2.6 Level 1B Products
2.6.1 ASAR Level 1B Algorithms
2.6.1.1 ASAR Level 1B Algorithm Physical Justification
2.6.1.1.1 Introduction The purpose of this section is
to provide a basic understanding of the sensor geometry and SAR
signal, in order to understand the algorithm
descriptions in the succeeding sections.
2.6.1.1.2 Radar Geometry An imaging radar such as a SAR is
sidelooking. (see also "Imaging
Geometry" in the Geometry
subsection in the Glossary ). That is,
the radar antenna
beam is pointed
sideways, typically nearly perpendicular to
the flight direction of the spacecraft. The
basic sidelooking geometry is illustrated
in figure2.14 below. As the
transmitted pulse propagates from the
radar, it is reflected from scatterers, or
targets, located at increasing distances
from the radar along the ground. The
received echoes from a single transmitted
pulse form one line of SAR data, as a
function of the time delay to the
scatterers. Thus, one dimension in a radar image is the
distance from the radar to the scatterer,
called range. This is not the
same as distance along the ground, since the
radar is located at some altitude above the
ground. Thus the dimension in the image is
called slant range, as
opposed to ground range, as shown
in the figure. The spacecraft
transmits pulses and receives echoes
periodically. Because the spacecraft is
moving, a pulse is transmitted at
different locations along the flight
path. Thus, the alongtrack direction,
or azimuth, is the other dimension
in the radar image. Note that the speed
of light at which the pulse propagates
is much faster than the spacecraft
velocity, so the echo of the pulse from
the ground is assumed to be received at
the same spacecraft position at which
the pulse was transmitted.

Figure 2.14 Sidelooking radar geometry and slant range dimension in line of SAR data. 
2.6.1.1.3 Pulse Compression In collecting the SAR data, a
longduration linear FM pulse is transmitted.
This allows the pulse energy to be
transmitted with a lower peak power. The
linear FM pulse has the property that, when
filtered with a matched filter, the
result is a narrow pulse in which all the
pulse energy has been collected to the peak
value. Thus, when a matched filter is
applied to the received echo, it is as if a
narrow pulse were transmitted, with its
corresponding range resolution and
signaltonoise ratio.
Mathematically, in complex form, a
linear FM signal can be represented as:
  eq 2.2 
where
i
is the square root of 1, is time within the pulse,
and
K
is the frequency rate. Note the
signal has quadratic phase, the derivative
of which gives the linearly increasing
instantaneous frequency. Matched
filtering is illustrated in the
following figure. The linear FM pulse
has an instantaneous frequency that
increases linearly with time, as
shown by the increasing rate of
oscillation in the linear FM pulse in
the figure2.15 below. The
matched filter can be thought of as a
filter with a different time delay
for each frequency component of the
signal passing through the filter. That
is, the different frequency components
of the linear FM signal are each delayed
so that they all arrive at the same time
at the output of the filter. This way,
all the signal energy is gathered
into a narrow peak in the compressed pulse.

Figure 2.15 Illustration of pulse compression. 
2.6.1.1.4 SAR Signal The sequence of received
pulses are arranged in a twodimensional
format, with dimensions of range and azimuth, to form the SAR
signal. The signal is typically described by
the considering the signal received from a
single scatterer on the ground, or point
target. In this case, at each azimuth
position, a single pulse echo from the point
target is received. The delay of the
received pulse depends on the distance from
the radar to the target, and this distance
varies as the spacecraft travels along
the flight path. Also, pulses are received
for as long as the target is illuminated by
the antenna beam. This illumination time
determines the azimuth extent of the raw SAR
signal from a point target, or synthetic aperture.
The changing range, synthetic aperture, and
resulting SAR signal are illustrated in the
following figure.

Figure 2.16 Illustration of changing range to scatterer, synthetic aperture, and 2d SAR signal 
In the twodimensional array,
the received signal from a point scatterer
(target) follows a trajectory that depends
on the changing range to the scatterer,
as shown in figure2.16 above The
changing range to the scatterer also
affects the phase of each received
pulse. The antenna actually transmits a
high frequency, sinusoidal carrier
signal, that has been modulated by the
transmitted pulse. This carrier signal
is reflected by the target, received by
the antenna, and demodulated to get the
received pulse echo. The delay of the
carrier signal due to the range to the
target leaves a phase change in the
demodulated, received pulse. The coherent demodulation
in the radar preserves this phase
from pulse to pulse, resulting in an
azimuthvarying phase in the SAR signal.
Let
t
be time along the flight path
(azimuth time) and
R(t)
be the distance to the
target as a function of azimuth time.
Also, let
be the range time within a received
pulse (also called fast time), relative
to the pulse transmission time.
Then, the twodimensional SAR signal can
be expressed as
  eq 2.3 
where
is the transmitted pulse,
c is the speed of light,
and
is the wavelength of the carrier.
That is, the delay of the pulse is the
roundtrip distance divided by the speed of
light. Similarly, the phase of the
signal due to the delay of the carrier is
the roundtrip distance divided by the
wavelength. For a straight flight
path, the range
R(t)
is a hyperbolic function.
Typically, it can be approximated by a
quadratic function over the length of
the synthetic aperture. Thus, the SAR
azimuth signal is approximately linear
FM, and has an instantaneous
frequency that varies with azimuth time.
This property allows the azimuth signal
to be compressed, as described above for
pulse compression, which makes possible
the focussing of SAR images.
The varying instantaneous frequency of
the azimuth signal is analogous to the
varying Doppler frequency
shift of the carrier signal. The Doppler
frequency depends on the component of
satellite velocity in the
lineofsight direction to the target.
This direction changes with each
satellite position along the flight
path, so the Doppler frequency
varies with azimuth time. For this
reason, azimuth frequency is often
referred to as Doppler frequency.
2.6.1.1.5 Continuous versus Burst Data Different ASAR modes collect the
SAR data in different ways. The fundamental
distinction, from the point of view of SAR
processing, is whether the data from a swath on the ground is
collected continuously or burstwise. The
continuous model of SAR data applies to Image Mode (IM) data
processed with the rangeDoppler 2.6.1.2.3. algorithm.
The burst model of SAR data applies to Alternating Polarisation
(AP) and ScanSAR (i.e. Wide Swath (WS) and Global Monitoring
(GM)) data, and Image Mode data
processed to mediumresolution with the SPECAN 2.6.1.2.4. algorithm.
Essentially, in continuous mode, all
of the received echoes at each azimuth
position are used in the focussing of
the SAR data. In burst mode, only a
certain number of echoes are collected
at a time, i.e. bursts of data. The
following figure illustrates the SAR
signals from several point targets
distributed in the azimuth direction,
and the data that is actually collected
in continuous mode and burst mode.

Figure 2.17 Illustration of continuous and burst SAR data 
Note that the time between
bursts is smaller than the synthetic
aperture, so that the signal from one point
target shows up in at least two bursts.
Thus all the information needed to form an
image is still available in the SAR data.
Each burst of data can be compressed in the
azimuth direction, forming overlapping burst
images of the ground.
2.6.1.1.6 Image Cross Spectra The cross spectra of images
formed from different azimuth frequency
bands are used to estimate the ocean wave
spectra from Wave Mode products.
A SAR image of the ocean typically
has a striped appearance, corrupted by
speckle, as shown in the figure in the
Level 1b Wave
Products 2.6.2.1.2. section. The stripes
correspond to ocean waves. The
twodimensional spectrum of the image
will contain energy at the frequency
components corresponding to the stripes
in the image, and this could be
transformed to an estimate the ocean
wave spectrum. A problem, however is the
180 degree ambiguity in the direction of
the ocean waves. This ambiguity can
be resolved by using different look
images in the estimation of the ocean
wave spectra.
In the SAR signal described above, the
azimuth signal has a linear FM
character, which means the instantaneous
azimuth frequency is related to the
azimuth time in the signal. At different
azimuth times, the satellite is in
different azimuth positions and is
viewing a particular point on the
earth from different directions. Thus,
there is a relationship between azimuth
frequency and the viewing angle of
the point on the earth.
Given a complex SAR image, a set of
images can be formed from different
azimuthfrequency bands, or looks. These
lookimages correspond to different
viewing directions of the ocean surface,
which allows the ambiguity in the ocean
wave direction to be resolved. Instead
of using the spectrum of the
original SAR detected image, the cross
spectra between the look images is
transformed to an estimate of the ocean
wave spectrum. This method also allows a
noise reduction in the derived spectrum.
2.6.1.2 ASAR Level 1B Algorithm Descriptions
In general, the algorithms can
be broken down into the following categories:

Preprocessing: ingest
and correct the raw ASAR data

Doppler Centroid
Estimation: estimate the centre
frequency of the Doppler spectrum of the
data, related to the azimuth beam centre

Image Formation:
process the raw data into an image
(using rangeDoppler 2.6.1.2.3. and SPECAN 2.6.1.2.4. algorithms)
These steps are shown with
the flow of the imagery data in the flow
diagram below:

Figure 2.18 Flow of imagery data during processing 
The particular algorithms
used in each step depend on the type of
image product. Table 1 below summarises the
algorithms usedfor each product. (See Table 2.39 "Level 1B
Products", in Section "Level 1B
HighLevel Organisation of Products"
for the complete table of Level 1B products
and the corresponding product type description.)

Table 2.32 Processing steps used during generate products

Stripline processing is
applied to mediumresolution products
and to Wide Swath (WS) Mode
and Global Monitoring (GM)
Mode products.
Note 1: For Wave Mode (WV)
Level 1B products, Wave Mode processing
includes Doppler Centroid
Estimation 2.6.1.2.2. (DCE) and a rangeDoppler 2.6.1.2.3. algorithm to
form the imagette for crossspectra estimation.
2.6.1.2.1 Preprocessing
Preprocessing is applied to all raw data before
the other parameter estimation and
image formation steps are performed.
During preprocessing, PFASAR performs raw data analysis,
noise processing, and the processing of
the ASAR calibration
pulse data. Chirp replicas and antenna pattern gain
factors are obtained from the
calibration pulse processing. The
signal data for all modes is FBAQ decoded; prior to
processing (from 2, 3 or 4 bits back
to 8 bits).
Preprocessing includes:

Ingest and validation of raw
ASAR data. Check the
data packet headers.

Block Adaptive Quantisation
(BAQ) decoding.
Decompress the data from the 2 or 4
bit coded representation to the 8
bit representation. A table lookup
operation is used to perform BAQ
decoding. This is described further
in the Level 0
Packets 2.5.2. section.

Raw Data Analysis.
Complex data is collected in
inphase (I) and quadraturephase
(Q) channels. Receiver electronics
may introduce biases or
crosscoupling (nonorthogonality)
between the I and Q channels. This
can be estimated by collecting
statistics of the I and Q
channels. These statistics are
collected by accumulating the sums
I, Q, I squared and Q squared . Only
a fraction of the data set is used.

Raw Data
Correction. I/Q bias
removal, I/Q gain imbalance
correction, I/Q nonorthogonality correction.

Replica Construction and
Power Estimation. A
replica of the transmitted pulse is
extracted. This process is also
described in Characterisation 2.11.3.2. .

Noise Power
Estimation. Analysis of
noise packets.
2.6.1.2.2 Doppler Frequency Estimator
Doppler centroid estimation is a key
element in the processing of ASAR data.
A full discussion of the various methods
used for different product types is
given below
2.6.1.2.2.1 Introduction The Doppler centroid
frequency of the Synthetic Aperture
Radar (SAR) signal is related to
location of the azimuth beam centre,
and is an important input parameter when
processing SAR imagery. The Doppler
centroid locates the azimuth signal
energy in the azimuth (Doppler)
frequency domain, and is required so
that all of the signal energy in the
Doppler spectrum can be correctly
captured by the azimuth
compression filter, providing the
best signaltonoise ratio and
azimuth resolution. Also, for modes
processed by SPECAN 2.6.1.2.4. , accurate
knowledge of the azimuth beam pointing
angle is required in order to correctly
apply radiometric
compensation for the azimuth beam
pattern. Even with the use of
yawsteering, and an accurate
knowledge of the satellite position
and velocity, the pointing angle
will have to be dynamically
estimated from the SAR data in order
to ensure that radiometric
requirements of the PFASAR processor
are met.
The azimuth pointing angle translates
into a Doppler centroid frequency
that must be estimated to an
accuracy on the order of 25 Hz (for
burstmodes including Alternating
Polarisation (AP), Wide Swath
(WS) and Global
Monitoring (GM)), and 50 Hz for
all other modes.
There exist a number of algorithms to
estimate the Doppler centroid
frequency. For the PFASAR, the key
challenge is to define
techniques that will yield
sufficiently accurate estimates for
all processing modes. In particular,
the burstmode data
products require attention, since
their products are inherently more
sensitive to Doppler centroid errors.
The Doppler centroid varies with both
range and azimuth.
The variation with range depends
on the particular satellite attitude
and how closely the illuminated
footprint on the ground follows an
isoDoppler line on the ground,
as a function of range. The
variation in azimuth is due to
relatively slow changes in satellite
attitude as a function of time.
The azimuth signal in SAR is sampled
by the pulse repetition
frequency (PRF). As with
all sampled signals, there is an
ambiguity in the location of
frequency spectrum, by a multiple of
the sampling rate. For this reason,
the Doppler centroid is written as
  eq 2.4 
where the absolute
Doppler centroid frequency
is composed of two parts. The
fine Doppler centroid frequency is sometimes referred
to as the fractional part, and is
ambiguous to within the azimuth
sampling rate,
Fa
. ( the PRF). The other part
is an the integer multiple of the
azimuth sampling rate. Doppler centroid
estimation often refers to the
estimation of the fine Doppler centroid
frequency, which is limited to the
range, where
Fa
is the pulse repetition
frequency (PRF). The Doppler
ambiguity,
M
, is an integer in the set
{..., 2, 1, 0, 1, 2, ...} and the
method used to determine this value
is known as the Doppler Ambiguity
Resolver (DAR). In general, as ENVISAT is
yawsteered, the expected value of
M
is 0; however, the true value
may be larger. Usually the two
components of the unambiguous
Doppler centroid frequency are estimated
independently. Because of the
rangevariation of the Doppler
centroid, the Doppler centroid is
estimated at different ranges in the
data, and a polynomial function
of range is fit to the measurements.
The Doppler centroid may also be
updated in successive azimuth blocks.
The following sections will discuss
the Doppler Centroid Frequency and a
number of different methods used to
determine both the absolute and
the fine Doppler Centroid
frequencies. The method used will
depend on the type of product being generated:
 For
Global
Monitoring (GM) mode:

this is similar to
WS mode except the Look Power
Balancing method is not used
because the amount of data in
each azimuth look of GM mode
is too little for the Look
balancing method.
2.6.1.2.2.2 Madsen's Method Madsen's method is
used to estimate the fractional PRF part
of the Doppler centroid, or the fine
Doppler centroid. It is very simple to
implement and it is well suited to the
processing of ScanSAR data. The
Madsen method works by estimating
the autocorrelation function of the
data, and using the fact that the phase of the
autocorrelation function relates to a
Doppler shift in the Power Spectral
Density function. Madsen's
method is very efficient, as it does not
require any Fast Fourier
Transforms (FFTs) or peak
searches to be performed. It also works
in raw data or on range compressed data.
Madsen's method will be used to
derive an initial Doppler centroid
estimate in ScanSAR modes, to be refined
by the more computationally expensive,
but more accurate look balancing
techniques described below.
Madsen's method works by
calculating the first lag of the
autocorrelation function of the data
in the azimuth direction. This is
the Cross Correlation
Coefficient (CCC) of adjacent
azimuth samples. The phase of the
CCC,, is related to the
Doppler centroid by
  eq 2.5 
Note that the phase of the
CCC is only known within one cycle of
2 pi. From the
above equation, one cycle of
2 pi
corresponds to one interval of
the azimuth sampling rate, so that the
method actually estimates the fine
Doppler Centroid. In the
estimation algorithm, the CCC is
estimated over the azimuth line for
each range gate. The result is a
complex coefficient for each
range gate. The range gates are then
divided into groups, and
coefficients are then averaged over
each group of range gates. This
gives an Average Cross
Correlation Coefficient
(ACCC) for each group of range
gates, whose argument is used to
calculate the Doppler centroid. The
result is an array of fine Doppler
centroid estimates as a function of range.
2.6.1.2.2.3 MultiLook Cross Correlation
(MLCC) Method The MLCC Doppler centroid
estimator estimates both the integer and
fractional parts of the Doppler centroid
from the SAR data. The estimator
determines the absolute Doppler
frequency, without aliasing, obtaining
both of the parameters as a function of
range. Knowledge of the beam
pointing angles is not needed by the
algorithm, except as a cross check.
The MLCC algorithm described here
uses the property that the absolute
Doppler centroid is a function of
the frequency of the transmitted
signal. The MLCC algorithm is based
on computing the CCC from images
formed from two range frequency
bands, or looks. That is, the range
frequency spectrum is divided into
two parts, and Inverse Fourier
Transforms (IFFTs) are
taken to form complex images
corresponding to each of the two
range frequency looks. Then, in
the range time domain, the
ACCC's of two images are
computed, and the phase difference
between them,
  eq 2.6 
is used to compute the absolute
Doppler centroid. The
absolute Doppler centroid
frequency is given by:
  eq 2.7 
where
denotes the centre transmitted
frequency. The accuracy of this absolute
Doppler centroid is by itself not
sufficient, but it can be used to
estimate the Doppler ambiguity, as
discussed below. The fractional PRF part
is obtained from the phases of each of
the ACCC's of the two
looks,and:
  eq 2.8 
Note that the phases
measured from each look can be in
different
2 pi
cycles. That is, one or both of
the ACCC angles,, may have been wrapped
around. To correct for this, a simple
discontinuity detector is used to
set
to within the interval
. Similar to Madsen's
method, the error tolerance in the two
autocorrelation coefficient angles is
relatively high; an error of 5° in
the autocorrelation coefficient
angles causes an error of only 0.014 F
a in. Using this method
as the Doppler Ambiguity Resolver
(DAR), the Doppler ambiguity
estimated is:
  eq 2.9 dop_eq_3_4 
where
is the system offset frequency
discussed below. Finally, ginen
the fine Doppler estimate, the
ambiguity
M
is used to calculate the
value of the absolute Doppler
centroid
. Basically, the sum
of the two ACCC angles gives the
fractional PRF part, and the
difference gives the Doppler
ambiguity.

Figure 2.19 The MLCC Algorithm for Doppler Centroid Estimation 
The system offset
frequency,, must be subtracted from
the measured value of. This offset frequency
depends upon the antenna
characteristics, and must be calibrated for each
satellite. It originates from the
dependence of the antenna pointing angle
on transmitted frequency. While the
transmitted pulse sweeps across the
total bandwidth, the
instantaneous antenna pointing angles
vary. Since the Doppler centroid
frequency depends upon the antenna
pointing angles, the instantaneous
Doppler centroid frequency varies across
the pulse. (The SAR processor requires
the average Doppler centroid frequency.)
In the MLCC algorithm, the two range
looks are extracted each with a
different effective carrier frequency.
This means that each look has a slightly
different Doppler centroid. This effect
can be corrected by determining the
offset frequency and removing the
resulting bias. The system offset
frequency can be measured by processing
sensor data with a known Doppler
ambiguity. The value, before rounding,
can be compared to the correct
ambiguity. An appropriate offset
frequency can be determined by
minimising the error between the
nonrounded value and
M
. These values are output
by the processor into a debug file to
facilitate this calibration operation.
The system offset frequency may be beam
dependent.
2.6.1.2.2.4 Look Power Balancing Image products that are
formed with the SPECAN algorithm are
formed from periodic bursts of SAR data.
The burstiness may arise because of
the way the data is collected, as in
Alternating Polarisation mode or ScanSAR
modes (Global Monitoring, Wide Swath),
or bursts may be extracted during
processing as in the formation of
mediumresolution products. Typically a
burst is much shorter than the time it
takes the azimuth beam to pass over
a point on the ground (synthetic
aperture time). Thus, during a burst,
the portion of the antenna beam that
illuminates a target depends on the
target position. That is, targets at
different azimuth positions are
illuminated through different parts
of the antenna beam. The azimuth antenna
gain pattern then causes an azimuth
variation in target intensity in the
processed image. The known antenna
pattern can be used to compensate for
this radiometric variation, but the
Doppler centroid, or azimuth beam
pointing direction, must be known
very accurately. Figure2.20 illustrates
the effect of radiometric correction
with the antenna pattern, when the
Doppler Centroid is incorrect. The
result is an intensity variation
across the burst image. When the burst
images are stitched together to form the
image product, the periodic intensity
variation is seen as scalloping in the
image. ( For a further discussion of
this topic, refer to thesection entitled
"Descalloping
" 2.6.1.2.4.2. ). Typically, there
are at least two bursts within a
synthetic aperture length, so that
multiple bursts contain some data
that is received from the same
points on the ground. Thus, the
images formed from each burst
correspond to different looks of
overlapping areas on the ground.
In forming the final image product,
the detected burst images are added
together to reduce speckle.

Figure 2.20 Effect of Doppler Centroid Error on Image Intensity 

Figure 2.21 Radiometric Error Resulting from Doppler Centroid Error 
Figure2.21 illustrates the
residual scalloping error due to a fine
Doppler centroid error for the case of
two azimuth looks (the graph has
been computed assuming a RADARSAT antenna
pattern, although the trends are
applicable here). Note that a Doppler
centroid accuracy on the order of 25 Hz
is required to keep the total scalloping
error below 0.4 dB (+/ 0.2 dB from
the mean image intensity). The Doppler
centroid estimators described above are
not expected to meet the accuracy
that is required for radiometric
correction. Here we will describe a
technique for Doppler centroid
estimation from burst data which,
like those proposed in reference
Ref. [2.2 ]
, compares the energy in
the two extracted looks. The algorithm
is based on the fact that the two look
images are of the same patch of ground,
even they were formed through
different parts of the antenna beam. In
taking the ratio of the two images, the
scene content cancels. Also, the ratio
is averaged to reduce the speckle
noise. The result is an estimate of the
ratio of the azimuth antenna patterns
through which the two looks were taken,
which depends on the antenna azimuth
pointing angle or Doppler centroid.
To describe the method in more
detail, we may express the image
from lookas:
  eq 2.10 
where the dimensions of
the images are range time, range
time, and azimuth time. Thedenotes the weighting
applied by the antenna azimuth beam
pattern on the image
, as a function of azimuth time.
The weighting depends on the azimuth
time at which the burst
occurred,, and the azimuth
beam offset time from the zeroDoppler
direction (which is related to the The
Doppler centroid), is the reflectivity of
the ground patch being imaged that is
common to both looks, andis a multiplicative
(speckle) noise component, assumed
independent from one look to the
next. As,,and theare known, an estimate
ofmay be derived, with
accuracy subject to the constraints
imposed by the noise terms.
Taking the ratio ofto removes the dependence
on the reflectivity, and taking
the log of the ratio makes the
function better behaved:
  eq 2.11 
The noise term,
B, can be reduced by
averaging. The basic approach is to
estimateby forming an estimate of
A() from the look
ratio of the data, and compare it to a
set of template functions that has
been precomputed using the known
antenna pattern, to find the value
of that gives the best fit.
Specifically, we begin with an
assumed value ofand process the data,
including application of the
descalloping function ( see note: 1)
If the assumed value of
is correct, and if
B is small relative
to A(), then the
expected value of A() is 0. In
general, there will be an offset
between the assumed and actual
values of
which is what the algorithm
estimates. The estimate of is generated from the
unbiased estimate of A() by comparing
with a number of template
functions, each formed by
calculating the expected log ratio
of the data with a given Doppler
centroid offset between the data and
the applied descalloping function.
The look power balancing technique
requires a sufficiently accurate
initial estimate of the fine Doppler
centroid in order to work, as it
has limited "pullin".
Madsen's method or the MLCC
method is used as described above to
provide the initial estimate.
Computation of the Look Ratio
The raw SAR data is fully range and
azimuth compressed, and the descalloping
correction is applied, using the initial
Doppler centroid. Two azimuth looks
are extracted from the data; each burst
of data produces a Look 2 segment which
corresponds to the Look 1 segment from
the previous burst. Once the
processor has the two looks
corresponding to the same patch on
the ground, a boxcar type average of
length
in azimuth is applied to
the data of each look in order
to reduce the variance of the data.
After the azimuth averaging, the
ratio of the two looks is taken, and
the logarithm of the result is
generated. In addition to the boxcar
filtering performed in azimuth,
variance reduction is performed in
the range dimension by averaging
the resulting log ratios over a
number of adjacent range cells,. The result, as
function of azimuth sample, is
denoted
L(n)
and it is this
function that we attempt to match to
the A() term.
Generation of Template
Functions A family of reference
functions is defined by calculating the
expected radiometry for the individual
looks as a function of the offset
frequency between the real Doppler
centroid and that assumed in calculating
the descalloping 2.6.1.2.4.2. function:
  eq 2.12 
where where
n
is the azimuth sample index
(the output from the Specan process),
and
k
is the azimuth offset
error, in azimuth samples, which
corresponds to the Doppler centroid
error . The reference functions are
averaged in azimuth with the same
boxcar filter. Finally, the logarithm of
the ratio is taken resulting in a family
of template functions for
comparison. Note that for
k
= 0, there is no Doppler
centroid error, and as expected the
template is flat with a value of 0.
The magnitude of the template functions
increases monotonically with
k
.
Estimation of the Doppler Centroid
Error The look ratio
L(n)
is computed for a number of
bursts in azimuth. The results are then
averaged to obtain the average look
ratio, which is then compared against
each template function by computing the
difference between the functions. An
estimate of the Doppler centroid
error is determined by the error sample,
, which yields the minimum
difference between the compute look
ratio and the corresponding template
function. The error sample is then
converted to an estimate of the Doppler
Centroid Error (measured in Hertz):
  eq 2.13 
Where
LFFT is the length of
the FFT in the SPECAN algorithm.
The development thus far has
assumed that only 2 looks are
extracted per burst of data
processed. For the 3 and 4 look
modes, the method can be easily
generalised by comparing the outer
pair of looks. These looks are
chosen because the variation in the
beam pattern is highest at the edges.
This method gives the estimate of the
Doppler centroid error at a single
range. A loworder polynomial is
then fit to the values ofat a number of ranges
. When combined with the
polynomial producing the initial
Doppler centroid it yields the
absolute value of the Doppler centroid.
2.6.1.2.2.5 PRF Diversity Method In the Wide Swath and
Global Monitoring Modes in which ScanSAR
is used, a different PRF is used at each
of the different range subswaths.
This PRF diversity can be used to derive
the correct Doppler Ambiguity .
Ref. [2.4 ]
Recall the Doppler
Ambiguity number,
M
, is the number of PRF's
that must be added to the fine
Doppler centroid in order to obtain the
absolute Doppler centroid, as described
in the Introduction. When different
PRF's are used at different
range subswaths, the values of the
Doppler ambiguity number,
M,
for the various subswaths must
be such that the absolute Doppler
centroid is a smooth function of range.
In order for this technique to work, the
fine Doppler centroid must be known
to a resolution less than the difference
in PRF's between the subswaths.
Consider two measurements of the fine
Doppler centroid, each taken with a
different PRF. If the absolute
(i.e., actual unambiguous)
Doppler centroid is the same for
each measurement, then
  eq 2.14 
where
is the unambiguous Doppler centroid
frequency,is the pulse repetition
frequency for thebeam, andis the fine Doppler
centroid, again for the beam. The goal of Doppler
Ambiguity Resolution is to determine the
correct value of
Given a prior knowledge that the
satellite is yawsteered limits the
allowable values of to be in the set
{2,1,0,1,2}. (see note: 2) For
a sufficiently large difference
between the PRFs, and sufficiently
accurate estimates of
, the Doppler ambiguity may be
derived. The specific algorithm
for deriving the Doppler ambiguity
is as follows:
 derive the fine Doppler
estimates, for a series of
ranges,
r
, across the swath
 for each possible value of
M
= 2,1,0,1,2:
 Calculate the unambiguous
Doppler centroid for all
ranges in the nearest
subswath using the
appropriate PRF. Ensure the
continuity of the
absolute Doppler centroid
frequency as a function
range by incrementing or
decrementing
M
for each
range as necessary.
 For the next subswath,
calculate the absolute
Doppler centroid frequency,
again using the appropriate
PRF. Ensure that the correct
value of
M
is used across the
subswath boundary by
ensuring that the
unambiguous Doppler centroid
frequency is
reasonably continuous
between the beams (i.e. the
value of
M
may be different
than the one used in the
step above.)
 Continue outwards through
all the ranges for all the
subswaths, unwrapping as
necessary to ensure a smooth
function of the
unambiguous Doppler centroid
frequency as a function of range.
 Calculate a low order
polynomial through the
resulting absolute Doppler
centroid estimates. Measure
the distance between the
resulting polynomial and the
data points.
 Pick the value(s) of
M
that provides the best
fit between the polynomial and
the data points. Note that
ambiguity errors will cause a
step in the data that is
noticeable, and the polynomial
will not be able to do a smooth
fit except for when the
ambiguity is correct.
2.6.1.2.2.6 Notes
1. We assume here that
the descalloping or look weighting
function is the inverse of the
azimuth beampattern. This ensures that
the effective number of looks is a
constant for all targets. Other
approaches exist which lessen the
radiometric sensitivity to Doppler
centroid estimation errors and ensure
constant noise level, but not constant
speckle reduction. Note however,
that even if an alternate look weighting
algorithm were chosen, the inverse
beampattern could be used to derive the
Doppler centroid estimate as
described herein.
2. If the satellite
is not yawsteered, the range of
acceptable values will not be
centred about 0; however,
knowledge of the satellite position
and velocity yield an expected value
of the Doppler ambiguity, and the
uncertainty about that expected
value will be +/ 2 prfs.
2.6.1.2.2.7 References
F.
H. Wong, I. G. Cumming, A Combined SAR
Doppler Scheme Based Upon Signal
Phase, IEEE Trans. Geoscience and
Remote Sensing, 34 (3):
696707, May 1996.
. M.Y. Jin. Optimal
Range and Doppler Centroid
Estimation for a ScanSAR System,
submitted to IEEE Transactions
on Geoscience and Remote Sensing.
. S.N. Madsen.
Estimating the Doppler Centroid of
SAR Data. IEEE Transactions on
Aerospace and Electronic
Systems , Vol 25(2):pp 134140,
March 1989.
C.Y. Chang and J.C. Curlander.
Application of the Multiple PRF
Technique to Resolve Doppler
Centroid Estimation Ambiguity for
Spaceborne SAR. IEEE
Transactions on Geoscience and
Remote Sensing, Vol.
GE30(5): pp 941949, September, 1992.
