiver
Depth (cm)
20
10
0
-10
-20
Non-Metallic Mine
-30
-40
60
80
100
120
140
Transverse Position (cm)
160
180
Non-Metallic Mine Scattered Field 10 cm
Deep - Smooth Surface
0.4
-------
++++++
oooooo
xxxxxx
Relative Amplitude
0.3
Air
Dry Sand
Non-Dispersive Loam 20% moisture
Dispersive Loam 20% moisture
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
1000
2000
3000
4000
5000
6000
Time (ps)
7000
8000
9000
10000
Non-Metallic Mine Scattered Field (about
10 cm burial) - Rough Surface
0.4
-------
++++++
oooooo
xxxxxx
Relative Amplitude
0.3
Air
Dry Sand
Non-Dispersive Loam 20% moisture
Dispersive Loam 20% moisture
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
1000
2000
3000
4000
5000
6000
Time (ps)
7000
8000
9000
10000
Relative Amplitude
Non-Metallic Mine Scattered Field 10 cm depth
a) Flat Surface, b) Rough Surface
0.1
0.05
0
Relative Amplitude
-0.05
-0.1
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
7000
8000
9000
10000
0.06
Non-Dispersive Loam 20% moisture
Dispersive Loam 20% moisture
0.04
0.02
0
-0.02
-0.04
0
1000
2000
3000
4000
5000
6000
Time (ps)
Shape Determination of Buried Non-Metallic
Targets, Multiple Single-Frequency Observations
Sandy soil: s = 2.5, s = 0.01
Target: m = 2.9, m = 0.004
Circular Target
Air
Air
20 cm
d
11.28 cm
Soil
Square Target
20 cm
d
10 cm
Soil
10 cm
60 cm
80 cm
60 cm
80 cm
Different Buried Test Target Shapes
Height (cm)
Square
Circle
Diamond
0
0
0
-5
-5
-5
-10
-10
-10
-15
-15
-15
-20
-10
-20
-10
-20
-10
-5
0
5
10
Star
0
-5
-10
-10
-15
-15
-5
0
5
10
-20
-10
0
5
10
5
10
Blob
0
-5
-20
-10
-5
-5
0
Horizontal Position (cm)
-5
0
5
10
Scattered Field - Real Part
Height (cm)
Square
Circle
Diamond
20
20
20
0
0
0
-20
-20
-20
-40
-40
-40
-60
-40
-20
0
Star
20
40
-60
-40
-20
0
Blob
20
40
-60
-40
0.04
0
0
-20
-20
-40
-40
0.02
0
-0.02
-0.04
-60
-40
-20
0
20
40
-60
-40
-20
0
0
20
40
0.06
20
20
-20
20
40
Horizontal Position (cm)
-0.06
500 MHz,
depth = 5 cm
Scattered Field - Real Part
Square
Height (cm)
20
Circle
20
20
0
0
0
-20
-20
-20
-40
-40
-40
-60
-40
-20
0
20
40
Star
20
-60
-40
-20
0
20
Diamond
40
-60
-40
-20
-20
0
-40
-40
-0.1
0
20
40
-60
-40
20
40
0.2
0
-20
0
Blob
20
0
-60
-40
-20
0.1
-20
0
20
40
Horizontal Position (cm)
-0.2
1000 MHz,
depth = 5 cm
Surface Field - Magnitude
500 MHz, depth = 5cm
0.04
1000 MHz, depth = 5cm
0.05
square
circle
0.04
0.035
diamond
Intensity
star
blob
0.03
square
0.03
circle
0.02
diamond
0.025
0.01
star
blob
0.02
-40
0
-20
0
20
40
-40
-20
Horizontal Position (cm)
0
20
40
Scattered Field - Aspect Ratio Dependence
Height (cm)
20
0
Circle, r = 5.64 cm
10 x 10 cm
20
-20
0
-20
-40
-40
-60
-40 -20 0 20 40
-60
-40 -20 0
20
7.5 x 13.3 cm
20 40
5 x 20 cm
20
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
20
0
0
0
-20
-20
-20
-40
-40
-40
-60
-40 -20 0 20 40
-60
-40 -20 0
20
20
13.3 x 7.5 cm
20 x 5 cm
20 40
0
0
0
-20
-20
-20
-40
-40
-40
-60
-40 -20 0
20 40
 = 2.5,  = 0.01
freq = 500 MHz
depth = 5 cm
2.5 x 40 cm
-60
-40 -20 0
20
Sandy Soil
20 40
40 x 2.5 cm
-60
-60
-40 -20 0 20 40
-40 -20 0
Horizontal Position (cm)
20 40
Distinguishing Shapes of 3D Buried Objects
under Rough Surfaces: Geometry
Point Source
10 cm
Rough Surface
5 cm
Soil
Mine
4 cm
10 cm
Total Ex Field from an x-Directed Point Source,
with a Buried Non-Metallic Square Target
Total Ex Field from x-Directed Point Source, with
a Buried Non-Metallic Square Target (back view)
Comparison of Total Ex Field for Buried NonMetallic Square and Circular Targets
Comparison of Scattered Ex Field for Buried
Non-Metallic Square and Circular Targets
Soil Packing Affects Greatly Scattering:
3D FDFD with Short Cylindrical Target
Relative Height 30
TNT in 26% moist Bosnian soil at 960 MHz
Surface Scattering Clutter Increases with Frequency.
Example: 4 GPR Freq., PRCL 10% moisture, 1.4 g/cc
-5
Depth (cm)
0
Air
5
10
15
Soil
Non-Metallic Target
20
25
30
35
-20
-15
-10
-5
0
5
10
15
Transverse Position (cm)
20
Display Format for each of
Four Frequencies
Mine scattered field:
smooth surface
Scattered field:
rough surface with mine
Scattered field:
rough surface only
Mine scattered field:
rough surface
480 MHz
Depth (cm)
0.1
10
0
20
-0.1
30
-0.2
-20
-10
0
10
20
Scattered field: rough surface with mine
0.2
0
0.1
10
0
20
-0.1
30
-0.2
-20
-10
0
10
20
Transverse Position (cm)
Transverse Position (cm)
Scattered field: rough surface only
Mine scattered field: rough surface
10
0
20
-0.1
30
-0.2
-20
-10
0
10
20
Transverse Position (cm)
0
Depth (cm)
Depth (cm)
0.1
0.2
0.1
10
0
20
-0.1
30
-0.2
-20
-10
0
10
20
Transverse Position (cm)
Amplitude Relative to Incident
0
Amplitude Relative to Incident
0.2
Amplitude Relative to Incident
0
Amplitude Relative to Incident
0.2
Depth (cm)
Mine scattered field: smooth surface
960 MHz
Scattered field: rough surface with mine
0.1
10
0
20
-0.1
30
-20
-10
0
10
20
-0.2
0
Depth (cm)
Depth (cm)
0
0.2
0.1
10
0
20
-0.1
30
-20
Transverse Position (cm)
10
0
20
-0.1
30
10
20
Transverse Position (cm)
20
-0.2
-0.2
Mine scattered field: rough surface
0.2
0
Depth (cm)
Depth (cm)
0.1
0
10
0.1
10
0
20
-0.1
30
-20
-10
0
10
20
Transverse Position (cm)
-0.2
Amplitude Relative to Incident
0
Amplitude Relative to Incident
0.2
-10
0
Transverse Position (cm)
Scattered field: rough surface only
-20
-10
Amplitude Relative to Incident
0.2
Amplitude Relative to Incident
Mine scattered field: smooth
surface
1920 MHz
Mine scattered field: smooth surface Scattered field: rough surface with mine
10
0
20
0
Depth (cm)
Depth (cm)
0.05
0.1
0.05
10
0
20
-0.05
30
-10
0
10
20
-0.1
30
-20
Transverse Position (cm)
-10
0
10
20
0.05
10
0
20
Mine scattered field: rough surface
0.1
0
0.05
10
0
20
-0.05
30
-10
0
10
20
Transverse Position (cm)
-0.1
Amplitude Relative to Incident
0
Amplitude Relative to Incident
0.1
-20
-0.1
Transverse Position (cm)
Scattered field: rough surface only
Depth (cm)
-0.05
Depth (cm)
-20
Amplitude Relative to Incident
0
Amplitude Relative to Incident
0.1
-0.05
30
-20
-10
0
10
20
Transverse Position (cm)
-0.1
3840 MHz
Mine scattered field: smooth surfaceScattered field: rough surface with mine
0
20
30
-20
-10
0
10
20
0
Depth (cm)
Depth (cm)
10
0.05
10
0
20
30
-0.05
-20
Transverse Position (cm)
-10
0
10
20
Mine scattered field: rough surface
0
-20
-10
0
10
20
Transverse Position (cm)
-0.05
0
Depth (cm)
Depth (cm)
10
0.05
10
0
20
30
-20
-10
0
10
20
Transverse Position (cm)
Amplitude Relative to Incident
0
Amplitude Relative to Incident
0.05
30
-0.05
Transverse Position (cm)
Scattered field: rough surface only
20
Amplitude Relative to Incident
0
Amplitude Relative to Incident
0.05
-0.05
Short Pulse GPR Interaction with Rough,
Dispersive Ground / Mine
Air
Soil
PMN-1A Non-Metallic AP Mine
Geometry
From MineFacts, version 1.2, National Ground Intelligence Center
Snapshot of Total Time Domain EField (with Target)
Air
Soil
Snapshot of Background Time
Domain E-Field
Air
Soil
Snapshot of Scattered Time
Domain E-Field (Mine Only)
Air
Mine
Soil
Effect of Rough Ground of Bistatic
GPR Signals
Height (cm)
Transmitter
12
Receiver
-
0
-12
-48
-24
0
24
Signal Amplitude
Rough Ground (cm)
0.5
0.5
Mean
48 Height variation
h= 6cm
Correlation distance
between surface peaks
lc= 15cm
0
0.0
-0.5
-0.5
0
0.0
100
100
200
200
Time Step (t = 20ps)
300
300
Rough Ground Clutter Signal
Characterization
• Signals from rough ground vary considerably
– Pulse shape depends on roughness and TR
position
– Peak depends on particular TR position
– Overall amplitude varies
• Monte Carlo simulation can model following
relevant features
– 2D FDTD model
– Real measured impulse GPR excitation and
dispersive soil
– 500 different rough surface realizations
Monte Carlo Analysis
• Run many simulations
• Vary each run
– Change geometry
– Change signal
• Compute statistics
– Mean values
– Standard deviations
• Conclude “typical” behavior
– Determine likelihood of given test
• Set threshold and count number of occurrences
of detection or false alarm --> ROC curve
Computational Geometry
Transmitter
Receiver
24.5 cm
Z = 28cm
Z=0
Z = depth
mine
soil
L = 294 cm
2
0
-2
-4
Relative Amplitude
4
Impulse Ground Penetrating
Radar Specifications
0
100
200
Time Step (t=20ps)
300
Original Signal Averages
Obscure Mine Signal
No Mine
500 computed signals
Mine
Physics-based Signal Processing
flowchart
Raw Signals
Cross-correlate
with reference
Shift and scale
raw signals and
take average
Shifting
Scaling
Compute different
velocity in soil, shift
to line up the target
feature
Subtract shifted and
scaled average from
each raw signal
Ground Clutter Signal Removal
No Mine
Mine
Realigning Signals to Presumed
Mine Position
No Mine
Mine
Average Mine Scattered Signals
h=3cm
lc=10cm
lc=3cm
h=1cm
ROC Curves for Mismatched
Target Depths
8.5
9.8
h= 1cm lc= 10cm
4.8
3.6
6.1
2.4
Trail depth=8.5cm
test depth= 2.4cm
3.6cm
4.8cm
6.1cm
8.5cm
9.8cm
Soil Moisture Change with Wetting
Water movement in a vertical column of a
medium is described by the advectiondispersion equation in the z-direction, as:
d
d
d
d

( D ( )
)
K ( )
dt
dz
dz
dz
Where:
 = moisture content
z = depth [L]
D = dispersion coefficient of water [L/t2]
K = hydraulic conductivity [ L/t]
t = time [t]
Time Response Due to Saturating
Soil Surface
Moisture Profile
Ksat = 0.2 cm/min
Moisture Content (%)
0.25
0.1 Minute
1 Minute
2 Minutes
3 Minutes
4 Minutes
0.20
0.15
0.10
0.05
0.00
0
10
20
30
40
Depth into Soil (cm)
50
60
Rough Surface with Buried Non Metallic
Mine and Point Source Geometry
-100
-80
-60
Air
-40
Source
-20
Ground Surface
0
20
40
60
-100
Non-Metallic Target
Soil with Varying Moisture Content
-80
-60
-40
-20
0
20
40
Testing geometry
60
80
100
Planar Ground Surface,
5% Uniform Moisture
Planar Ground Surface,
20% Uniform Moisture
Planar Ground Surface,
5 - 20% Moisture Profile
Rough Ground Surface,
5% Uniform Moisture
Rough Ground Surface,
20% Uniform Moisture
Rough Ground Surface,
5 - 20% Moisture Profile
Summary
• Realistic soil media complicates the sensing of
subsurface objects
– Loss affects penetration depth and makes surface clutter
more dominant
– Rough interfaces produce additive uncertain clutter and
distort transmitted signals
– Moisture variations cause huge propagation differences
• Small contrast differences makes detection/
imaging more challenging
• Shapes of underground dielectric object are
hard to distinguish
• Multistatic wideband GPR can provide much
more information than monostatic