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2.6 Level 1B Products

2.6.1 ASAR Level 1B Algorithms ASAR Level 1B Algorithm Physical Justification 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. Radar Geometry

An imaging radar such as a SAR is side-looking. (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 side-looking 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 along-track 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 Side-looking radar geometry and slant range dimension in line of SAR data. Pulse Compression

In collecting the SAR data, a long-duration 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 signal-to-noise ratio.

Mathematically, in complex form, a linear FM signal can be represented as:

$p(\tau) = exp (i \pi K \tau^2)$ eq 2.2
where i is the square root of -1,$\tau$ 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. SAR Signal

The sequence of received pulses are arranged in a two-dimensional 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 2-d SAR signal
In the two-dimensional 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 azimuth-varying 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 image be the range time within a received pulse (also called fast time), relative to the pulse transmission time. Then, the two-dimensional SAR signal can be expressed as

image eq 2.3
where image is the transmitted pulse, c is the speed of light, and image is the wavelength of the carrier. That is, the delay of the pulse is the round-trip distance divided by the speed of light. Similarly, the phase of the signal due to the delay of the carrier is the round-trip 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 line-of-sight 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. 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 burst-wise. The continuous model of SAR data applies to Image Mode (IM) data processed with the range-Doppler 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 medium-resolution with the SPECAN 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. 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 section. The stripes correspond to ocean waves. The two-dimensional 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 azimuth-frequency bands, or looks. These look-images 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. 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 range-Doppler and SPECAN 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 High-Level 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
Processing Steps IMM and IMB IMP or IMS IMG APM and APB or APP APS APG WSM and WSB GM1 and GMB WVI, WVS and WVW
Doppler Centroid Estimation (Single Swath)
Doppler Centroid Estimation (MultiSwath)
Doppler Refinement
Range Doppler Image Formation
SPECAN Image Formation
Wave Mode Processing
(see note 1)

Stripline processing is applied to medium-resolution 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 (DCE) and a range-Doppler algorithm to form the imagette for cross-spectra estimation. Preprocessing

Preprocessing is applied to all raw data before the other parameter estimation and image formation steps are performed.

During preprocessing, PF-ASAR 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 look-up 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 in-phase (I) and quadrature-phase (Q) channels. Receiver electronics may introduce biases or cross-coupling (non-orthogonality) 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 non-orthogonality correction.
  • Replica Construction and Power Estimation. A replica of the transmitted pulse is extracted. This process is also described in Characterisation .
  • Noise Power Estimation. Analysis of noise packets. 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 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 signal-to-noise ratio and azimuth resolution. Also, for modes processed by SPECAN , 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 yaw-steering, 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 PF-ASAR 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 burst-modes 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 PF-ASAR, the key challenge is to define techniques that will yield sufficiently accurate estimates for all processing modes. In particular, the burst-mode 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 iso-Doppler 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

$f_{\eta_c} = f'_{\eta_c} + MF_a$ eq 2.4
where the absolute Doppler centroid frequency$f_{\eta_c}$ is composed of two parts. The fine Doppler centroid frequency$f_{\eta_c}$ 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$MF_a$. Doppler centroid estimation often refers to the estimation of the fine Doppler centroid frequency, which is limited to the range$\pm F_a / 2$, 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 yaw-steered, 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 range-variation 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. 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,$\phi$, is related to the Doppler centroid by

$F_{\eta_c} = \frac{-F_a \phi}{2\pi}$ 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. Multi-Look 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,

$\delta \phi = \phi_{L_2} - \phi_{L_1} = \frac {2\pi(K_{a_2} - K_{a_1}) \eta_c}{F_a}$ eq 2.6

is used to compute the absolute Doppler centroid. The absolute Doppler centroid frequency is given by:

$f_{\eta_c} = \frac{-f_0 F_a \delta \phi} {2\pi\delta f}$ eq 2.7
where image 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,imageandimage:
$f'_{\eta_c} = \frac{-F_a}{2\pi} \left( \frac{\phi_{L_1} + \phi_{L_2}}{2} \right)$ 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 anglesimage,image, may have been wrapped around. To correct for this, a simple discontinuity detector is used to set image to within the interval image . 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 inimage.

Using this method as the Doppler Ambiguity Resolver (DAR), the Doppler ambiguity estimated is:

$M \; = \; round\; \left( \frac{f_{\eta_c} - f'_{\eta_c} - f_{offset}}{F_a} \right)$ eq 2.9 dop_eq_3_4
where image 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 image .
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,image, must be subtracted from the measured value ofimage. 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 non-rounded 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. 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 medium-resolution 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 " ).

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 look$i (i=1,2)$as:

$X_i(\tau, \eta) = W (\tau, \eta - \eta_i - \eta_c) \sigma (\tau, \eta) \gamma_i (\tau, \eta)$ eq 2.10
where the dimensions of the images are range time, range time$\tau$, and azimuth time$\eta$. The$W(\tau, \eta - \eta_i - \eta_c)$denotes the weighting applied by the antenna azimuth beam pattern on the image image , as a function of azimuth time. The weighting depends on the azimuth time at which the burst occurred,$\eta_i$, and the azimuth beam offset time from the zero-Doppler direction (which is related to the The Doppler centroid),$\eta_c \cdot \sigma(\tau, \eta)$ is the reflectivity of the ground patch being imaged that is common to both looks, and$\gamma_i(\tau, \eta)$is a multiplicative (speckle) noise component, assumed independent from one look to the next. Asimage,image,imageand theimageare known, an estimate ofimagemay be derived, with accuracy subject to the constraints imposed by the noise terms.

Taking the ratio of$W,\tau, \eta$, $\eta_i$, $\eta_c$, $X_1$, $X_2$toimage removes the dependence on the reflectivity, and taking the log of the ratio makes the function better behaved:

$log \left( \frac{X_1}{X_2} \right) = log \left( \frac{W(\tau, \eta - \eta_2 - \eta_c)}{W(\tau, \eta - \eta_1 - \eta_c)} \right) + log \left( \frac { \gamma_2(\tau, \eta)}{\gamma_1(\tau, \eta)} \right) = A(\eta_c) + B$ eq 2.11
The noise term, B, can be reduced by averaging. The basic approach is to estimateimageby forming an estimate of A(image) from the look ratio of the data, and compare it to a set of template functions that has been pre-computed using the known antenna pattern, to find the value ofimage that gives the best fit.

Specifically, we begin with an assumed value ofimageand process the data, including application of the descalloping function ( see note: 1) If the assumed value of image is correct, and if B is small relative to A(image), then the expected value of A(image) is 0. In general, there will be an offset between the assumed and actual values of image which is what the algorithm estimates. The estimate ofimage is generated from the unbiased estimate of A(image) 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 "pull-in". 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 lengthimage 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,image. The result, as function of azimuth sample, is denoted L(n) and it is this function that we attempt to match to the A(image) 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 function:

$y_f(k,n) = \left( \frac{W_f(n)}{W_f(n+k)} \right) ^2$ 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, image , 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):
$F_{offset} = k_{min} \frac{F_a}{L_{FFT}}$ 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 low-order polynomial is then fit to the values ofimageat a number of ranges image . When combined with the polynomial producing the initial Doppler centroid it yields the absolute value of the Doppler centroid. 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

$f_{\eta_c} = M_i F_{a_i} + f'_{\eta_{c,i}}$ eq 2.14
where image is the unambiguous Doppler centroid frequency,imageis the pulse repetition frequency for theimagebeam, andimageis the fine Doppler centroid, again for theimage beam. The goal of Doppler Ambiguity Resolution is to determine the correct value of image Given a prior knowledge that the satellite is yaw-steered limits the allowable values ofimage 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 image , the Doppler ambiguity may be derived.

The specific algorithm for deriving the Doppler ambiguity is as follows:

  • derive the fine Doppler estimates,image 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. 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 yaw-steered, 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. References

Ref 2.1
F. H. Wong, I. G. Cumming, A Combined SAR Doppler Scheme Based Upon Signal Phase, IEEE Trans. Geoscience and Remote Sensing, 34 (3): 696-707, May 1996.

Ref 2.2
. M.Y. Jin. Optimal Range and Doppler Centroid Estimation for a ScanSAR System, submitted to IEEE Transactions on Geoscience and Remote Sensing.

Ref 2.3
. S.N. Madsen. Estimating the Doppler Centroid of SAR Data. IEEE Transactions on Aerospace and Electronic Systems , Vol 25(2):pp 134-140, March 1989.

Ref 2.4
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. GE-30(5): pp 941-949, September, 1992.