Corso di Dottorato in Ingegneria dei Sistemi Elettronici

Ciclo XXIII

Advancements on interferometric radar: studies and applications.

Coordinatore: Prof. Carlo Atzeni

Tutor: Prof. Massimiliano Pieraccini

Candidato: Andrea Barucci

Settore Disciplinare

ING-INF/01

Firenze, 31 Dicembre 2010

List of Publications of the author published during the doctoral course

International journal

1. Barucci, G. Macaluso, D. Mecatti, L. Noferini, D. Fanelli, A. Facchini, M.

Materassi, M. Pieraccini and C. Atzeni, Universal fluctuations in tropospheric

radar measurements, EPL, 89 (2010) 20006, DOI 10.1209/0295-

5075/89/20006

International workshop

1. A. Barucci et al., Universal fluctuations in tropospheric radar measurements –

Geosciences Union General Assembly 2010, Vienna, Austria, 2 - 7 maggio

2010.

2. M. Pieraccini, A. Barucci, D. Mecatti, L. Noferini, G. Macaluso and C.

Atzeni, Attenuation of Large Bandwidth Microwave Signals in Water and Wet

Sand – XIII International Conference on Ground Penetrating Radar, Lecce, 21

- 25 giugno 2010.

3. D. Mecatti, G. Macaluso, A. Barucci, L. Noferini, M. Pieraccini and C.

Atzeni, Monitoring Open-Pit quarries by Interferometric Radar for Safety –

European radar Conference 2010, Parigi, Francia, 30 settembre – 1 ottobre

2010.

Table of Contents CHAPTER1SYNTHETICAPERTURERADAR .......................................................................... 7

1.1 CONTINUOUS WAVE STEPPED-FREQUENCY RADAR ....................................................................71.1.1 Mathematical signal description .................................................................................................. 8

1.2 SAR IMAGING.........................................................................................................................................101.2.1 Synthetic Aperture: the azimuth direction ............................................................................. 111.2.2 SAR projection.................................................................................................................................. 131.2.3 SAR processing................................................................................................................................. 131.2.4 Implementation................................................................................................................................. 14

1.3 SAR INTERFEROMETRY ......................................................................................................................151.3.1 The Interferograms ......................................................................................................................... 161.3.2PhaseUnwrapping......................................................................................................................... 17

1.4 REFERENCES ...........................................................................................................................................21CHAPTER2MONITORINGOPEN­PITQUARRIESBYINTERFEROMETRICRADARFORSAFETYPURPOSES ..............................................................................................................24

2.1 GB-SAR FOR TERRAIN DISPLACEMENT MAPPING ....................................................................242.2 MONITORING AN OPEN PIT QUARRIES BY A GB-SAR ...............................................................252.2.1TheRadarSystem........................................................................................................................... 262.2.2 The Field Survey .............................................................................................................................. 292.2.3 The RCS image ................................................................................................................................. 31

2.3 DATA PROCESSING................................................................................................................................332.3.1 Pixels selection as permanent scatterers ............................................................................... 342.3.2 Atmospheric phase screen............................................................................................................ 372.3.3 Total displacement: cumulative interferogram.................................................................... 38

2.4 RESULTS ...................................................................................................................................................382.4.1 Pixels displacements ...................................................................................................................... 382.4.2 Interferograms .................................................................................................................................. 402.4.3 Simulation of displacement by a Corner Reflector target ............................................... 422.4.4 Digital Elevation Model ............................................................................................................... 43

2.5REFERENCES .............................................................................................................................................47CHAPTER3UNIVERSALFLUCTUATIONSINTROPOSPHERICRADARMEASUREMENT .............................................................................................................................50

3.1 THE EFFECT OF THE ATMOSPHERE ON THE GB-SAR.................................................................503.1.1 The landslide of Formigal............................................................................................................ 51

3.2 RADAR- ATMOSPHERE INTERACTION..............................................................................................533.2.1 Spectrum of the wind...................................................................................................................... 533.2.2 A brief description of turbulence............................................................................................... 553.2.3 Radar echo as a global quantity................................................................................................ 56

3.3 EXPERIMENTAL TESTS .........................................................................................................................573.4 THE EXPERIMENTAL TEST AT THE FACILITY OF THE FACULTY OF ENGINEERING ..........57

3.4.1 Instrumentation ................................................................................................................................ 593.4.2 Results.................................................................................................................................................. 61

3.5 THE EXPERIMENT AT FIRENZUOLA..................................................................................................653.5.1 The location ....................................................................................................................................... 663.5.2 Instumentation .................................................................................................................................. 663.5.3 Data pre-processing ....................................................................................................................... 703.5.4 Data analysis I.................................................................................................................................. 723.5.5 Data analysis II ................................................................................................................................ 773.5.6 Conclusions ....................................................................................................................................... 81

3.6 ACKNOWLEDGEMENTS ........................................................................................................................813.7 REFERENCES ...........................................................................................................................................81

Preface This doctoral thesis was written at the Technologies for Cultural Heritage Laboratory (TCH Lab) which is part of the Electronic Department of the University of Florence. It is a multidisciplinary laboratory in the Information Technology area that is dealing with developing and testing of innovative techniques for conservation and monitoring of Cultural and Environmental Heritage.

This thesis deals with advanced data processing in Interferometric Synthetic Aperture Radar for unstable slope monitoring. Mainly the attention of the author is turned to the data processing algorithms for imaging the meaningful content of the specific radar observations. The particular acquisition system used in each type of observation was developed at the TCH Lab and was based on a Continuous Wave Stepped Frequency radar architecture.

Now GB-SAR interferometry is generally recognized as a powerful tool for detecting the displacements of large structures such as dams or buildings, but also for the terrain monitoring, exactly as the space-born SAR interferometry. But, in respect to space-born systems, for monitoring localized landslide phenomena (of the order of 1 km x 1 km area), GB-SAR seems to be more suitable as it can be specifically set up for a particular scenario geometry and in according to the dynamic characteristics of the landslide.

While conventional interferometric synthetic aperture radar is a very effective technique for measuring surface deformation, almost any interferogram includes large areas where the signals decorrelate and no measurements are possible. Consequently, most SAR studies focuses on areas that are dry and sparsely vegetated. A relatively new analysis technique (PS method) overcomes the decorrelation problem by identifying resolution elements whose echo is dominated by a single scatterer in a series of interferograms.

After an introduction on radar and SAR systems, in this Thesis we focused on the application of GB-SAR to monitor an open pit quarry using the PS method, and on the influence of the atmospheric disturbances, in particular boundary layer turbulence, on ground based radar measurements.

As far as the turbulence disturbances concerns, in this thesis an experimental survey was planned in order to deeply understand how radar signal, and in particular his statistic, can change in presence of such phenomena. The results obtained by using these procedures on real data acquired in a experimental facility are discussed too. The agreement between theoretical and real data confirms the validity of the used statistic, confirming the “Universal distribution” as a powerful tool to understand differend kind of complex phenomena.

Chapter 1 deals with Synthetic aperture techniques for imaging and interferometric applications, while Chapter 2 describes a measurement campaign in an open pit quarry near Florence, and his results are presented and discussed. Chapter 3 presents two experiments devoted to the study of the influence of atmospherical turbulence on radar signals.

Ringraziamenti Durante i tre anni di dottorato ho avuto l’onore di lavorare con persone meravigliose: Alessandro Spinetti, Daniele Mecatti, Devis Dei, Filippo Parrini, Gilberto Grazzini, Giovanni Macaluso, Linhsia Noferini e Matteo Fratini. Le qualità lavorative e morali di questo gruppo di persone sono state e saranno per me un esempio da seguire. A tutti loro va il mio rispetto e la mia amicizia. Daniele mi ha aiutato tantissime volte nel corso di questi anni, progettando e costruendo molta della strumentazione utilizzata per le misure da me svolte, non ho parole per ringraziarlo. Lavorare con lui è stato un piacere ed un divertimento. Con Linhsia ho condiviso la maggioranza del mio tempo, discutendo infinite volte dei problemi che via via ci trovavamo ad affrontare, cercando di capire ed imparare. Linhsia è stata per me un’insegnante ed un’amica impareggiabile, sostenendomi, aiutandomi, sopportandomi e consigliandomi, per questo avrà sempre la mia più grande stima ed il mio più profondo rispetto. A Lei devo moltissimo, senza il suo prezioso contributo niente di quanto descritto in questa tesi sarebbe stato possible. Un ringraziamento al Prof. Carlo Atzeni ed al Prof. Massimiliano Pieraccini che mi hanno dato l’opportunità di lavorare nel Laboratorio di Tecnologie per i Beni Culturali ed Ambientali, dandomi sempre libertà di ricerca ed ampio spazio di lavoro, e ricordandosi sempre che sono un fisico. Infine un ringraziamento alla mia famiglia, che nonostante i tanti problemi che la vita ci porta ad affrontare, mi ha sempre sostenuto ed incoraggiato a fare ciò che mi piaceva. A loro devo tutto.

Chapter 1 Synthetic

Aperture Radar

This chapter provides the basic concepts on synthetic aperture radar imaging and

interferometry (InSAR).

The first part presents the essential characteristics of a continuous wave stepped-

frequency radar with a mathematical description of the transmitting waveform.

Section 2 is dedicated to SAR principles and images generation and interpretation

including issues that impact processing algorithms, while Section 3 provides a

consistent mathematical representation of InSAR principles and phenomenology

associated with interferometric phase measurements.

1.1 Continuous Wave Stepped-Frequency Radar

Since 1960’s in developing pulsed radar systems it was evident that the simple pulsed

radar was not suitable for radar applications requiring ultimate performance in both

range (high transmitted pulse power) and resolution (short pulse width). To overcome

the difficulty of combining these requirements, alternative waveforms were

developed.

Pulse compression waveforms are specially coded to allow the transmission of longer

pulse widths which allow higher mean powers, while retaining wide bandwidths for

high resolution. The system used in this thesis exploits a frequency compression

technique known as stepped frequency technique.

Figure 1-1: Block diagram of the stepped-frequency radar. The transmitted waveform x(t), received waveform y(t) and the complex baseband waveform I(t)+jQ(t) are shown.

The stepped-frequency radar technique transmits a sequence of single frequency tones

or pulses stepped in frequency at intervals of Δf across a bandwidth B. The block

diagram of the stepped-frequency radar is shown in Figure 1-1. Each frequency step

received from reflecting targets is mixed with the transmitted waveform for down-

conversion to baseband by means of a quadrature mixer. The quadrature mixer

produces a baseband signal comprising of two orthogonal components to preserve the

magnitude and phase of the received tone. These components, known as the in-phase I

and in-quadrature Q components, are sampled using a high precision, low frequency

(audio) dual channel analog to digital converter. After I and Q were stored, the next

frequency step is transmitted. Once the radar has completed the steps across the

bandwidth an IDFT (Inverse Discrete Fourier Transform) is applied to the digitised I

and Q frequency domain arrays to produce the synthesised time profiles of the targets.

The IDFT is the digital equivalent of the analog matched filter.

1.1.1 Mathematical signal description

The time waveform x(t) transmitted by a stepped-frequency radar can be represented

as a continuous train of N stepped-frequency pulses. Each pulse in the train has a

width of T in time and a carrier frequency which is stepped at increments of Δf from

pulse to pulse starting from the lower frequency bound f1 to

!

f2 = f1 + N "1( ) # $f .

Generally, there are many cycles of the carrier frequency within each pulse envelope.

It is assumed for a coherent system that the amplitude and phase of each stepped-

frequency pulse in the train is constant for all frequency steps. The switching delay

between frequency steps is assumed negligible.

The stepped-frequency pulse train x(t) is mathematically represented by:

!

x t( ) = Re exp j2" f1 + i # $f( )t( )[ ]i=0

N %1

& rectt % i #T % T /2

T' (

) *

Eq. 1-1

Consider a single point target with a reflection coefficient of

!

A = A exp j"( ) and an

associated delay τ in a lossless medium. It is assumed that the permittivities of the

target and medium are constant with frequency so that A and τ can be treated constant

as well. The reflected signal y(t) from this single point target is represented by:

!

y t( ) = Re Aexp j2" f1 + i # $f( ) # t % &( )( )[ ]i=0

N %1

' # rectt % & % i #T % T /2

T( )

* +

Eq. 1-2

A reference version of the transmitted signal is coupled to the quadrature mixer of the

receiver. The complex baseband signal generated by the mixing process and low-pass

filtering (it is assumed

!

T >> " ) is:

!

I t( ) + j "Q t( ) = Aexp # j2$ f1 + i " %f( )&( )i=0

N #1

' " rectt # i "T # T /2

T( )

* +

Eq. 1-3

The sampling of the baseband signal in Eq. 1-3 produces a complex array C(k) whose

elements correspond each to the value taken by the baseband signal at roughly the

middle of the relative pulse interval:

!

C k( ) = Aexp " j2# f1 + k $ %f( )&( ) with k = 0,1,2,..,N "1 Eq. 1-4

The elements of this array can be viewed as the baseband signal content at the

specified frequencies. In order to reconstruct the typical time profile of the target a

Fourier transformation can be carried out. The limited number of frequencies

considered on the transmitted bandwidth brings with it a limitation on the longest

delay observable by the system after transformation from frequency to time domain.

The support of the Fourier transformed function of Eq. 1-4 is the time interval

!

0,1 "f[ ) , indeed. No target, even if it really is, appear farther than 1/Δf. Aliasing

effects occur when targets so far from the sensor are illuminated by the system.

In the frequency domain, the resulting function can be supposed to approximate the

analytic function:

�

D f( ) = A ⋅ rectf − fc

B⎛

⎝

⎞

⎠ exp − j2πfτ( )

Eq. 1-5

where fc is the centre frequency of the transmitted bandwidth. By applying the inverse

Fourier transform to Eq. 1-5, the time profile of the baseband signal results:

�

d t( ) = FT −1 D f( )[ ] = A ⋅B ⋅ sinc πB t − τ( )( ) ⋅exp j2πfc t − τ( )( ) Eq. 1-6

This function is the synthesised time profile containing the target’s reflection

coefficient and delay time.

The IDFT algorithm is a pulse compression filter which compresses the received time

waveform y(t) to form the high-resolution synthesized time waveform d(t).

Applying the Rayleigh’s criterion to Eq. 1-6 it is obvious that the time resolution of

the stepped-frequency radar is exactly 1/B if the two way path toward and back from

the target is taken into account and the velocity in the medium is known. If there are

multiple targets in the synthesised time profile separated peaks appear in

correspondence of the time bin of those targets that are farther from each another than

the system range resolution.

1.2 SAR Imaging

In synthetic aperture radar (SAR) remote sensing, the region to be imaged is observed

from a sequence of spatial sampling positions and the received signals are combined

using SAR algorithms to obtain high resolution along the antenna track direction. The

generated images generally reproduce the location of a target in a two-dimensional

coordinate system, with one axis along the antenna track direction (“azimuth” or

“cross range” direction) and the other orthogonal to the first axis and pointing toward

the radar line of sight direction (“cross-track” or “range” direction).

1.2.1 Synthetic Aperture: the azimuth direction

Generally a radar sensor resolves targets in the range direction by measuring the time

that a radar pulse takes to propagate to the target and return to the radar. Targets that

differ from each other in their azimuth coordinates only, generate overlapping radar

echoes and thus they cannot be distinguished. Conceptually azimuth location can be

achieved by changing the viewing angle of a very directive antenna and processing

the range information relative to each angle separately. SAR techniques just

accomplish the latter task by globally processing the data provided by an antenna that

moves across a defined series of positions.

The antenna motion, realizing a synthetic antenna array, assures the required antenna

directivity since the system is equivalent to a physical antenna twice the length of the

synthetic array. The main lobe of the physical antenna exactly is two times larger than

that of the synthetic aperture of the same length.

The series of antenna positions along track is geometrically limited by the fact that

from each reached position the antenna aperture is assumed to comprise the area to be

investigated. In practice this assumption provide a down limit to the achievable

azimuth resolution. If an azimuth resolution ΔX is required for a target at distance R

from the center of the rail, the synthesized antenna length L must satisfy:

!

L "#

2 $ %XR Eq. 1-7

where λ is the transmitted wavelength and the factor 2 is due to the round trip from

the radar to the target and back to the radar. The assumption that the physical antenna

aperture effectively comprises the synthesized length from a distance R where the

target is located, means that:

!

L "#

!R Eq. 1-8

where is the azimuth length of the physical employed antenna. By comparing Eq.

1-7 and Eq. 1-8 it results that

!

"X #!

2 Eq. 1-9

i.e. the SAR azimuth resolution cannot be better than half the length of the employed

antenna no matter the distance.

Once the synthetic length L was chosen, the number N of antenna positions along the

track must guarantee a suitable spatial sampling of the received signal. The spatial

sampling period is provided by the step between two successive acquisitions and is

given by

!

d = L N "1( ) . To avoid ambiguity in the measured azimuth profile, the

following equation must be satisfied:

!

d "#

4 $ sin %0( )

Eq. 1-10

where

!

2"0 is the full angular extent of the illuminated scenario. A three-dimensional

view of the resolution cell of a conventional SAR system within the physical antenna

elevation aperture indicated in the figure, is shown Figure 1-2.

Figure 1-2: Three-dimensional view of the resolution cell of a conventional SAR. The red points depicted in the resolution cell cannot be distinguished in SAR images.

1.2.2 SAR projection

Data backscattered by the illuminated scenario are processed in range and cross range

only. The third dimension is lost and the effect on SAR images depends on the

relative geometry of the SAR aperture and the illuminated surface. Then target

locations are distorted relative to a planimetric view and their identifications could be

very difficult. For many applications, this altitude-dependent distortion adversely

affects the interpretation of the imagery. To obtain the three-dimensional information,

an additional measurement of elevation angle is needed. Interferometry using two or

more SAR images separated by a spatial baseline provides a means of determining

this angle.

1.2.3 SAR processing

The antenna employed in the present work is mounted on a ground based linear rail

and moves along the rail one step further after each signal acquisition. The transmitted

waveform is the stepped frequency continuous wave. So the in-phase ( ) and

quadrature (Q) components of the received signal are measured for each frequency

and for each antenna position on the rail.

The implemented SAR processor provides the range and cross-range synthesis

combining the Fourier transform efficiency with the ultimate resolution obtainable

from the received data. The Fourier transform is used only for getting the range

profile from the measurement relative at each antenna position. This is a natural step

and no approximations are needed. The backscattering coefficient of an hypothetic

point scatterer that would be located in a well defined point in the three-dimensional

space within the antenna aperture is estimates first by picking up from the Fourier

transformed data relative to each antenna position the element corresponding to the

distance from the point and the antenna position, then summing all these antenna

contributions each multiplied by a right factor to compensate the phase exactly.

It is worth to outline that, although the SAR processor that will be presented works

with points in the three-dimensional space, the elevation angle has not been resolved

yet. Referring to Figure 1-2 the two point-wise scatterers cannot be distinguished and, if

focused, in the images they appear with the same backscattering coefficient equal to

the sum of their individual coefficient. Anyway the capability of focusing three-

dimensional points is very useful when a DEM (Digital Elevation Model) is available

in order to obtain radar image much more similar to that observed and hence facilitate

image interpretation.

1.2.4 Implementation

Here a detailed description of the used SAR processing is presented. Let

�

Ei,k be the

measured signal at frequency and antenna position on the rail. The value of the

complex radar image I(Pn) at point

!

Pn = xx,yn,zn( ) is obtained by adding all signal

contributions, taking in account their phase history:

!

I(Pn ) =1

n f np

Rnk2 Eike

j4"fi

cRnk o

i=1

n f

#k=1

n p

# = I(n)e j$ (n ) Eq. 1-11

where is the number of frequencies, is the number of positions on the rail, c is

the speed of light, Rn,k is the distance between the point and the position on the

rail.

Eq. 1-11 can be evaluated rather fast by remarking that it is the inverse spatial

Fourier transform of the measured data. So for each antenna position, the inverse

discrete Fourier transform (IDFT) of the measured data is calculated versus index i.

Samples of the discrete IDFT are regularly spaced in range by step of

!

"r =c

2Bn f #1( )

n f

. In order to avoid further resolution degradation, the signal is

generally expanded by zero padding before IDFT is applied.

The algorithm works considering all the geometric points to be focused for each

antenna position and every time for each point. The sample of the IDFT whose range

distance is the nearest to distance between the point and the antenna position is

selected and added to the image value relative to the point. At the end of iteration a set

of values distributed in the space is obtained. The obtained images are complex

bidimensional images whose amplitude represents the distributed radar cross-section

(RCS) and whose phase is the matter of interferometry.

1.3 SAR Interferometry

Interferometric synthetic aperture radar (InSAR) is potentially a unique tool for low-

cost precise digital elevation models (DEM) generation, and large-coverage surface

deformation monitoring. It has applications for geophysical monitoring of natural

hazards, for example earthquakes, volcanoes and landslides, and also in structural

engineering, in particular monitoring of subsidence and structural stability. The

technique involves interferometric phase comparison of SAR images gathered with

different baselines and at different times and has the potential to provide DEMs with

meter accuracy and terrain deformations with millimetric accuracy.

Being the dimension of the resolution cell of a SAR survey much greater than the

wavelength of the impinging radiation, the signal corresponds to the algebraic

combination of all the reflections from independent scatterers within the cell, with

their proper amplitudes and phases. This superposition of effects is only approximate

as one should consider not only the primary reflections (i.e. radar → target → radar),

but also multiple ones (e.g. radar → tree trunk → ground → radar). Then, the signal is

a combination of many independent reflections, with the possibility that some of them

are much higher than others.

1.3.1 The Interferograms

With the aim of determining the occurred range variations, in SAR interferometry two

radar images of the same surface area are acquired and differenced in phase, forming

a radar interferogram. This is usually implemented by cross multiplying the complex

reflectivity at each point of one image by its corresponding conjugate value in the

second image, so that the interferogram also preserves useful information about the

signal amplitudes. The complex product effectively cancels the common

backscattering phase in each resolution element leaving the phase term proportional to

the difference between the path delays.

Until now the radar echo from the elemental scatterers within a resolution element has

been considered a deterministic quantity as it is calculable from knowledge of the

precise locations of each elemental scatterers. Of course, it is no feasible to go on this

way.

Because the size of the resolution element is typically many wavelengths, many

elemental scatterers can be supposed randomly distributed within the resolution

element and each with its individual backscattering amplitude statistically

independent from its phase. With these assumptions the real and imaginary part of the

resolution element backscattering coefficient are independent, zero-mean, Gaussian

random variables with equal variances. These really are the properties that lead to the

Rayleigh probability density function for the amplitude of SAR image pixels. They

also allow to calculate a probability density function for SAR image phase, and hence

for the interferometric phase itself, in an analytically closed form.

It is worth to introduce another very important quantity, i.e. the interferometric degree

of coherence, or simply coherence.

The interferometric coherence is the amplitude of the correlation coefficient

between the two complex SAR images forming the interferogram:

!

" =I1 # I2

*

I1 # I1 I2 # I2

= ce j$

Eq. 1-12

where and represent the two SAR images, the brackets mean the average value

of the argument and is the corresponding interferometric phase.

The coherence provides an estimation of the achievable quality of an interferogram

since it determines the peak width of the probability density function around that is

strictly related to the measurable phase variance. Physically, a loss of coherence

depends on how the elemental scatterers move within resolution cell. If the

displacements randomize the positions of each scatterer with respect to others, the

pair of echoes will be less well related and the coherence of the interferogram will

decrease resulting in a noise that may mask the underlying phase signature.

1.3.2 PhaseUnwrapping

The interferometric phase signatures vary relatively smoothly from point to point in

the interferogram and may be inverted to recover surface topography or displacement

fields. However, the phase observables are measured modulo 2π; that is, the integral

number of phase cycles on each measurement is lost. It is said that the phases linear

with range variation are wrapped in the interval [-π, π) by the measurement process.

Consequently, if the surface displacement in a scene determines a phase excursion

greater than one cycle, or if the combination of interferometric baseline and surface

topography yields more than one fringe of topographic signature, the interferogram

cannot be uniquely inverted without a procedure to recover the missing cycles. We

refer to such procedures as phase unwrapping.

The aim of phase unwrapping (PU) is to recover the integer number of cycles to be

added to the wrapped phase so that the unambiguous phase value can be finally

obtained for each image pixel.

1.3.2.1Mathematicalformulation

In general, if no a priori information about the wrapped phase is available, i.e. no

constraint is given to the solution, PU is an ill-posed problem and therefore an infinite

number of different solutions can be found, all honouring the data.

The most straightforward PU procedure would be a simple integration of the phase

differences, starting from a reference point. However, because of phase

discontinuities, it is not always accurate.

Almost all PU algorithms are based on the assumption that the true unwrapped phase

field is “smooth” and varies “slowly”. More precisely, neighbouring phase values are

assumed to be within one-half cycle (π radians) of one another. Though this

hypothesis is often valid for most of the image pixels, the presence of some phase

discontinuities (i.e. absolute phase variation between neighbouring pixels of greater

than π radians) causes inconsistencies, since integration yields different results

depending on the path followed. This feature is evident whenever the sum of the

wrapped phase differences (the integral of the estimated phase gradient) around a

closed paths differs from zero. To be consistent, a gradient field must be irrotational,

i.e. the curl of ∇ϕ should be zero everywhere. Whenever this condition is verified

over the whole interferogram, we have a “trivial PU problem”. Unfortunately, this is

almost never the case in InSAR data processing.

Phase unwrapping algorithms share then this common initial approach: the relative

phases, or gradient, between adjacent pixels in the interferogram are correctly

evaluated, so the obtained connected phase field is adjusted by an overall constant

multiple of 2π. The basic theory of all the unwrapping procedures starts with the

following equation:

�

∇ϕ x,y( ) = s x,y( ) + n Eq. 1-13

where ∇ϕ is the gradient of the absolute interferometric phase, that is the value

linearly related to the range variations, s(x,y) is the measurable variable and n is the

noise included in the measurement. For the noise-free state

!

n = 0, Eq. 1-13 must be

completely observable, meaning that there is a unique solution for ϕ and the value of

ϕ must be derived at every point

!

x,y( ) if the vector s is known. For the noise free-

state a necessary and sufficient condition that s must satisfy for a unique

determination of ϕ from Eq. 1-13 is easily derived from the potential theory. This

condition is given by:

!

s " d!C# = 0$%& s = 0 Eq. 1-14

The integration is performed along an arbitrary enclosure in the

!

x,y( ) plane, and

!

d! = (d! x,d! y ) is the infinitesimal length element along the path. In other words,

must be conservative.

This formulation can be extended to a generic noisy case by simply assuming that a

correct absolute phase gradient estimation can be accomplished anyway by removing

all the points where the measured gradient doesn’t satisfy irrotational requirement.

These points are generally called residues.

The occurrence of these points lead to an inherent 2π ambiguity in the phase (due to

circulations around them) and, consequently, self-consistent phase-unwrapping

methods do not succeed for most objects.

Obviously residues are located where Eq. 1-14 is not satisfied. In practice the phase

field is measured over a two-dimensional pixel array

!

N " M , so the discrete version

of the above equations must be considered:

!

si, jl

C" = 0, with l = 1,2 and

si, j1 = # i, j $# i$1, j p

i = 2,..,N; j = 1,..,M

si, j2 = # i, j $# i, j$1 p

i = 1,..,N; j = 1,..,N

%

& '

( '

Eq. 1-15

where ψi,j is the principal value on the interferogram rectangular grid and is an

arbitrary enclosure. The minimum enclosure in the discrete system is the set of 2x2

pixels, called a node. Then residues can be located by evaluating the sum of principal

value differences around each node in the interferogram. A residue or inconsistent

node may be either positive or negative, depending on the outcome of the summation

and the chosen integration direction (clockwise or counterclockwise).

1.3.2.1Phaseunwrappingmethods

All of the commonly used phase unwrapping algorithms relate the phase values by

first differentiating the phase field and subsequently reintegrating, adding back the

missing integral cycles to obtain a more continuous result. The many algorithms

proposed for phase unwrapping over the past few years fall into three basic classes:

• path-following algorithms based on identification of residues in the wrapped

phase field and cuts, or ‘‘trees,’’ connecting a group of residues to limit the

integration paths;

• algorithms that derive a smooth field by integrating the gradient of the

observations subject to smoothness constraints as determined by least-squared-

difference criteria;

• networking algorithms;

Basically all the existing phase unwrapping techniques start from the fact that it is

possible to estimate the neighboring pixel differences of the unwrapped phase when

these differences are less than π. From these, the unwrapped phase can be

reconstructed up to an additive constant. The methods differ in the way they

overcome the difficulty posed by the fact that this hypothesis may be somewhere

false, which cause the estimated unwrapped phase differences to be inconsistent, that

is their "integral" depends on the integration path.

Branch cuts methods unwrap by integrating the estimated neighboring pixel

differences of the unwrapped phase along paths avoiding the regions where these

estimated differences are inconsistent. The problem of building cuts delimiting these

regions is very difficult and the resulting phase unwrapping algorithm is very

expensive computationally.

In least squares methods, unwrapping is achieved by minimizing the mean square

deviation between the estimated and the unknown neighboring pixel differences of the

unwrapped phase. Least squares methods are very efficient computationally when

they make use of fast Fourier transform techniques. But the resulting unwrapping is

not very accurate, because least squares procedures tend to spread the errors that are

instead concentrated on a limited set of points. To overcome this problem a weighting

of the wrapped phase can be useful. However, the weighted least squares algorithms

proposed are iterative and not as efficient as the unweighted ones. Moreover, the

accuracy of the results depends on the weighting mask used.

The networking algorithms exploit the fact that the neighboring pixel differences of

the unwrapped phase are estimated with possibly an error which is an integer multiple

of 2π. This leads to formulate the phase unwrapping model as the problem of

minimizing the deviations between the estimated and the unknown neighboring pixel

differences of the unwrapped phase with the constraint that the deviations must be

integer multiple of 2π. This constraint prevents the errors from spreading, so a

weighting of the data is not necessary. In any case, weighting is allowed in this

formulation (without loss of efficiency), and can be useful when there are large

regions of noisy data. Even if minimization problem with integer variables are usually

very complex computationally, recognizing the network structure underlying the

problem makes available very efficient strategies for its solution. In fact, the problem

can be equated to the problem of finding the minimum cost flow on a network, for the

solution of which there exist very efficient algorithms.

1.4 References

C. Cafforio, C. Prati, F. Rocca, “L’Elaborazione di segnale nel radar ad apertura sintetica”, L’Elettrotecnica, vol. 9, pp. 787–795, Settembre 1989.

C. Cafforio, C. Prati, E. Rocca, “SAR Data Focusing Using Seismic Migration techniques”, IEEE Trans. on aerospace and electronic systems, vol. 27, no. 2, pp. 194-205, march 1991.

C. W. Chen, Statistical – cost network – flow approaches to two-dimensional phase unwrapping for Radar Interferometry, Ph. D. Thesis, Stanford University, 2001.

M. Costantini, “A Novel Phase Unwrapping Method Based on Network Programming”, IEEE Trans. Geosci. Remote Sensing, vol. 36, no. 3, May 1998.

J. C. Curlander and R. N. McDonough, “Synthetic Aperture Radar - Systems and Signal Processing”, John Wiley & Sons, Inc., New York, NY, USA, 1991.

D. Dei, “Monitoraggio dinamico di strutture architettoniche mediante sensori a microonde”, Ph. D. Thesis, University of Florence, 2010.

J. P. Fitch “Synthetic Aperture Radar”, New York; Springer-Verlag,1988.

M. Fratini, “Sistemi RADAR per il monitoraggio di beni artistici ed ambientali”, Ph. D. Thesis, University of Florence, 2005.

A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res., vol. 94, no. B7, pp. 9183–9191, July 1989.

D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods”, J. Opt. Soc. Am., vol. 11, no.1, pp. 107-117, January 1994.

R. M. Goldestein, H. A. Zebker, and C. L. Werner, “Satellite Radar Interferometry: Two-dimensional Phase Unwrapping”, Radio Scienc, vol. 23, no. 4, pp. 713-720, July 1988.

L. C. Graham, “Synthetic interferometric radar for topographic mapping”, Proc. IEEE, vol. 62, pp. 763–768, June 1974.

G. Fornaro, G. Franceschetti, and R. Lanari “Interferometric SAR Phase Unwrapping Using Green’s Formulation”, IEEE Trans. Geosci. Remote Sensing, vol. 34, no. 3, pp. 720-727, May 1996.

D. Leva, G. Nico, D. Tarchi, J. Fortuny, and A. J. Sieber, “Temporal Analysis of a Landslide by Means of a Ground-Based SAR Interferometer”, IEEE Trans. Geosci. Remote Sensing, vol. 41, no. 4, pp 745-752, April 2003.

G. Luzi, et al., “Ground-Based radar interferometry for landslides monitoring: atmospheric and instrumental decorrelation sources on experimental data”, IEEE Trans. Geosci. Remote Sensing, vol: 42 Issue: 11 , pp: 2454 –2466, Nov 2004.

G. Macaluso, Radar interferometrici basati a terra ed integrazione dati in ambiente GIS, Ph. D. thesis, 2007.

D. Massonnet et al., “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature, vol. 364, pp. 138–142, July 8, 1993.

D. Mecatti, “Radar interferometrici per il monitoraggio da terra di versanti franosi”, Ph. D. Thesis, 2005.

D. L. Mensa, “High Resolution Radar Cross-Section Imaging”, Artech House.

L. Noferini, Processing techniques of microwave data acquired by Continuous Wave Stepped Frequency Radar, Ph. D. thesis, 2005.

D. A. Noon, “Stepped-Frequency Radar Design and Signal Processing Enhances Ground Penetrating Radar Performance”, Ph. D. Thesis, University of Queensland, January 1996.

M. Pieraccini, D. Tarchi, H. Rudolf, D. Leva, G. Luzi, and C. Atzeni, “Interferometric radar for remote monitoring of building deformations”, in Electronics Letters, vol. 36, no. 6, pp. 569–570, March 2000.

M. Pieraccini, D. Mecatti, L. Noferini, G. Luzi, G. Franchioni, and C. Atzeni, “SAR interferometry for detecting the effects of earthquakes on buildings”, in NDT&Intrenational, vol. 35, pp. 615–625, August 2002.

M. Pieraccini, N. Casagli, G. Luzi, D. Tarchi, D. Mecatti, L. Noferini, and C. Atzeni, “Landslide Monitoring by Ground-Based Radar Interferometry: a field test in Valdarno (Italy)”, Int. J. Remote Sensing, vol 24, no 6, pp 1385-1391, Sept. 2002.

P. A. Rosen, S. Hensley, I. R. Joughin, F. K. Li, S. N. Madsen, E. Rodriguez and R. M. Goldstein, “Synthetic Aperture Radar Interferometry”, Proc. IEEE, vol. 88, no. 3; pp. 333–382, March 2000.

M. I. Skolnik, “Radar Handbook”, McGrawHill, New York, NY, USA, 1990.

U. Spagnolini, “2-D phase unwrapping and instantaneous frequency estimation”, IEEE Trans. On Geoscience and Remote Sensing, vol. 33, no. 3; pp. 579–589, May 1995.

H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference”, J. Opt. Soc. Am. A, vol. 5, no. 3, pp. 416-425, March 1988.

D. Tarchi, H. Rudolf, G. Luzi, L. Chiarantini, P. Coppo, and A. J. Sieber, “SAR interferometry for structural changes detection: A demonstration test on a dam”, in Proc. IGARSS, 1999, pp. 1522–1524.

F. T. Ulaby, K. Sarabandi, A. Nashashibi, “Statistical Properties of the Mueller matrix of distributed targets ”, Proc. IEEE, vol. 139, no. 2, pp. 136-142, April 1992.

D. R. Wehner, “High Resolution Radar”, Artech House, Norwood, MA, USA, 1995.

H. A. Zebker and R. M. Goldstein, “Topographic mapping from interferometric SAR observations”, J. Geophys. Res., vol. 91, pp. 4993–4999, 1986.

Chapter 2 Monitoring

Open-Pit Quarries by

Interferometric Radar for

Safety Purposes

This chapter focuses on a measurement campaign carried out to monitor an open pit

quarry of pietra serena using a Ground-Based SAR.

The work described has been carried out in the framework of the PRIN 2007 project

“Application of advanced technologies to the safety, environmental protection,

optimization of management and yield in quarry: geomining characterization and

monitoring methodologies”, founded by the Italian Research and University Ministry.

2.1 GB-SAR for Terrain Displacement Mapping

Today Ground-Based Synthetic-Aperture Radar (GB-SAR) interferometers have

become powerful and effective tools for remote sensing of movements in many

application fields. These sensors have been successfully employed in landslide and

glaciers monitoring, as well as in avalanche localization and detection of snow cover

changes.

With respect to airborne and spaceborne SAR systems, GB-SAR can offer a

continuous temporal coverage and, because of its versatility, an optimal illumination

geometry in almost all cases. Usually in applications for ground displacement

mapping the imaging geometry of all the SAR observations is maintained identical,

avoiding the baseline error.

A GB-SAR interferometer works by acquiring images of the observed scenario from

remote, each image containing phase and amplitude informations of the received

echoes, at a high time rate (typically more than one image per hour). The phase values

of each pixel of the image are related to the distance between the sensor and the pixel

itself, whereas the phase differences between two images acquired at different times

are related to the eventual radial displacements occurred in the observed scenario.

Therefore calculating the differences between the pixel phase values in two images,

the measurement of the range variations in each pixel is directly obtained, providing a

direct measurement of the along of sight displacements. The multiple images acquired

can be thought of as “time-lapse” imagery.

2.2 Monitoring an open pit quarries by a GB-SAR

Slope failure accidents are nowadays responsible for a large number of fatalities and

machinery damages in open pit quarries. Monitoring of slope stability in excavation

areas is then a crucial issue in order to optimize the production efficiency and to

ensure the safety of the operators.

Because large rock collapses may show minor precursor movements starting hours or

days in advance, the early detection of these small deformations could be of

paramount importance in an early warning system.

In this context, the idea was to set up and test the available GB-SAR system for open-

pit quarries monitoring. The field test took place at a stone quarry in Tuscany, Italy

(Figure 2-1) and lasted 39 days.

Figure 2-1: a picture of the pit, it is located about 50 km north of Florence.

2.2.1 TheRadarSystem

The employed ground based radar instrumentation was entirely developed at the

Technology for Cultural Heritage Laboratory (TCH Lab), a research group within the

Departement of Electronics and Telecommunications of the University of Florence.

The radar system is an evolution of the sensor developed at the University of Florence

in collaboration with IDS SpA (Pisa, Italy), working in the licensed Ku band. The

system was adapted to this application through several improvements in the

mechanical and electronical components.

The measurement geometry in a quarry is often unfavourable for a standard GB-SAR

system, due to the need of imaging a wide area looking from a close location, thus

requiring a wide viewing angle.

For this reason the radar system was improved adding the capability to rotate the

antennas around a vertical axis: the system was able to observe a very wide area by

simply acquiring several different SAR images at different viewing angles and

combining them in post processing. With this improvement the view angle of the

radar sensor was extended up to 150 degrees (Figure 2-2).

Figure 2-2: a detail of the radar system showing the capability to rotate the antennas along the vertical axis, in order to be able to image all the very wide area of interest.

This solution allows to perform the measurements using high gain antennas, thus

maintaining a higher signal to noise ratio, instead of using wide angle – low gain

antennas. The drawback is a longer acquisition time needed to acquire the sub-images

at the different angles.

A quarry is typically an area subject to morphological changes due to the continuous

mining activity. Therefore, when the interferometric measurements were performed,

an updated Digital Elevation Model (DEM) of the area was not available.

As the Digital Elevation Model is necessary to correctly generate the SAR image, in

the novel GB-SAR system it was added the capability to realize interferometric

DEMs, simply allowing the antennas to move up to 30 centimeters along a vertical

axis.

In order to be able to operate day and night even where the electric network was not

available, the radar system was equipped with an autonomous power supply system

using solar and wind energy (Figure 2-3).

Figure 2-3: the power supply of the GB-SAR system using solar and wind energy.

Finally, a GPRS or long range WIFI radio link allows the remote control of the

instrument and the real time data download. A picture of the system is shown in Figure

2-4.

Figure 2-4: the GB-SAR system for open pit quarries monitoring. In foreground the rail with the radar head and the antennas, behind the boxes of electronics and in background the wind

generator for the power supply of the system.

2.2.2 The Field Survey

The novel radar system described above was tested during a field survey 39 days long

from February 16 to March 25, 2009.

The test site was located at a stone (‘pietra serena’) quarry in Firenzuola (Firenze,

Italy). The aim of the survey was to check the capabilities of the system to detect

possible small displacements across the steep slope, overlooking the excavation area.

The radar was installed over a hill in front of the quarrying area, 800 meters faraway,

at about the same quote. The measurement scenario is shown in Figure 2-5 and Figure

2-6.

Figure 2-5: the radar sensor was located at a distanc of 800 meters over a hill in front of the area of interest.

Figure 2-6: the stone (“pietra serena”) quarry in Firenzuola (Firenze-Italy) where the radar system was tested.

At the test site the GB-SAR system worked in the site automatically day and night, in

all weather conditions. Data were uploaded real time to the automatic processing

station located at the University of Florence through a local radio link and the Internet

network. Displacement maps were generated automatically and were available half an

hour after the end of each acquisition. The measurement parameters are listed in Table

2-1.

Central Frequency 16.75 GHz

Bandwidth 300 MHz

Tx Power 26 dBm EIRP

SAR scan length 2 m

SAR acquisition time 10 minutes for each sub-image

SAR acquisition rate 2 sub-images every hour

Range Resolution 0.50 m

Cross Range Resolution 0.5°

View angle 150°

Power consumption < 40 W Table 2-1: measurements parameters.

2.2.3 The RCS image

A RCS SAR image of the quarry is shown in Figure 2-7, with superimposed the map of

the same area. The reference system is a local system with the origin in the radar

location. This wide angle map originates from two sub-images acquired sequentially

at two different angles.

Figure 2-7: RCS SAR image of the quarry area with superimposed the topographic map. The reference system is a local system with the origin in the radar location.

In Figure 2-7 three different zones can be distinguished: (a) the mining area, between

600 and 700 meters in range, (b) the rocky wall, between 700 and 800 meters in

range, and (c) the ravaneto1 between 450 and 600 meters. The zone (a) was

characterized by continuous movement of mining machinery, the zone (c) consisted of

rock debris resulting from quarrying operations, and the zone (b) was the area of

interest with a limited extension in range due to its prevalent vertical slope.

1Inquarriesravanetoisanareaofaccumulationofquarryingdebris,usuallyaravine.

2.3 Data Processing

Displacement maps were generated, all over the survey period, processing more than

five hundred SAR acquisitions.

One of the main difficulties in interferometric data processing is the mix of several

different types of information in a given signal. Dielectrical, atmospherical and

displacement informations are all wrapped together in the same signal. Consequently,

the measurement accuracy is not driven by the characteristics of the radar system

(power, resolution, etc.) as much as by the possibility of separating the various

component of the signal.

A short review is here presented of the main results to establish notation and to

discuss the different physical signals that contribute to the interferometric phase. For

the sake of simplicity, phase term due to noise, image misregistration, wrong focusing

parameters, etc. will be neglected. The analysis outlines the rationale behind a multi-

image approach for surface displacements estimation.

The phase of a pixel in a SAR image can change due to:

1. changing in the radar-scatterer relative position

2. possible temporal changes of the target

3. atmospheric variations

Considering K+1 SAR images of the same area, the phase of the pixel x=[x,y] (x and

y being cross-range and range coordinates respectively) of the k-th focused SAR

image is the sum of different contributions:

!

" k (x) =4#$% rk (x) +& k (x) + ak (x)

Eq. 2-1

where r is the radar-target distance, σ is the scatterer reflectivity phase, and a is the

atmospheric phase contribution.

Let us now consider one of the K+1 images as the “master” acquisition m. The phase

difference of the generic “slave” image k with respect to the master will be indicated

with φk(x):

!

"k (x) =# k (x) $# m (x) =4%&' rk (x) $ rm (x)[ ] + ( k (x) $( m (x)[ ] + ak (x) $ am (x)[ ]

Eq. 2-2

In GB-SAR interferometry, we can express rk as follows:

�

rk = rm + Δµk Eq. 2-3

where Δµk is the possible target motion in the direction of the radar line-of-sight,

occurring during the time interval between the two acquisitions, the order of

magnitude of this term usually being of millimeters.

The interferometric phase is then a blend of several signals that depends on the

acquisition geometry (radar postion and topography), terrain motion, scattering

changes and atmospheric inhomogeneities:

!

"k (x) = "µk (x) + "ak (x) + "#k (x) Eq. 2-4

where we posed

!

"µk (x) =4#$% &µk

"ak (x) =4#$%[ak (x) ' am (x)]

"(k (x) =4#$%[( k (x) '( m (x)]

)

*

+ + +

,

+ + + Eq. 2-5

2.3.1 Pixels selection as permanent scatterers

The main drawback of the InSAR technique is that it is limited to the estimation of the

signals relative to only a restricted ensamble of points scatterers which are

caracterized by long-term stable backscattering properties and extremely good signal

to noise ratio. Pixels in the radar image, that are supposed to represent these stables

and coherents scattereres over the entire observation interval, are called permanent

scatterers (PSs).

In this thesis the method used to identify possible PS points is based on the analysis of

the time series of their amplitude values. This approach is due to the fact that an exact

evaluation of the phase stability is possible only after estimation and removal of the

atmospheric contribution, while amplitude values are almost insensitive to the

atmospheric phenomena. Consequently, pixels showing a stable time sequence of

amplitude values will be considered PSs.

The PS extraction procedure starts with the estimation of the amplitude dispersion

index, which is in turn a measure of the phase stability, relative to a temporal

sequence of SAR images. For each image pixel, the dispersion index is defined by:

!

DA =" A

mA

Eq. 2-6

where mA and are the mean and the standard deviation of the pixel amplitude

value A trough the sequence of images. PSs are then selected by computing the

dispersion index of the amplitude values relative to each pixel in the area of interest

and considering only those pixels exhibiting DA values under a given threshold

(typically DA ≤ 0.25).

At the test site, the area of interest for the safety of people and machinery was the

rocky wall, being the mining activities in progress just below it. In order to perform a

robust interferometric analysis over a long time, not all the pixels were considered,

but a selection based on double criterion was made. The first selection was based on a

PS approach, analyzing amplitude variation of the pixels, choosing a threshold of 0.2.

The second criterion was a threshold on SNR, selecting only pixels showing a SNR

greater than 30 dB. Beginning with an image with about 900000 pixels, about 50000

pixels were selected as permanent scatterers after the first selection. The second

selection further reduced the number of pixels available for the elaboration to about

20000 (Figure 2-8). Most of the pixels were selected in the wall, being it rock without

vegetation (Figure 2-9). It is however to be noted that the wall was not perfectly

vertical, being there a sort of terraces delimited by a metallic net, dug out in order to

keep possible fall of debris. This fact was certainly a problem for radar measurements,

because this net was not fixed very well and it moves with the wind and with

temperature variations. This is the reason because these areas are not completely

covered by PSs.

Figure 2-8: Dispersion Index of a restricted area of the pit near the wall.

Figure 2-9: the wall on the left and a detail of the net on the right.

It’s worth to outline that the number of PS depends strongly on the number of SAR

images used. However, using a number of images greater than 50, the number of PS

becomes almost stable, decreasing very slowly. Altough more rigorous and

computationally intensive statistical analyses can be adopted, this very simple

approach turns out to be enough for our purposes.

2.3.2 Atmospheric phase screen

Selected points were then processed in order to remove the atmospheric effects. A

brief discussion on how this step was performed is given here.

Due to the morphology of the area under investigation and to the seasonal

characteristics, frequents, suddens and severe meteorological changes occurred (sun,

rain, fog, snow and so on) during the survey. Therefore interferometric phases must

be compensated for atmospheric effects.

Atmospheric inhomogeneities create an atmospheric phase screen (APS)

superimposed on each SAR image that can seriously compromise accurate

deformation monitoring. APS exhibits a low-wavenumber spectral behavior

(according to th atmospheric water vapor distribution in the troposphere).

The APS mean effect is usually approximated as a phase ramp:

!

"ak (x) = Ak # x + Bk # y + Ck Eq. 2-7

where Ak, Bk and Ck are the parameters of the atmospheric plane of the k-th

interferogram.

Specifically given H pixels is possible to define the functional Γk determining the

APS as follows:

!

"k =1H# exp j # $kh (x) % Ak # xh + Bk # yh( )[ ]{ }

h =1

H

&

˜ A k, ˜ B k{ } = argmaxAk ,Bk"k{ }

˜ C k = ' ˜ " k

(

)

* * *

+

* * * Eq. 2-8

where φkh is the interferometric phase relative to the h-th pixel selected in the k-th

interferogram.

2.3.3 Total displacement: cumulative interferogram

For our purpose it results advantageous to calculate the cumulative interferogram

instead of the standard interferogram. Let

!

I1,I2,I3,...,IN be a set of radar images

acquired at

!

t1< t2 < t3,...< tN times respectively. The cumulative interferogram of this

set is defined as:

!

"1,N= #i=1

N $1

% Ii+1(&)Ii*(&)( ) = #

i=1

N $1

% e j(" i+1 (&)$" i (&))( ) Eq. 2-9

where “*” is the complex operator and

!

" Ik( ) = #k .

This approach offers the advantage that considering phase changes between two

consecutive SAR images, phase wrapping is most unlikely. By assuming that the

dielectric characteristics of the pixel point n-th are the same at each observation,

supposing having removed the atmospheric contribution, the displacement Δrn of this

point at the time of the nth observation can be recovered from the cumulative

interferogram by the following equation:

!

"rn =#1,n

4$% Eq. 2-10

where is the central wavelength.

2.4 Results

2.4.1 Pixels displacements

As long as the GB-SAR system was working at the quarry, interferometric

measurements did not detect significant movement in zone (b).

Figure 2-10 reports the measured displacement of a selection of 50 pixels located across

the wall over a time period of 23 days. It is worth to outline that in the period a

maximum displacement of 0.2 millimeters was observed. The trend which is visible in

the graph probably depends on a residual error of the meteorological correction. The

gaps are due to the lack of energy of the system when the system switched off.

Figure 2-10: cumulative displacements for 50 pixels in the wall during a time period of 23 days.

A zoom relative to a 3 days long period taken from the previous figure is shown in

Figure 2-11. Here it is well evident that none of the selected pixels was really moving.

Figure 2-11: a zoom of the cumulative displacements graph over a period of 3 days.

In Figure 2-12 the measured displacements versus time are shown for three selected

points in the wall and for two different days during the survey. The fluctuations are

again due to instrumental noise and to residuals of the atmospheric compensation

procedure. However the cumulative displacements are below the 0.1 mm, meaning

that this point must be considered as stable, being the value of 0.1 mm considered as a

threshold.

Figure 2-12: cumulated displacements versus time for three points in the rock wall during two days of measurement.

2.4.2 Interferograms

Aimed to a complete interferometric characterization of the scenario, it is interesting

to study the interferogram relative to all the pixels, excluding only the dead zone of

the antennas (shown in red in the Figure 2-13).

In zone (a) the interferometric analysis is not significant because of the continuous

movements of machinery for mining activities. In zone (c) sudden and consistent

movements were observed due to discharging of debris into the ravaneto. However in

this area the interferometric analysis cannot be performed meaningfully, as the rapid

accumulation of debris into the ravine is generally a non coherent process and often

originates phase ambiguity. Nevertheless it is interesting to examine these

interferograms qualitatively, as the areas where the previous phenomena happen can

be clearly identified.

Figure 2-13: (a), (b) and (c) cumulative interferometric maps corresponding to daytime during quarrying and (d) to night time without quarrying (dimensions in meters).

Figure 2-13 shows a sequence of four cumulative interferometric maps corresponding

to a time lapse of 8 hours during a period of intense quarrying activities. The paths of

debris along the ravine are well evident in interferograms (a), (b) and (c).

Interferogram (d), corresponding to the evening when quarrying was just finished,

shows a different phenomenon: in all the upper part of the ravaneto fresh debris

seems to settle down giving rise to typical interferometric fringes.

2.4.3 Simulation of displacement by a Corner Reflector target

As no significant movement was detected in the wall during the 39 days long survey,

in order to check the capabilities of the instrument to detect small displacements at

long distance, a simulation of rock movement was set up as shown in Figure 2-14.

A trihedral corner reflector (CR), 50 cm side, was installed on a micrometric device at

a distance of 800 meters from the radar, on the top of the vertical wall.

The reflector has been moved along the radar line of sight in three small steps, as

shown in Table 2-2. For each position of the CR one single SAR image was acquired,

and the interferograms were calculated. Results, shown in Table 2-2 and Figure 2-15,

prove a good accuracy in the measurement of displacements, but this is with a high

SNR due to the high RCS of the CR and with a quite stable atmosphere.

Figure 2-14: trihedral corner reflector and micrometric device for the simulation of rock movement.

Figure 2-15: corner reflector actual displacement versus measured displacement.

Table 2-2: Results of the simulation of rock movement.

Step n. 1 2 3

Displacement of CR

(mm)

2.00 1.00 0.50

Measured displacement

of CR (mm)

2.05 0.89 0.53

It is clearly evident that in the short term (few hours) and in optimal SNR conditions

an accuracy of 0.1 millimeters is guaranteed.

2.4.4 Digital Elevation Model

A Digital Elevation Model of the terrain can be addressed using the developed GB-

SAR. By raising the antenna at different levels along the vertical direction, images

with different viewing angles can be collected.

Because the same antenna is moved firmly, the achieved baseline exactly corresponds

to the geometric shift of the antenna along the vertical direction.

Referring to the measurement geometry depicted in the Figure2‐16, the relationship

between the unwrapped interferometric phase ϕ and the quote z along the axes

defined by the antenna vertical motion is

!

" =4#$

(r % r(B)) =4#$

r(1% 1% 2zBr2 +

B2

r2 ) Eq.2‐11

where λ is the central transmitted wavelength, B is the baseline, and r and r(B) are the

ranges to a target on the illuminated slope when the antenna is positioned at z = 0 and

z = B, respectively.

Because the GB-SAR generally works in a configuration such that

!

B /r is smaller

than 1, the Taylor’s expansion of the square root in terms of

!

B /r can be truncated to

the linear term and converted into the elevation z:

!

z ="

4#rB$

Eq.2‐12

The latter equation provides the basic phase-to-elevation conversion.

Figure2­16:GB-SAR measurement geometry to DEM generation.

A Digital elevation Model of the open pit was generated. Figure 2-17 shows the result,

where a little portion of the all scenario observed was reconstructed. Figure 2-18 shows

instead all the open pit quarries as reconstructed by a laser scanner. The discrepancy

between the two DEMs are evident, the laser scanner DEM being more detailed (a

spatial resolution of 0.25 m x 0.25 m compared with the 4 m x 1 m of the GB-SAR)

and more similar to the optical image. Some differences are due to the wrapping

errors, specially in the GB-SAR reconstruction of the wall area due to the high slope.

However in the waste area and in the working area accuracy is better, with an error of

about five meters compared to the laser scanner DEM.

In Figure 2-19 an amplitude SAR image is focused using the laser scanner DEM, where

the red spot represents the Corner Reflector Target observed by the radar.

Figure 2-17: DEM generated by the GB-SAR system.

Figure 2-18: DEM of the open pit quarry generated with a laser scanner.

Figure 2-19: an example of an amplitude SAR image focused using the laser scanner DEM of the

open pit quarry. The red spot represents the Corner Reflector Target observed by the radar.

2.5 References

G. Antonello, N. Casagli, P. Farina, D. Leva, G. Nico, A. J. Sieber, D. Tarchi,. “Ground-based SAR interferometry for monitoring mass movements”, Landslides, 1, pp. 21–28, February 2004

P. Bernardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms”, IEEE Trans. Geosci. Remote Sensing, vol. 40, n. 11, November 2002.

A. Ferretti, C. Prati, and F. Rocca, “Permanent scatterers in SAR interferometry”, IEEE Trans. Geosci. Remote Sensing, vol. 39, pp 8-20, Jan. 2001.

A. Ferretti, C. Prati, and F. Rocca, “Nonlinear subsidence rate estimation using Permanent Scatterers in differential SAR interferometry”, IEEE Trans. Geosci. Remote Sensing, vol. 38, pp 2202-2212, Sept. 2000.

A. Ferretti., C. Colesanti, D. Perissin, C. Prati, F. Rocca, “Evaluating the Effect of the Observation Time on the Distribution of SAR Permanent Scatterers”, Proc. FRINGE ‘2003 Workshop, Frascati, Italy, 2003.

R. F. Hansen, Radar interferometry. Data Interpretation and Error Analysis, Kluwer Academic Publishers, 2001.

R.F. Hansen, “Atmospheric heterogeneities in ERS Tandem SAR Interferometry”, Delft Univ. Press, 1998.

A. Hooper, Persistent Scatterer Radar Interferometry for crustal deformation studies and modeling of volcanic deformation, Ph. D. Thesis, Stanford University, 2006.

A. Hooper, H. A. Zebker, P. Segall, and B. Kampes, “A new method for measuring deformation on volcanoes and other natural terrain using InSar persistent scatterers”, Geophysical Research Letters, vol. 31, 31 (23), 5, doi:10.1029/2004GL021737, 2004.

D. Leva, G. Nico, D. Tarchi, J. Fortuny, and A. J. Sieber, “Temporal Analysis of a Landslide by Means of a Ground-Based SAR Interferometer”, IEEE Trans. Geosci. Remote Sensing, vol. 41, no. 4, pp 745-752, April 2003.

A. Martinez-Vazquez, J. Fortuny-Guasch, “Feasibility of snow avalanche volume retrieval by GB-SAR imagery,” in Proc. IGARSS 2006, July 31 – August 04 2006, Denver (CO).

A. Martinez-Vazquez, J. Fortuny-Guasch and Urs Gruber, “Monitoring of the snow cover with a ground-based synthetic aperture radar,” in Proc. EARSeL eProceedings vol. 4, n. 2, 2005.

G. Macaluso, Radar interferometrici basati a terra ed integrazione dati in ambiente GIS, PhD Thesis, 2007.

D. Mecatti, L. Noferini, G. Macaluso, M. Pieraccini, G. Luzi, C. Atzeni, A. Tamburini, “Remote sensing of glacier by ground-based radar interferometry”, in Proc. IGARSS 2007, Barcelona, Spain, 23-28 July 2007.

D. Mecatti, Radar interferometrici per il monitoraggio da terra di versanti franosi, PhD Thesis, University of Florence, 2005.

D. Mecatti, Monitoraggio di movimenti del terreno e di grandi strutture: sviluppo di prototipi di radar interferometrici, Bachelor Thesis, University of Florence, 2001.

L. Noferini, D. Mecatti, G. Macaluso, M. Pieraccini, C. Atzeni, M. Ripepe, A high speed microwave interferometer used for monitoring Stromboli volcano, IEEE International Geoscience and Remote Sensing Symposium, IGARSS, 2009.

L. Noferini, T. Takayama, M. Pieraccini, D. Mecatti., G. Macaluso, G. Luzi, C. Atzeni, “Analysis of Ground-Based SAR Data With Diverse Temporal Baselines”, IEEE Trans. Geosci. Remote Sens. Letters, vol. 46, Issue 6, 2008.

L. Noferini, M. Pieraccini, D. Mecatti, G. Macaluso, G. Luzi, C. Atzeni, “DEM by Ground-Based SAR Interferometry”, IEEE Trans. Geosci. Remote Sens. Letters, vol. 4, Issue 4, 2007.

L. Noferini, M. Pieraccini, G. Luzi, D. Mecatti, G. Macaluso, C. Atzeni, “Monitoring of Belvedere glacier using a wide angle ground-based SAR interferometer”, Proc. FRINGE 2007 (Advances in SAR Interferometry from ENVISAT and ERS missions), ESA ESRIN Frascati, Roma, Italia.

L. Noferini, M. Pieraccini, D. Mecatti, G. Macaluso, G. Luzi, C. Atzeni, Ground-baesd Radar Interferometry for Monitoring Unstable slopes, IEEE Conference on Geoscience and Remote Sensing Symposium, IGARSS 2006, Denver, CO, USA.

L. Noferini, M. Pieraccini, G. Luzi, D. Mecatti, C. Atzeni, A. Tamburini, and Broccolato M., “Permanent Scatterers Analysis for Atmospheric Correction in Ground-Based SAR Interferometry,” IEEE Trans. on Geosci. Remote Sens., vol. 43, n. 7, pp. 1459–1471, July 2005.

M. Pieraccini, M. Fratini, F. Parrini, C. and Atzeni, “Dynamic Monitoring of Bridges Using a High-Speed Coherent Radar”, IEEE Trans. on Geosci. Remote Sens., vol. 44, n. 11, November 2006, pp. 3284-3288.

M. Pieraccini, N. Casagli, G. Luzi, D. Tarchi, D. Mecatti, L. Noferini, and C. Atzeni, “Landslide monitoring by ground-based radar interferometry: A field test in Valdarno (Italy)”, Int. J. Remote Sens., vol. 24, no. 6, pp. 1385–1391, Sep. 2002.

M. Pieraccini , G. Luzi and C. Atzeni "Terrain mapping by ground-based interferometric radar", IEEE Trans. Geosci. Remote Sens., vol. 39, no. 10, 2001 .

L. Pipia, X. Fabregas, A. Aguasca, J. J. A. Mallorqui, “Comparison of different techniques for atmospheric artifact compensation in GB-SAR differential acquisitions”, Proc. IGARSS, Denver, Colorado (USA), pp. 3739 – 3742, August 2006.

C. Werner, U. Wegmuller, T. Strozzi, and A. Wiesmann, “Interferometric point target analysis for deformation mapping”, Proc. IGARSS, Toulouse, France, pp. 4362-4364, July 2003.

H. A. Zebker, P. A. Rosen, and S. Hensley, “Atmospheric effects in interferometric synthetic aperture radar surface deformation and topographic maps”, Journal of Geophysical Research, vol. 102(B4), pp 7547-7563, April 10,1997.

H. A. Zebker, J. Villasenor, and S. Hensley, “Decorrelation in Interferometric Radar Echoes”, Trans. Geosci. Remote Sens., vol. 30, no. 5, pp. 950–959, Sept. 1992.

Chapter 3 Universal

Fluctuations in

Tropospheric Radar

Measurement

This chapter deals with the influence of the atmospherical turbulence on the radar

signal, focusing on the statistical description of the amplitude and phase information

as a function of the air turbulence. Using the radar described in Chapter 2 an

experimental study was realized and the results, interpreted using the complex

systems theory, confirm a strong connection between turbulence and the radar signal

fluctuations and statistic.

3.1 The effect of the atmosphere on the GB-SAR

As the state-of-art in SAR interferometry moves toward a more quantitative and

localized approach, the influence of error budgets becomes more and more important.

It is therefore of paramount importance to have a clear control on all possible sources

of external disturbances, which could significantly alter the measurements response.

In the last decades many geophysical applications based on SAR Interferometry were

presented. Even if SAR systems work at frequencies minimizing the atmospheric

absorption, the interferometric phase is affected by a delay due to the propagation in

the tropospheric layer. An increase in the amount of atmospheric water vapour

fraction between the acquisition times affects the atmospheric index of refraction

which induces an increase in the propagation time of microwave radar pulses. All this

appears as an increase in the distance to the ground surface, indistinguishable from

topography or real ground deformation. However, a comprehensive approache to

mitigate atmospheric artefacts in InSAR data is still lacking.

3.1.1 The landslide of Formigal

From 2005 to 2008 the TCHLAB was involved within the activities of the

GALAHAD project, with the opportunity to monitor many different natural scenarios

in different weather conditions and for a long time. A large collection of ground based

radar data was then recorded and processed. In particular a landslide in Spain (Figure

3-1) was monitored continuously for 45 days, gathering one image about every hour.

The collected data set consisted of more than 1000 SAR images and provided a

unique opportunity for studying the temporal behaviour of the radar signal.

Figure 3-1: Formigal, Spain. GB-SAR monitoring of a landslide. In the top part of the figure the RCS is projected onto a DEM of the landslide. In the low part of the figure the radar system is

showed with in background the observed scenario.

It was in particular observed that occasionally, but in coincidence with hard windy

conditions, the radar interferograms were affected by an overall decorrelation that

could not be explained by the recorded weather conditions (no rainy events). Focusing

the analysis on rocky areas without vegetation, the behavior was the same.

Considering the coincidence between abrupt decorrelation and hard windy conditions,

it was hypothesized that the atmospheric turbulence could drastically influence the

radar propagation even over short ranges, an observation which however rested on

purely qualitative ground.

No further experimental realizations were in fact subsequently developed to scrutinize

the process and reach an unambiguous proof of concept.

3.2 Radar- atmosphere interaction

Conscious of the experimental results on Formigal we developed the idea that

atmospheric turbulence might be the process behind the radar signal anomalies. A

little introduction on this topic is given here.

When an electromagnetic wave propagates through the troposphere its amplitude and

phase are distorted by a number of mechanisms. These distortions may be placed into

three groups:

1. absorption, which causes a direct attenuation of the wave

2. long trend fluctuations due to the slow variations of the meteorological

variables, as pressure, temperature and specially humidity

3. fast fluctuations due to the turbulence

Long trend fluctuations can be well understood and recreated by the many models of

the air index of refraction presents in literature.

However atmospheric turbulence remains the best candidate to try to explain what

observed in the Formigal images, this phenomena having time and spatial features

comparable with those of the images generated by a ground-based radar.

3.2.1 Spectrum of the wind

Different scale processes occur within the atmosphere, generally referred to as

synoptic-scale processes (dozen of days), meso-scale processes (from day to hour)

and micro-scale processes (less than an hour). Turbulent phenomena interest the

micro scale, corresponding to distances shorter than 1 km.

Among the atmospheric phenomena interacting with microwaves, the turbulence has a

significant impact on radar system. By creating time-varying inhomogeneities of the

atmospheric refractive index, turbulence induces amplitude scintillations and phase

delays on the propagating signals.

Working in the surface boundary layer, the GB-SAR are especially interested from

these disturbances. It is worth to note that the time scale of turbulence phenomena in

troposphere and the acquisition rate of usually GB-SAR applications are comparable,

the same is true for the spatial scale.

Figure 3-2: power spectrum of wind speed (R. B. Stull, 1988).

Figure 3-2 shows the power spectrum of the wind speed near the ground (dotted line)

and in the free atmosphere, the former being of interest for its influence on GB-SAR

applications. In the graph five peaks are evident:

1. between 1 and 10 minutes: micrometeorological peak, associated with turbulent

phenomena;

2. 12 hours: half-day cycle;

3. 1 day: daily cycle related to the solar heating;

4. 4 days: macrometeorological peak, it is related to movement of a large-scale

weather system due to global circulation;

5. 1 year: annual cycle;

The frequency range between the first two peaks contains almost no energy. This

range is called the “spectral gap”, which separates the two different frequency ranges

of the wind fluctuation. The wind fluctuation in the micrometeorological peak should

be taken into account in evaluating the effects in the GB-SAR images.

3.2.2 A brief description of turbulence

The physical mechanisms generating turbulence in the the atmospheric surface

boundary layer are numerous and complex, being related to the atmospheric mixing

and vertical transfer processes. The irregularities of the terrain and the proximity of

forest or water can have a strong influence on the generation of turbulence, as well

differences in solar illumination conditions.

The atmosphere is an inhomogeneous medium, altough mean values of refractive

index usually have some horizontal homogeneity. The inhomogeneities are produced

and supported by turbulent motion. Turbulence may be defined as an irregular motion

which in general makes its appearance in fluids, when they flow over solid surfaces or

when thermal instability is present. While the wind has a mean speed and a mean

direction which are reasonably constant for periods of hours, the instantaneous speed

and direction at a particular point may differ widely from the mean values. There is no

clear idea of how or why turbulence arises, or of the exact nature of the eddies which

form in a fluid in turbulent flow, but it’s known that the local variations of velocity in

a fluid are associated with the variations of static pressure which travel downstream

with the mean velocity of the stream.

When a current of air moves over uneven ground we might be tempted to think of the

eddies which form in it as cylindrical eddies having their axes in the horizontal plane

perpendicular to the direction of the mean wind. However investigations indicate that

the mean-square fluctuations of wind speed are approximately equal for the three

cordinates components. It is usual to define turbulent velocity at a point as the

difference between the instantaneous velocity at the point and the slowly varying

mean velocity of the stream, and the intensity of turbulence as the ratio of turbulent

velocity to mean velocity. Experimental results show that the intensity of turbulence

may be as large as 2/5, with root-mean-square values of the order of 1/10 or 1/20. The

intensity results essentially independent from mean wind speed above about 5 meters

per second, with the turbulent velocities distributed Gaussianly. Generally turbulent

velocities occurring in the lower troposphere vary from a few tens of meters per

second down to a few centimeters per second.

Concluding, the atmosphere near the ground may therefore be pictured as subject to

accidental fluctuations in refractive index, these fluctuations being autocorrelated only

over distances which can varies from hundreds meters to less than centimeters,

depending on the stability conditions.

In this study turbulence intensity is quantified by the Turbulent Kinetic Energy

(TKE), which is defined in fluid dynamics as the mean kinetic energy per unit mass

associated with eddies in a turbulent flow. Physically, the TKE is characterised by the

measured root-mean-square velocity fluctuations. This latter is calculated from the

fluctuations of the three wind velocity components v'x, v'y and v'z, revealed by an

anemometer:

!

Kt = v'x2 + v'y

2 + v'z2

Eq. 3-1

where the symbol <…> stands for a time average over the inspected time window.

3.2.3 Radar echo as a global quantity

The fluctuations of the air refractive index due to the turbulence in the boundary layer

can then materialize in a recorded imprint on radar signal, which could therefore

encode an indirect signature of turbulence.

Indeed, the embedding medium can be ideally segmented in neighboring volume

cells, each contributing to the signal according to an overall trend and its own statistic.

The signal impinging on the junction between two adjacent cells is partly reflected

and partly transmitted and the data collected at the receiver is sensitive to the

sequence of scattering events occurred along the propagation path. In this sense, the

radar echo can be thought as a global quantity, indirectly representing the specific

state of the crossed medium, and turbulence can be thought as the binder between the

neighboring volume cells, creating the correlation in the behaviour of the cells.

A comprehensive interpretative framework for such phenomena is however still

lacking, following the inherent difficulties to accommodate for the intimate, highly

complex nature of radar-air interactions.

3.3 Experimental tests

Aimed to a deeper understanding of the radar-turbulence interaction and aware of our

results in many survey campaigns, we arranged two outdoor experiments, both

realized in controlled environments. The first experiment was arranged in a facility of

the Faculty of Engineering at the University of Florence, but it showed many

problems that will be discussed later, so we planned a second experiment in the open

pit quarry described in Chapter 2.

The second experimental test has given very interesting results, suggesting that the

footprint of atmospheric turbulence be really encoded in the radar signal statistics.

Both the experiments were performed using the radar system described in Chapter 2,

but not in SAR configuration. Working in simple radar configuration permits to

acquire radar image with an high rate (up to 100 images for second), further

neglecting the problems related to the SAR images construction.

3.4 The experimental test at the facility of the Faculty of Engineering

The first experiment was arranged in a facility of the Faculty of Engineering at the

University of Florence, from December 30 2008 to January 5 2009.

In order to follow the temporal evolution of the radar signal in different places of the

scenario observed from the system, trying to understand the temporal and spatial

evolution of atmospheric disturbances, the test site was equipped with twelve targets

constituted by nine corner reflector, two metallic discs and a small concrete wall. It’s

worth to outline that every target was strongly connected to the ground in order to

prevent oscillations.

An ultrasonic anemometer and seven weather stations were installed too, in order to

measure wind velocities, temperature, pressure and humidity. The radar and six

targets were equipped with a weather stations.

The observed scene was essentially a meadow ending with a thick wood, where the

only manmade targets present in the scene were the radar targets, which were

distributed over a range distance from 10 to 120 m (Figure 3-3). On the test site, if

vegetation is ruled out, there are no uncontrolled moving targets, so supposing

instrumentation errors are reduced to negligible values by the autocalibration

procedure, any amplitude and phase change eventually detected can be related to

atmospheric disturbances.

Figure 3-3: the test site at the Faculty of Engineering of the University of Florence. In foreground the radar system, in background the targets.

3.4.1 Instrumentation

The Ku radar described in Chapter 3 was installed on a tripod in the facility. The radar

didn’t work in SAR configuration, allowing a fast sampling (1 Hz) of the scenario

monitored for 7 days. The range resolution was 37.5 cm.

Figure 3-4: the radar system installed on a tripod along with a weather station the power supply battery and the electronic of control of all the system.

Each weather station was projected and realized in our laboratory. The energy is

supplied by battery and data are stored in a SD card (Figure 3-5).

The WindMaster 3D sonic anemometer was programmed to acquire data

synchronously with the weather stations, but with a 4 Hz rate, saving them in a SD

card (Figure 3-5).

Figure 3-5: Top: a corner reflector target along with the wind anemometer and a wheather

station. Bottom: details of the electronic of the weather station.

3.4.2 Results

In this section are exposed the results about the relationships between atmospheric

turbulence and the oscillations of the signals from the targets.

• The RCS image

The RCS of the observed scenario is shown in Figure 3-6, where 13 peaks are well

evidents above the threshold of -20dB. Twelve peaks representing the manmade

targets in the scenario (Figure 3-3), while the peak labeled with “T“ in the image (at a

distance of about 73 m from the radar) represents a tree presents in the scenario.

Instead, the small peaks in the range between 100 and 120 meters are due to the trees

in the wood at the end of the observed scenario.

Figure 3-6: radar test site scenario, 13 targets are evidents. Each targets has a label: CR – corner reflector, D – metallic disc, A – corner reflector equipped with the anemometer, T is a tree, W -

concrete wall. The labels with the “*” mark the targets integrated with the weather stations.

The targets equipped with the weather stations are shown in Figure 3-7.

Figure 3-7: the six manmade targets equipped with the weather stations.

Looking at the Figure 3-7 it’s possible to estimate the level of signal-to-noise ratio for

the manmade targets. The level of noise is at about -30 dB from the maximum of the

signal (0 dB, normalizing the y axis with the value coming from the target CR9).

Then the SNR ranging from about -10 dB (target CR1) to -30 dB (target CR9).

Due to a low level of SNR it’s then impossible to observe the effects of the

atmospherical turbulence in the fluctuations of the radar signal, being these

fluctuations expected at about -50 dB.

• Time series analysis

In order to find a connection between turbulence and the fluctuations in the radar

signal coming from a target we focused on the time series of amplitude and phase of

the corner reflector target equipped with the WindMaster 3D anemometer. We plot

the time series of the standard deviation of amplitude and phase along with the

Turbulent Kinetic Energy time series (Figure 3-8).

Figure 3-8: time series of the standard deviation of amplitude and phase of the corner reflector

target equipped with the anemometer, compared with the TKE time series.

The standard deviation are evaluated by choosing a moving temporal window 10

minutes long, but the same results can be found using temporal window ranging from

1 to 60 minutes.

The evaluation of the standard deviation of amplitude and phase and TKE are

performed in two steps:

1. detrending of the time series

2. evaluation of the standard deviation on the temporal window

The detrending of the time series must be performed in order to remove the long term

atmospherical trend. These trends can be evaluated using a moving average with a

width of half hour, thus keeping the atmospherical components due to the turbulence.

A window half hour long is a compromise between having a meaningful number of

samples and a short temporal window to correctly evaluate the turbulence.

In Figure 3-8 a strong connection between fluctuatons in radar signal and TKE is well

evident. The same conclusions are valid for the other targets on the scenario.

• Problems and conclusions

Being the turbulence primarily generated by mechanical drag of the wind on the earth

surface, in the resolution cell of the radar there could be many scatterers that can

potentially overtake or hide what we are looking for. This is especially true at the

facility of the University of Florence, where the SNR of the targets is very low. This

is due to the presence of grass in the soil and of some trees in proximity of the targets

in each radar resolution cell. The low SNR in each resolution cell is a strong limiting

factor for our conclusion on the influence of turbulence on radar signal, because we

have to focus in the fluctuations of the signal due to the only turbulence.

Another important aspect to outline is that if we are looking for the statistic of a

signal, in special way looking at the extremely rare events, we need a consistent

number of points in order to reconstruct the correct representative statistic. From the

theory on the statistic of rare fluctuations a number of about 10 thousand points is

sufficient is requested.

Sampling at 1 Hz means to have 3600 points per hour, but this is not sufficient for

statistical analysis of turbulence. Indeed turbulence phenomena are concentrated in

the range of frequency corrisponding to time less than 30 minutes, with the peak of

energy near few minutes.

So in our present configuration we need to use a longer temporal window, up to many

hours, but this could imply to mix different kinds of turbulence, because there is a

high probability that the Reynolds number changes in such a long period.

Furthemore the radar system and the targets were posed on tripods or supports, which

have natural oscillations, especially in presence of wind. These meccanical

oscillations are normally found in the range 2-15 Hz, so to eliminate this unwanted

oscillations, we need to sample our signal at least at 30 Hz.

All the problems exposed in this section didn’t allow to find a clear connection

between turbulence and radar signal.

3.5 The experiment at Firenzuola

Aware of the many problems encountered at the test site of the University of

Florence, and of the studies on turbulence statistics, we planned another experiment.

Aimed to infer the influence of turbulence on the radar signal we decided to change

the location of the experiment and to deepen the analysis. The analysis was performed

in two steps: as a first step we looked for a straight connection between turbulent

kinetic energy and amplitude and phase fluctuations of the radar signal, then as next

and newer step we focused on the study of the probability density function of the

radar signal looking for the footprint of the turbulence.

Then, the probability distribution function of the collected signal was obtained and

studied, focusing on the shape modification as function of the air wind condition. This

allowed us to return a statistical characterization of the radar fluctuations due to the

atmospheric propagation, up to momenta of arbitrarily high order.

As we shall demonstrate, strong wind conditions are associated to asymmetric

leptokurtic profiles, standard Gaussian distributions being instead recovered as the

wind strength decreases. This observation can be quantitatively cast in the form of a

phase transition, the wind strength acting as the external control parameter.

To establish a clear causality relation between the air turbulence, here encapsulated in

the wind velocity, and the radar signal statistics constitutes the primary goal of this

Paragraph. Equally important, as an additional result of our analysis, we will confirm

the adequacy of the so called generalized Gumbel (GG) distribution as the

paradigmatic PDF of global quantities in correlated, spatially extended system.

3.5.1 The location

The location was chosen in order to reproduce as close as possible an ideal situation,

in which the radar signal fluctuations would solely have depended on the propagation

medium. This means to minimize any other possible source of fluctuations, as those

due to the vegetation. Indeed, we observed that vegetation may produce further

unwanted scattering, blurring away any distinctive footprint of clear air turbulence in

the radar signal statistics.

The location was a plateau in an excavation area of a disused quarry of Pietra Serena

located near Firenzuola (Firenze, Italy). The observed scene was essentially a

rectangular rock plateau with three sides ending with a cliff, without any kind of

vegetation (Figure 3-9).

3.5.2 Instumentation

Figure 3-9: The radar sensor at the outdoor experimental facility. The site was equipped with a

trihedral corner reflector (highlighted on the right side of the picture), posed at about 65 m from the radar, and an ultrasonic anemometer (highlighted on the left).

• The radar system

The portable continuous wave stepped frequency ku-band radar (Figure 3-10) used in

the survey of the open pit quarry of Firenzuola (Chapter 2) was mounted on a plateau

looking at an artificial target at fixed distance. Measurement parameters used for the

reported experimentation are summarized in Table 3-1.

The radar was mounted on a tripod and connected strongly to the ground in order to

prevent any oscillations.

The data here presented were recorded from 22:45 LT on March 25, 2009, to 04:15

LT on March 26, 2009. Clear air conditions were fulfilled during the measurement

period.

Figure 3-10: the radar system mounted on a tripod and strongly connected to the ground.

Polarization VV

Transmitted Power 4 dBm

Central frequency 16.75 GHz

Half power beam width 13°

Bandwidth 150 MHz

Number of transmitted frequencies 600

Range resolution 1.00 m

Unambiguous range 600 m

Sampling frequency 50 Hz Table 3-1

• The radar target

The site was equipped with an artificial radar target, a trihedral corner reflector (Figure

3-11), posed at the edge of the cliff opposite to the radar side. The radar-target distance

was about 65 m.

It’s worth to outline as the resolution cell containing the radar target was rocky and

without vegetation. The corner reflector was mounted on a tripod and strongly

connected to the ground, in order to prevent any oscillations. Mechanical oscillations

of the radar and target supports are however always present, but being well indentified

in the frequency spectrum, it’s possible to remove them with a filter.

Figure 3-11: the radar target constitued by a corner reflector mounted on a tripod and strongly connected to the ground.

• The anemometer

The WindMaster ultrasonic anemometer, located near the target, was used to register

the tridimensional wind velocity (Figure 3-12). The anemometer was placed in the

observed scenario, but out of the principal lobe of the radar antennae, in this way

excluding or limiting the influence of such target on the corner reflector. The

sampling rate was set to 4 Hz.

Figure 3-12: the ultrasonic anemometer.

3.5.3 Data pre-processing

Radar data were collected at a rate of about 50 Hz, sampling a 6 hours long recording

window. Mainly they represent the amplitude and phase of the backscattered signal

from the corner reflector after propagation to the corner and back. The data to be

analyzed were arranged in the time series A(t) and P(t), each one of about one million

points.

To just focus on air's impact on the propagation, all other possible sources of

disturbances need to be identified and subsequently excluded. In the current case,

because of the bareness of the scene, signal fluctuations may depend on the

oscillations of the radar tripod and/or the corner reflector's support, and on the

intrinsic instrumental noise. Mechanical oscillations can be straightforwardly

characterized in the frequency domain (Figure 3-13), their contribution was removed by

a low-pass filter. The instrumental noise is thermal noise, which is known to be white

and with a SNR of about 60 dB.

Figure 3-13: spectrum of the time series of amplitude and phase. The peaks due to mechanical oscillations of the tripods are well evidents.

The residual signal as departing from the expected, a priori constant output, is

imputed to atmospheric turbulence affecting the propagation.

In order to remove the components due to meso-scale and longer scale processes from

the series A(t), a multiplicative detrending was applied. The detrended Adet(t) is

determined as

!

Adet (t) =A(t)A0(t) Eq. 3-2

where A0(t) is the trend obtained by a moving average filter, which in this case is

equivalent to a low-pass filter A(t) with a threshold frequency of 5.5*10-4 Hz.

The series P(t) instead was processed with a subtractive detrending:

!

Pdet (t) = P(t) " P0(t) Eq. 3-3

where P0(t) is the trend obtained by a moving average filter, which in this case is

equivalent to a low-pass filter P(t) with a threshold frequency of 5.5*10-3 Hz.

Once detrended, only micro-scale fluctuations are expected to contribute to the PDF.

Figure 3-14: amplitude (top) and phase (middle) time series after the pre-processing. TKE in the bottom of the figure.

Figure 3-14 shows the amplitude and phase time series after the pre-processing.

It’s worth to outline the difference between the threshold frequencies of amplitude

and phase. This difference comes from the intrinsically different sensibility of the two

physical quantity to the turbulence.

3.5.4 Data analysis I

The range profiles obtained with the radar represent the complex, amplitude and

phase, radar backscattered signal relative to the different resolution cells along the

range direction.

In the top part of Figure 3-15, which reports the amplitude of a range profile, the corner

reflector is the peak at 65 m distance from the radar. The value corresponding to this

peak was extracted from all the complex range profiles in order to analyse its

behaviour with time. A time series about one million points long, which represents the

temporal evolution of the corner reflector radar response, was finally formed.

Amplitude and phase of this series, shown in Figure 3-15 (middle and bottom parts)

respectively, were analysed separately.

Figure 3-15: range profile (top); amplitude in arbitrary unit (middle) and phase (bottom) time series.

After the pre-processing, non-stationary modulation can still affect the detrended

series, e.g., varying Reynolds regimes. The signal was then divided into sections as

stationary as possible by applying subsequent partially overlapping temporal windows

about half an hour long (Figure 3-16).

For each analysis window, the signal fluctuations were estimated with respect to an

overall trend due to the macroscopic variations of the atmosphere, and compared with

the current turbulent wind regimes, which was revealed via the turbulent kinetic

energy measured by the anemometer. It was found that the signal fluctuations increase

as the turbulent turbulence increases and with a high degree of confidence the

relationship has resulted to be linear. This means that, even at the short distances (<

100 m) here considered and in moderate turbulent conditions as those recorded during

the observation period, the turbulence might deteriorate the radar measurements

precision severely.

Figure 3-16: amplitude, phase and TKE time series after pre-processing, here showed with the

temporal windows used to the analysis.

Figure 3-17: Standard Deviation of the amplitude series normalized with respect to its mean plotted versus time together with TKE (gray line). The mean and the standard deviation are

calculated over half an hour long time windows.

Figure 3-18: Scatter plot of the Standard Deviation of the normalized amplitude and TKE.

Figure 3-19: Standard Deviation of the phase series plotted versus time together with TKE (gray

line). The mean and the standard deviation are calculated over 3 minutes long time windows.

Figure 3-20: Scatter plot of the Standard Deviation of the phase and TKE.

The linear proportionality between the two quantities is evident (correlation

coefficient 0.98 and 0.89 for the amplitude and phase, respectively), and, although the

phase data present some noisy features, the general agreement with the lines traced on

the figures clearly shows that, as the turbulence strength increases, radar

measurements become less precise (Figure 3-18, Figure 3-20).

The time behavior of the standard deviation of amplitude and phase in respect to TKE

are shown in Figure 3-17, Figure 3-19.

It is worth to note that, in the present case, where the radar operated in very moderate

turbulent conditions and the target was very close to the radar (65 m), the phase

uncertainty amounts to 0.01 rad. By extending these results up to more usual distances

(1-2 km), the short-term atmospheric variations due to turbulent seem to produce

errors nearly up to 1 radian.

3.5.5 Data analysis II

The probability distribution function of the collected signal was obtained and studied,

focusing on the shape modification as function on the air wind condition. This

allowed us to return a statistical characterization of the radar fluctuations due to the

atmospheric propagation, up to momenta of arbitrarily high order.

In this part of the analysis we focused on the time series of amplitude, the

interpretation of the results coming from the analysis of the time series of phase being

still under study.

A 30 minutes long temporal window was applied sequentially, in order to analyze

finite portions of the series, that are as stationary as possible, while still containing a

meaningful number of samples for statistical analysis. Each selected window consists

of about 105 points, which renders of some reliability the study of rare fluctuations.

Different windows overlapped partially, for about one third of their length. Wind

conditions associated to the different analysis windows, is quantified via the

Turbulent Kinetic Energy (TKE) indicator Kt.

To obtain an exhaustive representation of the signal fluctuations, beyond the second

momentum, we reconstructed the full PDF, within each selected time domain.

Imagine to assume an ideal setting where interference of the radar signal with the

wind can be neglected. Then, having eliminated other possible sources of systematic

errors, one expects to recover the theoretically predicted PDF for a pointwise scatterer

(the corner reflector) suffering from thermal noise. This is the well-known Rice

distribution, a right skewed continuous curve, which approaches the Gaussian for high

signal-to-noise ratio. Surprisingly, and at variance with what customarily believed, a

quite different distribution appears from our data analysis: the PDF of Adet is an

asymmetric, left skewed, distribution characterized by an exponential tail on the one

side and a rapid falloff on the other. Such a significant discrepancy can be ultimately

imputed to reflect the non trivial interplay between propagating signal and turbulent

dynamics of the embedding atmospheric medium, an effect which is not

accommodated for in the theoretical scenario yielding to the Rice profile.

Motivated by this working ansatz, we set down to analyze more closely the

experimental PDFs. To this end, and aiming at a quantitative representation of the

data, we considered a non-Gaussian, one parameter family of distributions, recently

invoked to describe the fluctuations of global quantities in turbulent context.

These are the so-called Generalized Gumbel (GG) distributions, proposed to

universally describe the fluctuation of global quantities in correlated many degrees of

freedom systems.

In this respect the GG constitutes the natural extension of the Gaussian distribution

which instead apply to uncorrelated variables, as a consequence of the central limit

theorem. In the last years, GG distributions proved to adequately adapt to distinct

realms of applications, ranging from e.g. the magnetization of the two dimensional

XY model close to the Koterlitz-Thouless transition, to the statistics of the level of the

Danube river, passing through the power injected in a turbulent flow and in

electroconvetion. Complexity and high correlation should be the common feature of

these systems, ending up with a GG distribution.

The GG distributions family of curves Ga is defined as:

!

Ga (x) =aa "b(a)#(a)

"exp a " b(a) " (x $ s(a))$ eb(a )"(x$s(a ))[ ]{ }

a%&+,b(a) =1' x

d2

da2 ln#(a),

s(a) = x +1

b(a)" lna $ d

daln#(a)(

) * +

, -

.

/

0 0 0

1

0 0 0

Eq. 3-4

When the real positive parameter a varies from large to small values, Ga(x) changes

its shape from that of a Gaussian distribution towards a more and more skewed PDF.

This feature of the distribution family Ga rendered it very popular in complex

dynamics, when the statistics of global quantities are dealt with. The parameter a has

been proposed to be inversely related to the correlation length of the system, and it

has been used for measuring the number of its effective degrees of freedom. As a side

observation, it is worth noting here that the GG distribution with approaches

the well known Bramwell-Holdsworth-Pinton distribution that was found for the

fluctuations of turbulent power consumption.

Back to the data collected in our experiment, properly rescaled fluctuations are

calculated and the normalized experimental histograms reconstructed. These latter are

then interpolated via Ga(x), a being adjusted as the best fit parameter. In Figure 3-21

three representative curves are displayed which testified on the adequacy of the

proposed ansatz. The general agreement between data and GG distributions is

remarkable. The value of the shape parameter a as resulted from the fitting procedure

is also reported on each panel (notice that Ga depends on the gamma function, which

limits the maximum accessible value fot the shape parameter to a = 143).

Figure 3-21: The PDF of the series Adet(t) (dot line) and the best-fit GG distribution (solid line) shown in the upper panels refer to three different analysis windows.

The shape of the distribution varies from Gaussian, for low TKE values (upper panel,

left), to Bramwell-Holdsworth-Pinton distribution for high TKE values (upper panel,

right). Distributions with shape similar to the ones reported in the panels are found for

TKE values within the intervals highlighted below the panels. This suggests a

relationship between the shape, and hence the a parameter values, and the wind

a = ! /2

fluctuations intensity. The scatter plot in the main panel of the figure reports the

inverse of the shape parameter, a-1, vs. TKE. The direct driving of the turbulent wind

in the medium toward asymmetric statistics, is stressed by the solid line representing a

fit based on the sigmoid profile:

!

"1 + "2 # tanh "3 # (x $ "4 )[ ], with λi free parameters.

Even more interestingly, the values of a as obtained from the fitting procedure appear

to be strongly correlated to the quantity Kt, representing the turbulent wind intensity

within the crossed medium, for every chosen (statistically stationary) data segment.

This finding, reported in the main panel of Figure 3-21, materializes in a genuine phase

transition for a-1 vs. Kt, showing the direct footprint of turbulence in the medium on

the non-Gaussian nature of the radar fluctuations.

The transition between Gaussian and a non-Gaussian regime is rather sharp and takes

place for a specific value of

!

Kt = Ktc" 6m2 /s2 , in this case.

The robustness of the scenario here depicted is also tested versus modulating the size

of the windows, in which the full time series is being decoupled: the transition point is

not smeared over a wider Kt interval, but conversely, it remains strictly positioned in

!

Ktc .

Notice however, that the value of

!

Ktc was obtained keeping fixed the geometry of the

experiment, in particular the radar-target distance: it would be interesting to vary this

distance and monitor the consequent changes on the estimated value of

!

Ktc.

These latter results entail the possibility of establishing a control flag in wind

intensity, when the device is being operated in weather conditions which are likely to

strongly influence the data acquisition process.

Indeed, measurements fluctuating according to skewed distributions could in principle

induced biased conclusions (e.g. position of the mean) without proper information.

Moreover, it can be speculated that the fitting parameter a returns an indirect measure

of the correlation degree among the volum cells, in which the emitter-detector

distance can be ideally segmented. When the wind fluctuations are pronunced,

correlations become certainly crucial over the finite emitter-detector separation

distance and

!

a"# /2 . Qualitative agreement is also found in the opposite limit,

when the wind fluctuations turs weak.

3.5.6 Conclusions

Summing up, in this chapter it is demonstrated that the fluctuations in time of a radar

signal propagating through the atmosphere, even over short ranges (less than 100 m)

and moderate turbulence conditions, are generally distributed according to non

Gaussian PDFs. Moreover, they generally agree with the GG distributions. These

distributions, have been recently shown to describe very well the fluctuations of

global quantities in correlated systems, turbulent systems and many other natural

phenomena. The conjecture of wind driven fluctuations in the radar measurements is

further strengthen by the observation that the shape parameter a of the GG

distribution fitting the radar data PDFs is directly related to the TKE of the crossed

medium.

All these facts clearly support the idea that the non-Gaussian fluctuations of radar data

are determined by the wind fluctuations strength. The developing of a consistent

model for radar signal transmission through a windy atmosphere, according to which

the GG distribution emerges from the physical mechanisms of scattering, will be the

focus of forthcoming works.

Considering the radar signal as a global quantity relative to the medium, and recalling

the meaning of the parameter a in complex system dynamics, it could be concluded

that the wind fluctuations, revealed as TKE, affect the correlation degree among the

air volume cells, increasing the medium correlation length as they get stronger.

3.6 Acknowledgements

I am deeply grateful to Prof. Duccio Fanelli from the Department of Energy

Engineering at the University of Florence, for his deep interest, guidance and

unfailing help throughout the course of my Ph.D. program. Without him the work

described in this chapter could not be possible.

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