Chapter - 2shodhganga.inflibnet.ac.in/bitstream/10603/119651/9/09_chapter-2.p… · Since it operates on the same principle as RAdio Detection And Ranging (RADAR), the lidar is also

Chapter - 2

Instrumentation and Data Analysis

Chapter-2 Instrumentation and Data Analysis

52

2.1. Introduction

The current knowledge of the atmospheric behaviour leads to predict its future

responses based on different meteorological parameters such as pressure, density,

temperature and the concentrations of various chemical species. The coupling

behaviour of the different atmospheric layers do not permit a single instrument to

probe all the layers simultaneously however a variety of instrumental techniques like

in-situ, ground based, air borne and space borne instruments helps to probe all the

atmospheric layers at different time and locations. Remote sensing is a technique for

measuring, observing, or monitoring an object or phenomenon without making any

physical contact with the object which is under observation. Basically, there are two

types of remote sensors to probe the atmosphere: (i) Active remote sensors which

include the energy source under control and (ii) Passive remote sensors which do not

include the energy source. Radar, lidar, sodar, sonar etc are the examples for active

remote sensors whereas the optical and radio telescopes, radiometers, photometers,

spectrometers etc are the examples for the passive remote sensors [Siva kumar, 2002].

The present chapter outlines the technical characteristics and measurement methods of

different ground based and space borne instruments that are used for obtaining the

information on atmospheric density, temperature, horizontal winds, chemical

constituents etc.

The data used for the present study can be summarized as follows:

The Rayleigh Lidar temperatures over Gadanki (13.5°N, 79.2°E)

The Medium Frequency (MF) Radar winds over Tirunelveli (8.7°N, 77.8°E)

The Rocketsonde winds over SHAR, Sriharikota (13.7°N, 80.2°E)

The Sounding of the Atmosphere using Broadband Emission Radiometry

(SABER) experiment onboard the Thermosphere-Ionosphere-Mesosphere

Energetics and Dynamics (TIMED) satellite temperatures, chemical heating

rates and volume mixing ratios of H, O, O3 (http://saber.gats-inc.com/ )

The TIMED Doppler Interferometer (TIDI) winds (ftp://tidi.engin.umich.edu)


53

The Mass Spectrometer Incoherent Scatter-Extended-1990 (MSISE-90) model

temperatures (http://ccmc.gsfc.nasa.gov/modelweb/models/msis_vitmo.php)

The European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-

Interim geopotential data (http://data-portal.ecmwf.int/data)

The present chapter mainly describes the technical details and data analysis

methods of Rayeigh lidar instrument at NARL, Gadanki. It also includes the major

technical details of the above data sets provided by various instruments. Finally,

different data analysis methods to extract the wave characteristics are presented in this

chapter.

2.2. The LIDAR Technique

LIght Detection And Ranging (LIDAR) is one of the most powerful remote

sensing techniques to probe the earth’s middle atmosphere. In this technique, a beam

of light (laser) is used to make range resolved remote measurements. A lidar emits a

beam of light that interacts with the medium or object under study. Some of this light

is back scattered toward the lidar receiver which is used to determine some properties

of the medium in which the beam propagated or the object that caused the scattering.

Since it operates on the same principle as RAdio Detection And Ranging (RADAR),

the lidar is also called as Optical Radar or Laser Radar. The principal difference

between lidar and radar is the wavelength (frequency) of the radiation used. Radar

uses wavelengths in the radio band where as lidar uses light that is usually generated

by lasers in modern lidar systems. The wavelength of the light used by a lidar

depends on the type of measurements being made and may vary from infrared

(0.7μm<λ<12μm) through visible (400nm<λ<700nm) and into ultraviolet

(225nm<λ<400nm). Atmospheric lidar relies on the interactions, scattering and

absorption of a beam of light with the constituents of the atmosphere. Depending on

the design of the lidar, they are now being used extensively in different parts of the

globe to study aerosols/clouds (Mie Scattering), atmospheric density and temperature

(Rayleigh Scattering), metallic ion species (Resonance Scattering), minor constituents

and trace gases (Differential absorption), composition (Raman Scattering) and winds

(Doppler Lidar). As shown in the Figure 2.1, lidars can be classified into monostatic


54

and bistatic configurations. The monostatic configuration has the transmitter and

receiver at the same location. Monostatic systems are subdivided coaxial (Figure

2.1a) systems where the laser beam is transmitted coaxially with the receiver’s field

of view (FOV) and biaxial (Figure 2.1b) systems where the transmitter and receiver

are located adjacent to each other. In bistatic configuration (Figure 2.1c), the

transmitter and receivers are separated by several kilometres of distance.

Figure 2.1: Field of view arrangements of different lidar configurations [Argall and Sica,

2002].

The major advantage of using monostatic lidar is that a complete altitude

scattering profile can be recorded depending on good signal to noise ratio whereas in

the case of bistatic configuration, scattering can be detected only from a small volume

of atmospheric layer at any one time and the detector must be moved many times to

obtain an altitude profile.

Figure 2.2: Block diagram of a generic lidar system [Argall and Sica, 2002].

FOV of receiver Laser beam

(a) Monostatic coaxial (b) Monostatic biaxial (c) Bistatic

Laser Beam expander (optional)

Laser transmitted into atmosphere

Light collecting telescope

Optical filtering for wavelength,

polarization and/or range

Transmitter Receiver

Optical to electrical transducer Electrical recording system

Backscattered light

Detector


55

A lidar consists essentially of five subsystems: (1) a transmitter, in all practical

cases a laser, mostly, but not necessarily pulsed; (2) a transmitter optics; (3) a receiver

optics; (4) a detector; and (5) an electronic system for data acquisition, processing,

evaluation, display, and storage [Fujii and Fukuchi, 2005] as shown in Figure 2.2.

The purpose of the transmitter is to generate light pulses and direct them into the

atmosphere. Generally, laser is used as the source for a lidar because of its three

special properties: low beam divergence, extreme narrow spectral width and short

intense pulses provide significant advantages over white light. Usually a pulsed laser

is used instead of continuous wave (CW) lasers in the transmitter because of the

pulsed properties of a pulsed laser allows ranging to be achieved by timing the

scattered signal.

The receiver system of a lidar (normally a telescope) collects and processes the

scattered laser light and then directs it onto a photodetector, a device which converts

the light signal to an electrical signal. The spectral filtering uses a narrowband

interference (IF) filter which significantly reduces the background noise. The signal

detection and recording of a lidar takes the light from the receiver and records the

measured intensity as a function of altitude. Usually photomultiplier tubes (PMTs)

are used as detectors in incoherent lidars that use visible and ultraviolet light. The

other detectors namely, multianode PMTs, multicathode PMTs, avalanche

photodiodes and CCDs can also be used depending on wavelength.

2.2.1. The Lidar Equation

The theoretical performance of any lidar system is governed by the lidar

equation. This equation is used to determine the signal level detected by a particular

lidar system. The expected received photon counts is approximately equal to the

product of the system efficiency, number of transmitted photons, probability that a

scattered photon is scattered and the probability that the scattered photon is received

[Gardner, 1989]. The assumptions for lidar equation are: (i) particles are separated

adequately and undergo random motions so that the contribution to the total scattered

energy by many particles have no phase relation, (ii) a photon is scattered only once

(single scattering) so that the multiple scattering is neglected.


56

The elastic back scatter lidar equation can be given as [Weitkamp, 2005]

R

drrRcRAPRP

020 ),(2exp),(

2),( (2.1)

where

),( RP power received from a distance R for the laser wavelength ,

0P average power of a single transmitted laser pulse,

system efficiency,

A telescope area,

c velocity of light,

temporal pulse width,

),(),(),( RRR aermol backscatter coefficient which is the sum of

molecular and aerosol backscatter coefficients )( 11 srm ,

),( R extinction coefficient )( 1m .

Figure 2.3: The simulated lidar signal for the lidar system at NARL, Gadanki.

The simulated lidar signal using equation (2.1) is shown in Figure 2.3. Since the

density decreases with height exponentially, the lidar backscattered signal also

decreases with height in the same manner. Lidars are used in variety of applications

in the field of atmospheric science. Depending on the scattering mechanisms, the lidar

systems are divided Rayleigh lidar, Mie lidar, Resonance lidar, Rayleigh Doppler

lidar (RDL), Raman lidar, Differential absorption lidar (DIAL) etc. The principal


57

scattering mechanisms and their applications are summarized in Table 2.1. Based on

the application and altitude region of interest, the lidar systems can be chosen for

different laser wavelengths.

Table 2.1: Dominant scattering/absorption mechanisms for different lidar techniques

[Sivakumar, 2002].

Technique

Cross-section ( cm2 )

Principle

Applications (studies)

Elastic Scattering

Rayleigh Scattering

10-25 – 10-24

Dimension of scatterer << laser wavelength.

Density of air-molecules

and temperature (above 35 km)

Mie Scattering

10–6 – 10-5

Dimension of scatterer ≥ laser wavelength.

Cloud, Smog, Dust, Aerosols (Below 35 km )

Resonance Scattering

(SODIUM)

10-8 – 10-6

Excitation with a laser produces emission of photon at same wavelength.

Trace Species, like K, Na, Li,

Ca for 80-95 km.

In Elastic Scattering

Raman Scattering

10-25 – 10-24

Wavelength of scattered radiation shifts with respect to the laser wavelength. Positive and negative shifts represent stokes and anti-stokes Raman Scattering, respectively.

N2, CO2, H2O (Temperature studies upto

20 km )

Fluorescence Scattering

10-14 – 10-12

Wavelength of laser coincides with absorption line or band in the sample material. Transmission to a lower state of higher energy occurs followed by emission of light at wavelength similar to or less than the laser wavelength.

Trace Species, mainly NO2,

SO2 upto 5 km

Absorption

Resonance Absorption

(DIAL)

10-15 – 10-14

Excitation with a laser at two wavelengths (one coincides with maximum and other with minimum absorption of the gas of interest) provides concentration of the constituent.

Trace Species, like O3, NO2,

CO2, CH4, CO, H2O for upto 50

km


58

2.2.2. System Description of Lidar at NARL, Gadanki

The state-of-the-art lidar system comprising Mie and Rayleigh lidars installed at

National Atmospheric Research Laboratory (NARL), Gadanki (13.5°N, 79.2°E) by

Department of Space (DOS), Government of India has provided an excellent

opportunity to undertake this study. This lidar system was established in April, 1998

(with 11 W laser power) jointly by Communication Research Laboratory (CRL),

Japan and Department of Space, Government of India. Since then it is in operational

mode continuously on all cloud free nights and was upgraded (with 30 W laser power)

in January, 2007 with new high power laser source (Table 2.2). The Indo-Japanese

lidar (IJL) system at NARL consists of (i) Transmitter, (ii) Receiver(s) and (iii) Data

acquisition system. The schematic of lidar system at NARL is shown in the Figure

2.4.

Figure 2.4: Block diagram of Mie-Rayleigh lidar system at NARL, Gadanki.

Discriminators

Online Computer

Offline Analysis

P

S

R

U Trigger

9 mm Beam

10× Beam Expander

90 mm Laser Beam

45°

Laser Control

Unit

Seeder

Water Recirculator

Remote Box

Nd:YAG Laser with

SHG Steering Mirror

Receiver Section Transmitter Section

(b)

+200 V

R U

-2000 V

-2000 V

Beam Splitter (90:10)

Pulse Generator

-2000 V

P S

Mie Receiver

-2000 V

+200 V

Rayleigh Receiver

PBS

Data Acquisition Section


59

Table 2.2: Major specifications of existing lidar system at NARL, Gadanki.

Parameters Specifications Transmitter Laser Source Nd:YAG Model PL 9050 (Continum, USA) [PL 8020]* Operating wavelength 532 nm Average energy per pulse 600 mJ [550 mJ]*

Pulse width 7 nsec Pulse repetition frequency (PRF) 50 Hz [20 Hz]*

Beam diameter 9 mm (Expanded to 90 mm) Beam divergence < 0.1 mrad (after expansion) Receiver Rayleigh Lidar Mie Lidar Telescope type Newtonian Schmidt–Cassegrain Diameter 750 mm 350 mm Field of view 1 mRad 1mRad Interference filter band width 1.07 nm 1.13 nm Maximum transmission 48% # 48% #

Photomultiplier tube (Head-on-type) Model Hamamatsu, R3234-01 Cathode sensitivity 64.2 mA/lumen Anode sensitivity 2170 A/lumen Gain 34 x 107 Dark current 50 pA Data acquisition

Software Four channel (P, S & R, U) PC-based data acquisition system with EG&G

ORTEC Multi Channel Scalar (MCS) software

Bin width 2 µsec (=300 m of height resolution) Scan length 1024 channels Integration time 250 sec (12500 laser pulses) (5000 laser pulses)*

* 1998-2006; # can be varied; values in [ ] are of PL8020

The brief description of NARL lidar system is given in the following sub-sections.


60

2.2.2.1. Laser Transmitter

The transmitter section is a place where the signal (laser) is produced and

transmitted into atmosphere. The lidar system at NARL employs a Q-switched solid

state (Nd: YAG) pulsed laser from Continuum, USA (Figure 2.5a). This system emits

the primary wavelength of IR radiation at 1064 nm with pulse repetition frequency

(PRF) of 50 Hz. Frequency doubling using non-linear crystals like Potassium (K) Di-

hydrogen Phosphate (KDP) converts this IR radiation into visible green coloured laser

of wavelength at 532 nm which has the diameter of 9 mm. Before sending into

atmosphere, this 9 mm beam will be expanded into 10 times. The beam expander is

small refractor type of telescope that expands the laser beam from 9 mm to 90 mm (10

times larger). Now the beam divergence is very much reduced from ~0.45 mRad

(before expansion) to less than 0.1 mRad (after expansion) so that the beam can reach

higher altitudes without much divergence so that it remains within the FOV of the

receiver system in all ranges of interest. The main specifications of transmitter system

are given in Table 2.2. The expanded beam of 90 mm diameter is made to fall on a

steering mirror (Figure 2.5b) which is hard coated flat type mirror with the diameter

of 154 mm and thickness of 25 mm. The mirror is provided with azimuth and

elevation controls to align the transmit beam axis to receive beam axis and the mirror

is inclined at an angle of ~45° to the axis of the beam so that the beam enters into

atmosphere (Figure 2.5c). Once the laser pulses (50 pulses per second) enter into

atmosphere, they interact with atmospheric constituents and the back scattered signal

in the form of photons is collected by the optical receivers or telescopes.

Figure 2.5: (a) Nd: YAG laser source at NARL, Gadanki, (b) beam reflecting from the

steering mirror, (c) expanded (90 mm) laser beam entering into the atmosphere.

90 mm Laser Beam

(a)

(c)

Steering Mirror

Nd: YAG Laser source

(b)


61

2.2.2.2. Receiver Sub-systems

The receiver section is a place where the back scattered signal is received and

detected. The NARL lidar receiver system consists of two sub-systems with two

independent receivers, namely Mie receiver and Rayleigh receiver to collect the

atmospheric back scattered signal in the form of photons. The Mie receiver collects

the back scattered signal mainly due to aerosols and clouds (Mie scattering) covering

the height region of ~4-30 km and the Rayleigh receiver is for collecting the signal

mainly due to air molecules (Rayleigh scattering) between 30 and 90 km. The main

specifications of these two receivers are given in Table 2.2. A detailed description of

these two receivers is given below.

2.2.2.2.1. Mie Receiver

The Mie receiver consists of a portable, compact and vertical Schmidt–

Cassegrain type telescope (Celestron, USA) with a diameter (primary mirror) of 350

mm (Figure 2.6a) along with a secondary mirror of diameter of 150 mm. The FOV of

this telescope is 1 mRad for the iris size of 4mm. The separation between the

transmitter (steering mirror) and Mie receiver is ~1.5 m in the lidar system at NARL,

Gadanki. A narrow band pass filter centred at 532 nm with Full Width Half Maximum

(FWHM) of 1.13 nm is used just before a polarizing beam splitter (PBS). The

polarizing beam splitter splits the beam into co-polarized and cross polarized

components of the received signal. The photo multiplier tubes (PMT) are used for

detecting the signal which convets the light signal into electrical signal. Two PMTs

along with optical attenuators are used for the detection of co-polarized (P-channel)

and cross-polarized (S-channel) signals respectively. The main specifications of these

PMTs are given in Table 2.2.

2.2.2.2.2. Rayleigh Receiver The Rayleigh receiver employs a vertical Newtonian type telescope (Figure

2.6b) with a field of view of 1 mRad for the iris size of 2mm. This telescope consists

of a concave mirror with a diameter of 750 mm (primary mirror) and a plane mirror


62

with a diameter of 250 mm (secondary mirror) which is oriented at angle of 45° to the

receive beam axis facing primary mirror. The separation between Rayleigh receiver

and the transmitter is ~2.5 m. The secondary mirror is located at a distance equal to

focal length of the primary mirror (2372 mm). The secondary mirror focuses the

received signal at field stop iris and then directs to a collimating lens. This collimating

lens makes the signal parallel and fall over an interference filter (IF). The IF with

FWHM of 1.07 nm rejects the back ground noise and allows only the signal of

wavelength, 532 nm. This band limited signal is divided into the ratio of 90:10

through a non-polarizing differential beam splitter and then these signals are allowed

to fall on two PMTs of different gains. The high gain (R-channel) and low gain (U-

channel) PMT are given with 90% and 10% of incoming signal respectively. The high

gain PMT is used to provide adequate sensitivity with 90% signal to cover the higher

altitudes (50-90 km) and the low gain PMT is used to cover the lower altitudes (30-50

km) with only 10% signal in order to avoid the signal saturation in lower height range.

m Figure 2.6: (a) Schmidt–Cassegrain type telescope as a Mie receiver, (b) Newtonian type

telescope as a Rayleigh receiver (not to scale), (c) electronics rack containing different

electronic systems including pulse generator, HT voltages, discriminators etc and (d) MCS

card in PC (not to scale) used in lidar system at NARL, Gadanki.

(a) (b) (c)

(d)


63

The PMTs operates at high tension (HT) voltages of -2000 V (P, S, R, U) for

cathodes and +200 V (R, U) at anodes. The PMTs for R and U channels are gated

(delayed) by duration of 80 µsec (equivalent to 12 km height) using pulse generator in

order to avoid the PMTs from getting saturation due to large number of photons back

scattered from lower atmosphere. A positive pulse of +200 V is applied for switching

off the PMTs (R, U) operation for an initial duration of (T+80) µsec in each transmit

pulse. The output of the PMTs is given to pulse discriminators (Figure 2.4) to

discriminate the dark current generated in the PMTs due to thermionic emission.

After noise reduction, the output of the pulse discriminators are connected to a

computer (PC) based photon counting data acquisition system (Multi Channel Scalar

– MCS) operating a EG & G MCS software.

2.2.2.3. Data Acquisition System

The MCS-plus is a full size plugin card (Figure 2.6d) that converts personal

computer into a powerful and flexible multichannel scalar (MCS). A multi channel

scalar records the counting rate of events as a function of time. When a scan starts, the

MCS plus begins counting the input events in the first channel of its digital memory.

At the end of pre selected dwell time, the MCS advances to the next channel of

memory to count the events. This dwell and advance process is repeated until the

MCS has scanned through all the channels in its memory. Presently the MCS-plus PC

based photon count system is working with four data acquisition channels, two from

Rayleigh receiver (R, U channels) and two from Mie receiver (P, S channels) for

simultaneous photon counting with 12500 (5000) shots average for the PL9050

(PL8020) laser corresponding to time resolution of 250 sec with range resolution

(dwell time=2µsec) of 300 m. For example, the raw photon counts taken for P and R

channel are plotted in Figures 2.7a and 2.7b respectively with time integration of 250

sec with dwell time as 2 µsec. It can be observed from these two figures that the

photon counts are decreasing with height matching with the simulated profile shown

in Figure 2.3. In R-channel, the photon counts are available up to 12 km and the

received counts after 100-150 km are considered as noise due to moon light, stars and

other system source which can be buried in the lidar signal also.


64

Figure 2.7: Photon count profiles with respect to height for (a) P-channel and (b) R-channel.

2.2.3. Rayleigh Lidar Temperature Determination Several algorithms have been suggested for the determination of a temperature

profiles from a measured photon count profile [Hauchecorne and Chanin, 1980;

Shibata et al., 1986; Gardner et al., 1989]. Here, the method to determine the

temperature profile from the Rayleigh back scattered photon counts closely follows

that given by Hauchecorne and Chanin, [1980]. In the height range above 30 km

where the aerosol (Mie scattering) contribution is negligible, the range corrected

signal and atmospheric transmission is proportional to the molecular number density.

The laser pulse sent vertically into the atmosphere is backscattered by the air

Noise level

(a)

(b)

P-channel

R-channel


65

molecules and by the aerosols. If the atmosphere is divided into a number of layers of

constant thickness, Δz then the backscattered signal in the ith altitude layer (zi-Δz/2,

zi+Δz/2) can be given by the lidar equation.

The lidar equation for the backscattered photon counts can be given as follows:

(2.2)

where

iN : Backscattered signal from ith altitude layer,

0N : Number of emitter photons,

)(&)( imir znzn : Air molecules and aerosols concentrations,

mr & : Rayleigh and Mie backscattering cross sections,

0z : Altitude of the lidar site,

A : Telescope area,

),( 02

izzT : Atmospheric transmission,

K : Optical efficiency of the lidar system,

z : Thickness of the layer.

The scattering ratio is used to identify the aerosol and molecular layers and it can be

defined as (2.3)

The temperature profile is computed from the density profile from by assuming:

(i) The atmosphere obeys ideal gas law and is in hydrostatic equilibrium,

(ii) The scattering due aerosols is negligible when compared to molecular

scattering,

(iii) The atmospheric turbulence does not affect the mean air density,

(iv) The telescope FOV is large enough to include the entire volume of the

scattered beam.

zznznzz

zzAKTNzN mimriri

ii

)()(

)(4),()( 2

0

02

0

rir

mimriri zn

znznzR

)(

)()()(


66

The scattering ratio is assumed to be unity above 30 km so that the contribution

from aerosol scattering is negligible. Also the atmospheric transmission term

),( 02

izzT due to ozone absorption and molecular extinction is considered to be

constant between 30 and 90 km. The atmospheric density )( and the backscattered

photons )(N are related as (asssuming the atmospheric composition is constant with

altitude)

2)()( zzNz

Therefore, the density (relative density) profile can be calculated by the following

equation [Keckhut et al., 1993]:

z

zNzzCz

)()()(2

0 (2.4)

where C is the normalizing constant which is altitude dependent and z is the

vertical spatial resolution (300 m). The value of C can be determined using CIRA 86

(COSPAR International Reference Atmosphere-1986) model density values at 40 km

and the absolute density profile is derived. The relative uncertainty on the density

profile is assumed to be equal to the statistical standard error:

)(

)()()(

2/1

i

mi

i

i

zNNzN

zz

(2.5)

where )( iz and )( iz are atmospheric density and its standard deviation in the thi

altitude layer and mN is the background noise. The constant mixing ratio of major

atmospheric constituents (N2, O2 and Ar) and the negligible value of the H2O mixing

ratio justify the choice of a constant value for the air mean molecular density (M).

According to the hydrostatic equation, the air pressure )(P , density )( and the

temperature )(T are related as:

M

zTzRzP )()()( (2.6)

and dzzgzzdP )()()( (2.7)

where R is the universal gas constant for dry air, )(zg is the acceleration due to

gravity and M is the mean molecular weight. Dividing the above two equations gives

the following:

))(()()(

)()( zPLogddz

zRTzMg

zPzdP

(2.8)


67

The pressure at the top and bottom of the thi layer are related as

zzRTzMg

zzPzzP

i

i

i

i

)()(exp

)2/()2/( (2.9)

The density profile is measured up to thn layer (90 km) and substituting the pressure

at top of this layer from CIRA-86 or MSIS-E-90 (Mass Spectrometer Incoherent

Scatter Extended-1990) model, )2/( zzP nm , the pressures at the top and bottom of

the thi layer are given by

)2/( zzP i

n

jjnmjj zzPzzgz

1

)2/()()( (2.10)

)2/( zzP i zzgzzzP iii )()()2/( (2.11)

Therefore the temperature profile )( izT can be derived using pressure and density

profiles and it can be expressed as

)1(

)()(XLogRzzgMzT i

i

where )2/(

)()(zzP

zzgzXi

ii

(2.12)

The statistical standard error on the temperature is

)1()1(1

1)()(

XLogXX

XLogXLog

zTzT

i

i

(2.13)

with 222

)2/()2/(

)()(

zzPzzP

zz

XX

i

i

i

i

and

n

jjnmjji zzPzzzgzzP

1

222 )2/()()()2/(

Initially the temperature profiles are computed from the density profiles using

the backscattered photon counts from R and U channels. Then, the integrated

temperature profile is constructed using low-sensitivity U-channel for 30-45 km

(some times 25-45 km) and high- sensitivity R-channel for 55-90 km. For both the

channels, the temperatures are determined for 45-55 km using a numerical weighting

function based on convex convergence technique [James and James, 1968;


68

Bhavanikumar et al., 2000; Sivakumar et al., 2003]. Therefore, the temperature and

standard error for the transition region (45-55 km) are expressed as

)()(

)()()()()()5545( zTzTzTzTzTzTzT

UR

RUURkm

(2.14)

and

)()(

)()()()()()5545( zTzTzTzTzTzTzT

UR

RUURkm

(2.15)

There is an upper limit of this Rayleigh lidar technique [Kent and Wright, 1970]

due to change in composition of the atmosphere above 90 km, this technique is used

to measure the atmospheric temperatures up to about 90 and 100 km and above this

region, this technique becomes invalid [Argall, 2007] . However, the NRLMSISE-00

(Naval Research Laboratory Mass Spectrometer, Incoherent Scatter Radar Extended

Model-2000) [Picone et al., 2002] can be used to simulate the lidar signal (photon

counts) taking into the effects of change in atmospheric composition above 100 km

[For further details refer to Argall, 2007]. The sensitivity of the retrieved temperature

profile to the reference atmosphere pressure is shown by Sivakumar et al., [2003]. A

change of ± 10% in atmosphere reference pressure introduces a change in temperature

of about ±5 K at 80 km which is 10 km below the reference level and at 90 km, the

error is found to be ~ 10-15 % of the derived temperatures.

The errors in the density or temperature calculated from Rayleigh lidar

backscattered photon counts can be reduced by increasing the number of photons

collected by increasing the laser power, the surface area of the receiver (power

aperture product= product of the laser power and area of the receiver) and the

detection efficiency and optic transmission. Figure 2.8 shows the standard error

profiles for different laser powers and different integration times observed on 19

January 1999 for 11 W laser power and 22 January 2007 for 30 W laser power

[Sridharan et al., 2010] observed at NARL, Gadanki. As both the nights are close to

new moon days, the background noise is less. The individual (250 sec integrated)

photon count profiles are averaged for 7, 15, 30, 45, 60 and 90 profiles and the

temperature and standard errors determined using the equations (2.12) and (2.13). It

can be observed from Figure 2.8 that there is a reduction in the standard error with


69

high power laser (30 W) when compared to low power laser (11 W) so that the height

converge for the temperature retrieval has been increased with high power laser by ~

5 km . Further, the standard errors observed to be ~20 K at 85 km with low power

laser and the same can be observed at 90 km with high power laser for 90 profiles

integration.

Figure 2.8: Standard errors obtained in the Rayleigh lidar temperature profiles with different

time integration of photon counts for low power laser (11 W) on 19 January 1999 (Red) and

high power laser (30 W) on 22 January 2007 (pressure seeded at: red, blue-90 km, black-100

km, green-105 km) [Sridharan et al., 2010].

2.3. Medium Frequency (MF) Radar Winds

The Medium Frequency/High Frequency (MF/HF) radars become the dominant

means to study the mesospheric dynamics through the partial reflection drift (PRD)

technique [Vincent, 1984]. Between around 70 and 100 km, the atmosphere is slightly

ionized (contains particles with either positive or negative charge) and is very

turbulent. The MF/HF radars can be used for the study of mesospheric dynamics due


70

to the presence of free electrons rendered the frequency dependence of the radar

backscatter in this region. The radio refractive index is expressed approximately as

c

e

NN

TP

Ten

2)1076.7()1.075.3(1

5

2

(2.16)

where

e is the partial water vapor pressures (mb), P is the atmospheric pressure (mb), T is

the temperature (K), eN is electron number density and cN is the critical plasma

density.

In the above equation, the first term is the contribution of water vapor and is

valid in troposphere, the second term is due to dry air which is applicable above

tropopause and the third term is due to free electrons which is dominant in ionosphere.

The scattering mechanisms responsible for the radar signals to return from various

layers of the atmosphere are (i) Bragg scattering, (ii) Fresnel reflection, (iii) Thermal

scattering. The first two mechanisms provide coherent scatter which results from

macroscopic fluctuations in refractive index associated with clear air turbulence. The

third arises from Thomson scattering by free electrons in the ionosphere and the signal

returned is characterized by the statistical fluctuations of electron density due to

random thermal motions of electrons and ions. At least two radar techniques are used

to measure winds and other parameters in the lower and middle atmosphere. The first

one is the Doppler Beam Swinging (DBS) technique which uses large arrays with

narrow polar diagrams. By using at least three beams pointing various directions, the

total wind vector can be determined. Normally, the vertical beam is to determine the

vertical wind and the other two beams are for determining horizontal (zonal,

meridional) winds. The second technique to measure the winds is the spaced-antenna

(SA) array technique or partial reflection drift (PRD) technique that uses a vertically

pointing transmitting beam and minimum three spaced receiving antennas. By cross

correlating the received signals from the receivers, the relative time displacement can

be determined which can be used to infer the horizontal winds.

2.3.1. System Description of MF Radar at Tirunelveli

The partial reflection radar at Equatorial Geophysical Research Laboratory

(EGRL), Tirunelveli (8.7°N, 77.8°E geographic; 0.3°N dip), India, was installed


71

during middle of the year, 1992 by the Indian Institute of Geomagnetism. The radar is

similar to that of ATRAD made MF radar at Christmas Island [Vincent and Lesicar,

1991]. The schematic diagram of this radar system and its field view are shown in

Figures 2.9 and 2.10 respectively.

Figure 2.9: Schematic showing the geometry of the transmitting and receiving antennas of

the MF radar system at Tirunelveli.

This partial reflection radar is operating with the frequency of 1.98 MHz. It is a

monostatic system as the centres of transmitting and receiving antenna array coincide.

The transmitting array is arranged in a square and consists of four centre-fed half-

wave dipoles approximately 75 m in length. Since the location of Tirunelveli is

situated close to the magnetic equator and the ordinary (O) rays are polarized parallel

to the magnetic filed and the extraordinary (E) rays are polarized at right angles to the

magnetic field respectively, the dipoles are arranged in such a manner that one set of

dipoles is arranged parallel to the Earth’ magnetic field while the other is arranged

180 m

75 m

75 m


72

orthogonal to it. The ordinary mode dipoles are used for day time transmission and

the extraordinary mode dipoles are used during night time. The important

specifications of this MF radar are given in Table 2.3.

Table 2.3: Major specifications of MF radar system at Tirunelveli.

Parameters Specifications Operating frequency : 1.98 MHz

Maximum transmit power : 25 KW RMS

Maximum mean transmit power : 200W

Pulse width : 30 µsec

Pulse repetition frequency : 80 Hz during day and 40 Hz during night

Figure 2.10: The field view of the MF radar system antennas at EGRL, Tirunelveli.

The major sub-systems of MF radar at EGRL, Tirunelveli can be described as

follows:

2.3.1.1. Transmitter Sub-system

The transmitter system is in a small rack containing a combiner unit, 10 power

amplifiers, a fan unit, a driver unit and a power supply unit. The radar is capable of

resolving the winds at every 4 km height intervals with the pulse repetition rate of 80


73

Hz with 32 point coherent integration during day time and 40 Hz with 16 point

coherent integration during night. This is mainly to avoid the difficulties arising from

multiple reflections from equatorial spread F.

2.3.1.2. Receiver Sub-system

The three receiving antennas are of the inverted V-type and are located at the

vertices of an equilateral triangle whose basic spacing is 180 m. The receiving system

is controlled y a PC-based microprocessor which controls both the transmitter and

data acquisition system. The 1.98 MHz transmitter pulses which are linearly polarized

can be derived from the maser oscillator and frequency synthesis module. The three

receivers are of the superheterodyne type and in each receiver, the receiver 1.98 MHz

signal is mixed with a 2.475 MHz local oscillator to produce a 495 KHz intermediate

frequency (IF) which is then fed to the signal processor modules. The signal

processors are phase sensitive and the 495 KHz IF signals are well mixed with in-

phase and quadrature local oscillators to produce in-phase and quadrture components,

which are then digitized to 16-bit resolution.

2.3.1.3. Data Acquisition System

The radar data acquisition system acquires and stores the data in its memory. A

data set comprises 256 data points of 16 successive height samples (68-98 km) at 2

km resolution providing a total of 30 km range coverage. Each data sample is the

result of integrating the digitized data over a number of consecutive transmitter

pulses. The transmitter pulses are coherently averaged to produce a mean data point

for every 0.4 second. Thus, a complete data set is acquired in 102.4 seconds and is

then transferred to the host computer. The recording information which is

programmable by the host computer includes the number of heights per sample,

transmitter pulse repetition frequency, integrations (transmitter pulses) per sample

point, and sample points per data set and receiver gains. The day time and night time

recording configurations are different and receiver gains are dynamically adjusted by

the analysis program before the start of each data set. After each data acquisition run,

the PC performs a full correlation analysis on the 256 point complex data set to

determine means winds.


74

2.3.2. Wind Analysis Procedure The MF radar winds can be derived using the correlation analysis technique. The

MF radar at Tirunelveli has been yielding continuous data in the height range of 68-98

km however the useful measurements at night are confined to altitudes above 80 km

with largest acceptance rate at around 86 km. Winds are recorded every 2 min at 2 km

height intervals, though the pulse width gives the height resolution of 4 km. The

signal backscattered from the mesospheric irregularities represented by an angular

spectrum of plane waves, forms a Fresnel diffraction pattern on the ground. The

diffraction pattern which is random has a horizontal motion with a velocity twice that

of scattering irregularities. Depending on spatial and temporal correlations of the

amplitude fading of the signal received at three spaced antennas, Briggs [1984] has

developed the following generalized method, referred to as full correlation analysis to

derive the velocity and certain characteristics of the irregularities.

The space-time correlation function ),,( tyxf of the received signal pattern on the

ground is given by

2),,(/),,(),,(),,( tyxftyxftyxf (2.17)

Assuming that the pattern is anisometric and the correlation surface form ellipsoids in

time and space, the correlation function for a stationary pattern can be written as

)2(),,( 222 HKBA (2.18)

Equation (2.17) implies that surfaces of constant correlation are defined as

HKBA 2222 constant (2.19)

For the pattern moving horizontally with a velocity ),( yx VVV then the correlation

function remains the same as that given in equation (2.18). Transforming it to the

fixed coordinate system, then equation (2.18) becomes as

))((2)()((),,( 222 yxyx VVHKVBVA (2.20)

The above equation can be written as

)222(),,( 222 HGFCBA (2.21)

where

FHVAV yx (2.22)

and GBVHV yx (2.23)


75

The coefficients in equation (2.21) can be determined from the auto and cross

correlation functions using the spaced antenna observations provided all the

information about the moving irregularities. The steady component of horizontal

velocity is given by the equations (2.22) and (2.23) the random component of the

velocity cV defined as the ratio of the irregularity size to its life time is expressed by

yxyx

c VHVBVAV

KVV 22

22

2

(2.24)

The equation for the characteristic ellipse is given by

25.0

22 2 CHBA (2.25)

The spatial properties of the irregularity pattern are specified by the parameters, minor

axis, axial ratio and the orientation of the major axis of the characteristic ellipse which

is given by the equation (2.25). For the determination of the vertical component of

the velocity along with the horizontal, it is essential that the system should operate as

phase coherent radar.

Rajaram and Gurubaran [1998] calculated the standard errors in MF radar

horizontal winds (zonal, meridional) at Tirunelveli and found that the overall standard

error is less than 0.5 ms-1 on most of the occasions (all seasons) for the height region

above 84 km and it is 0.4 ms-1 to 7.7 ms-1 at lower heights (70-80 km).

2.4. Rocketsonde (RH-200) Wind Measurements

The Indian Space Research Organisation (ISRO) launches the sounding rockets

for the purpose of meteorological and atmospheric studies. Rohini (RH) series of

sounding rockets belongs to this type of rockets which can carry the payloads of 2 to

200 kilograms. Presently ISRO uses RH-200, RH-300, RH-300 Mk-II and RH-560

Mk-II rockets, which are launched from the Thumba Equatorial Rocket Launching

Station (TERLS) in Thumba and the Satish Dhawan Space Center (SHAR) in

Sriharikota (Figure 2.11). The major specifications of RH-200 sounding rocket are

given in Table 2.4. In the name ‘RH-200’, the number 200 implies the diameter of the

sounding rocket in millimetre. It is a two-stage rocket, which uses chaff as a payload

to measure the atmospheric winds in the altitude region of ~20–65-km with a height

resolution of 1 km [Devarajan et al., 1984; Ramkumar et al., 2013].


76

Figure 2.11: Picture showing the launch of RH-200 sounding rocket by ISRO.

Table 2.4: Major specifications of RH-200 sounding rocket.

Parameters Specifications

First launch 1 January 1979

Height 3.6 m

Diameter 0.2 m

Mass 100 kg

Thrust 17 kN

Stages 2

Payload 10 kg

The chaff consists of a large number of very thin metallic strips usually copper

or aluminum or metalized nylon, cut to half the wavelength size of the radio wave

from the tracking radar. The middle atmospheric winds can be measured by tracking

the vertically descending chaff using ground based radars when the chaff is released at

the apogee point of the rocket trajectory. After it is released the chaff gradually

spreads in space and becomes a diffusive target for the ground based tracking radar.

The strongest portion of the diffused echo is tracked by the radar and the position

coordinates measured by the radar gives the densest part of the diffused chaff cloud.

The wind velocity is calculated from the radar measured position coordinates of the

chaff (R, θ, ϕ) as a function of time, where R, θ, and ϕ are the range, elevation and


77

azimuth angles respectively. The standard errors involved in the wind measurements

are 2.7 ms−1 in 20–30 km altitudes, 1.9 ms−1 in the 31–50 km and 3.8 ms−1 in the 50–

65 km altitude regions [Devarajan et al., 1984; Ramkumar et al., 2013].

2.5. Sounding of the Atmosphere using Broadband Emission Radiometry

(SABER) Instrument

The Sounding of the Atmosphere using Broadband Emission Radiometry

(SABER) experiment is one of the four instruments launched on NASA’s (National

Aeronautics and Space Administration) Thermosphere-Ionosphere-Mesosphere

Energetics and Dynamics (TIMED) satellite (Figure 2.12) on 7 December 2001

[Russell et al., 1999]. It was designed and built by Space Dynamics Laboratory, Utah

State University.

Figure 2.12: The four instruments onboard TIMED’s science payload [courtesy: NASA].

The primary goal of the SABER experiment is to provide the data needed to

advance our understanding of the fundamental processes governing the energetics,

chemistry, dynamics, and transport in the mesosphere and lower thermosphere

(MLT). SABER began routine operations in January 2002, and it continues to provide

measurements of the radiative and chemical sources and sinks of energy in the MLT

region ranging from about 60 to 180 km. In every one orbit period (97 minutes),

SABER will observe polar regions in one hemisphere to high latitudes in the opposite

hemisphere. Over the course of a day, SABER will finish 15 orbits, and making

measurement covering 15 longitude bands (Figure 2.13).

Solar Extreme Ultraviolet Experiment (SEE)

TIMED Doppler Interferometer (TIDI)

Sounding of the Atmosphere using Broadband Emission Radiometry (SABER)

Global Ultraviolet Imager (GUVI)


78

Figure 2.13: SABER daily latitude versus longitude coverage [courtesy: NASA].

SABER accomplishes this with global measurements of the atmosphere using a

10-channel broadband limb-scanning infrared radiometer covering the spectral range

from 1.27 µm to 17 µm. These measurements are used to provide vertical profiles of

kinetic temperature, pressure, geopotential height, volume mixing ratios for the trace

species O3, CO2, H2O, O and H concentrations, volume emission rates for 5.3 µm NO,

2.1 µm OH, 1.6 µm OH, and 1.27 µm O2 (1Δ), cooling and heating rates for many

CO2, O3, and O2 bands, and chemical heating rates for six important exothermic

reactions between H, O, O2, O3, OH, HO2 etc. The latitude coverage of SABER limb

observations extends from 52°N to 83° S or from 52°S to 83° N, so that the

measurements can always cover latitudes from 52°N to 52°S. In addition, TIMED’s

orbit precesses slowly and the satellite takes approximately 60 days to cover the full

local solar time (~24 h) by combining the ascending and descending nodes [Remsberg

et al., 2003; Gan et al., 2012].

The SABER instrument consists of version 1.07 (V1.07) and advanced version

of 2.0 (V2.0) containing Level 1B, Level 2A and Level 2B data sets. Level 1B data

consists of calibrated radiance profiles converted to radiance units with instrument

effects removed, geolocated and gridded to a uniform angle spacing. Level 2A data

consists of profiles of kinetic temperature, pressure and density, profiles of emission

rates of nitrogen oxide (NO), hydroxyl (OH), and oxygen (O2); mixing ratios of ozone

(O3), water vapor (H2O),carbon dioxide (CO2), oxygen (O), and hydrogen (H). Level

2B data consists of profiles of CO2, O and H mixing ratios. Also it consists of profiles

of cooling rates from CO2, NO, O3, and H2O; profiles of heating rates from O3, O2,


79

and CO2; profiles of heating rates from various reactions; profiles of emission and

heating efficiency from OH and O2; and profiles of geotropic wind.

Table 2.5: Different parameters of TIMED-SABER with their height range and

estimated accuracies.

Parameter Measurement range Estimated accuracy

Temperature (T) 10 - 100 km 0.5K, 15 - 65 km 1K, 65 - 75 km 2K, 75 - 100 km

O3 (9.6µm) 15 - 100 km 5%, 15 - 65 km 20%, 65 - 90 km O3 (1.27 m) 50 - 95 km 10%, 55 - 85 km (Day) 15%, 85 - 95 km H2O 15 - 80 km 10%, 20 - 65 km 25%, 65 - 80 km CO2 65 - 100 km 10%, 65 - 100 km Energy Loss Rates OH (ν), 1.06 m* 80 - 100 km 0.5%, 80 - 90 km OH (ν), 2.10 m* 5%, 90 - 100 km O2 (1), 1.27 µm* 50 - 105 km 0.05%, 50 - 70 km 0.2%, 70 - 80 km 1%, 80 - 90 km O3 (9.6 m) 15 - 100 km 0.5%, 50 - 70 km (Night) 2%, 70 - 90 km CO2 (15 m) 15 - 120 km 3%, 80 - 100 km NO* 90 - 180 km 3%, 100 - 150 km 5%, 150 - 170 km CO2 (4.3 m) 85 - 150 km 10%, 95 - 140 km (Day) *volume emission rate


80

The various SABER data products along with their height coverage and

measurement accuracies are given in Table 2.5. The data’s native vertical resolution

is approximately 2 km [Mertens et al., 2004]. Remsberg et al. [2008] assessed the

quality of version 1.07 temperature data and suggested that the systematic error is no

more than 2 K below 70 km, while in the upper mesosphere lower thermosphere

(UMLT) region the error increases with altitude from 1.8 K at 80 km to 6.7 K at 100

km. More information about the instrumentation and data sets can be obtained from

http://saber.gats-inc.com/ and Remsberg et al., [2003]; Mlynczak et al., [2007]; Gan

et al., [2012].

2.6. TIMED Doppler Interferometer (TIDI) The TIMED Doppler Interferometer (TIDI) is one of the four instruments

onboard the Thermosphere, Ionosphere, Mesosphere, Energetic, and Dynamics

(TIMED) satellite (Figure 2.12). This is developed by University of Michigan. TIDI

is a Fabry-Perot interferometer designed to investigate the dynamics and energetics of

the Earth’s mesosphere and lower-thermosphere-ionosphere (MLTI) from an altitude

of ~ 70 to 120 km [Yee et al., 1999; Killeen et al, 2006]. TIDI measurements will

allow us to obtain a global description of the vector wind and temperature fields, as

well as the important information on gravity waves, species densities, airglow and

auroral emission rates, noctilucent clouds, and ion drifts. TIDI will provide basic

information about global winds and temperatures with a vertical resolution 2.5 km at

the lower altitudes and with accuracy that approaches ~3 m/sec and ~3 K,

respectively, under optimum viewing conditions. Using limb scans of airglow

emissions through four orthogonally oriented telescopes, the TIDI simultaneously

measures the mesospheric neutral winds and views emissions from OI 557.7nm and

O2 (0–0) to determine Doppler wind in the TIMED altitude range.

TIDI consists of three major subsystems: four identical telescopes, a Fabry-Perot

interferometer with CCD detector, and an electronics box. Light from the selected

regions of the atmosphere is collected by the telescopes and fibre-optically coupled to

the detection optics. The four TIDI telescopes perform limb scans through the

terrestrial airglow layers throughout the satellite orbit. TIDI obtain these scans

simultaneously in four orthogonal directions: two at 45º forward but on either side of


81

satellite’s velocity vector and two at 45º rearward of the satellite (Figure 2.14). These

four views provide the measurements necessary to construct the horizontally resolved

vector winds as a function of altitude within the MLT region along two parallel tracks,

on either side of the spacecraft. Each vertical scan consists of individual views with

2.5º (horizontal, along the limb) by 0.05º (vertical, normal to the limb) as angular size.

The vertical altitude resolution of the instrument is 2.5 km, but the altitude spacing

between the views can be adjusted to yield a measurement vertical resolution of half a

scale height throughout the limb scan. TIDI observes emissions from OI at 557.7 nm

and rotational lines in the O2 (0-0) Atmospheric band at 762 nm to determine the

Doppler shift and hence the wind velocities. The three O2 (0-0) band filters are P15

(765.07 nm), P9 (763.78 nm), and broadband (764.00 nm).

Figure 2.14: Geometrical view of TIMED Doppler Interferometer (TIDI) [Source:

http://tidi.engin.umich.edu/].

Currently, the TIDI instrument samples the altitude range from 70 to 120 km

during the day time and from 80 to 103 km during the night. The daytime mode

includes the three O2 (0-0) band filters along with few samples of the O 557.7 nm

(green) at high altitudes. The altitude step size for three O2 (0-0) band filters is 1.25

km and 1.5 km for O 557.7 nm. The night mode also includes the same filters with

altitude step size of 2.5 km for all emissions. As mentioned earlier, the TIDI samples

on the two sides of the satellite. For each side, two telescopes sample the same

location 9 min apart as the satellite moves along its orbit at 625 km altitude. The two

telescopes view the same region in nearly orthogonal directions, which allows

measurements from two viewing directions that can be decomposed into meridional


82

and zonal wind directions. The simple way to calculate the wind vector is to simply

pair the sampling locations of one telescope with nearest neighbours from the other

telescope. The distance between the two sampling locations should be less than 500

km. The pair of samples is then used to calculate the meridional and zonal wind

components at the first telescope sampling location. If no pairing sample can be

found, then no vector winds are calculated. More information about TIDI instrument

and data retrieval methods can be found in Killeen et al, [1999, 2006].

2.7. Data Analysis Methods to Extract Wave Characteristics

The atmospheric parameters like pressure, density and temperatures exhibit

signatures of gravity waves, planetary waves, tides etc. In order to extract the waves

from these background parameters, their perturbations should be subjected to spectral

analysis. Spectral analysis using the Fourier Transform technique has been proved as

a powerful tool in the analysis of wave activity in the atmosphere [VanZandt, 1982;

Dewan et al. 1992; Turnbull and Lowe, 1991; etc.]. One drawback of the Fourier

Transform is that it gives time-averaged spectrum. This is adequate for stationary time

series in which the characteristics of the time series do not change with time. The

geophysical data, however, are mostly non stationary in nature. The spectral content

of the time series changes with time, and the time averaged amplitudes found by

current methods are inadequate to describe such phenomenon. In recent years, the

time honoured technique of Fourier analysis has given away to more advanced

representations known as Short Time Fourier Transforms [Portnoff, 1980] to represent

the joint time-frequency representations. Wavelet Transform has gained popularity

since the 1980’s [e.g. Goupillaud et al., 1984]. More information on wavelet

transform can be found in Daubechies [1990], Young [1993], Rioul and Vetterli

[1991], Hlawatsch and Boudreaux –Bartels [1992], Cohen [1989] and Qian and Chen

[1999]. The following sub-sections include various data analysis techniques used in

this thesis to observe atmospheric waves.

2.7.1. Time Series (Fourier) Analysis

The Fourier analysis or harmonic analysis of a time series is a decomposition of

the series into a sum of sinusoidal components (the coefficients of which are the


83

discrete Fourier Transform of the series). In this analysis, the periods of the

oscillations of interest are assumed to be precisely known. For periodic components of

unknown period, the result may be very misleading. A sinusoidal function of

frequency, (in radians per unit time) or period /2 can be written as

)cos()( tRtf (2.26)

where R is the amplitude and is the phase.

The description of a time series in Fourier series expansion can be written as follows:

1

000 )2sin()2cos(

2)(

nnn tnBtnAATF (2.27)

Where 0 is the fundamental frequency, nA and nB are the Fourier coefficients. These

Fourier coefficients can be expressed as

1

2cos)(2n

n dtTnttF

TA and (2.28)

1

2sin)(2n

n dtTnttF

TB (2.29)

The amplitude )( nR and phase )( n of the Fourier harmonics can be determined from

2/122 )( nnn BAR and (2.30)

n

nn A

Ba tan (2.31)

This method is useful when the periodicity clearly exists in the data and needs to be

described exactly. But for an arbitrary set of data, the Fourier transform for a function

)(th can be written as

dtiftfHth )2exp()()( (2.32)

where t is measured in seconds and f is in cycles/sec or Hz.

In frequency domain, the equation (2.32) becomes as

dtiftthfH )2exp()()( (2.33)


84

The total power in a signal is same whether that is calculated either in time

domain or in frequency domain. This result is known as Perseval’s theorem which

can be expressed as:

Total power

dffHdtth 22 )()( (2.34)

The one-sided power spectral density (PSD) of the function, )(th is defined as

22 )()()( fHfHfPL where f0 (2.35)

If the function )(th goes from t , then its total power and power spectral

density will in general be infinite. Suppose N consecutive sampled values, kh ,

1.....,.........2,1,0 Nk so that the sampling interval is . The integrals (2.32),

(2.33) may approximated by Discrete Fourier Transform which can be written as

1

0

2expN

kk N

iknhH (2.36)

The discrete Fourier transform (DFT) maps N complex numbers of shk into N

complex numbers of sH n . The DFT computes 2N complex computations where as

the Fast Fourier Transform (FFT) computes NN 2log operations.

2.7.2. Wavelet Spectral Analysis

Wavelet may be seen as a complement to classical Fourier decomposition

method. Wavelet means a small wave (the sinusoids used in Fourier analysis are big

waves) and in brief, a wavelet is an oscillation that decays quickly. The term

“wavelets” can be used to describe a wide range of topics. It was first introduced by

Morlet [Morlet et al., 1982a, 1982b] in describing the Continuous Wavelet Transform

using Morlet wavelets.

Like Fourier transform, wavelet transform deals with the expansion of functions in

terms of a set of basis functions. Unlike Fourier transform, wavelet transform uses

generalized local base functions (wavelets) that can be stretched and translated with a


85

flexible resolution in both frequency and time. The flexible windows are adaptive to

the entire time- frequency domain, known as the wavelet domain, which narrows

while focussing on high frequency signals and widens while searching the low

frequency background (Figure 2.15). The different uses of wavelets in computational

mathematics, and in particular in computer graphics, are related with two facts: (i)

representation and reconstruction of functions and (ii) multi resolution representation.

The term “wavelet function” is used generically to refer to either orthogonal or

nonorthogonal wavelets. The term “wavelet basis” refers only to an orthogonal set of

functions. The use of an orthogonal basis implies the use of the discrete wavelet

transform; while a nonorthogonal wavelet function can be used with either the

discrete or the continuous wavelet transform [Farge, 1992].

Figure 2.15: Time-frequency windows used in Wavelet transform and their corresponding

time series represented in time space and frequency space [Lau and Weng, 1995].

The wavelet transform can be used to analyze time series that contain

nonstationary power at many different frequencies [Daubechies, 1990]. If one has a

time series, nx with equal time spacing δt and n = 0 … N − 1. Also considering if one

has a wavelet function, )(0 , which depends on a nondimensional “time”

parameter, . To be admissible as a wavelet, this function must have zero mean and

be localized in both time and frequency space [Farge, 1992]. For a Gaussian

modulated plane wave function, the Morlet wavelet equation can be given as

[Torrence and Compo, 1998]:

2/41

02

0)( eei (2.37)

where 0 is the nondimensional frequency. Though the Fourier analysis shows the

dominant frequencies of the simulated signal, it is impossible to tell at what times a


86

particular signal is generated or event takes place. The advantage of the wavelet

analysis over the Fourier analysis is that the time information of the signal is not lost.

----------- END OF CHAPTER -----------

Documents

Chapter - 2shodhganga.inflibnet.ac.in/bitstream/10603/119651/9/09_chapter-2.p… · Since it operates on the same principle as RAdio Detection And Ranging (RADAR), the lidar is also