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REPORT 15/2010 ISBN ISSN 1890-5226 SENTINEL-1 DOPPLER AND OCEAN RADIAL VELOCITY (RVL) ALGORITHM DEFINITION Author (s): Geir Engen (Norut), Harald Johnsen (Norut)
REPORT
15/2010 ISBN ISSN 1890-5226
SENTINEL-1 DOPPLER AND OCEAN RADIAL VELOCITY (RVL) ALGORITHM DEFINITION
Author (s): Geir Engen (Norut), Harald Johnsen (Norut)
Project Name: Sentinel-1 IPF Development Project No.: 355
Contractor(s): MacDonald Dettwiler and Associates (MDA) Contractors Ref.: SC-16131
Document No.: 15/2010
S1-TN-NRT-53-0658
Document Type: P5 Status: Confidential
ISBN: ISSN: 1890-5226 No. Pages: 59
Project Manager: Harald Johnsen Date: 09.05.2011
Author (s): Geir Engen (Norut), Harald Johnsen (Norut)
Title: Sentinel-1 Doppler and Ocean Radial Velocity Algorithm Definition
Resumé / Summary:
This document describes the algorithm implemented in the S1 L2 IPF for estimating the L2 Doppler and the Radial Velocity (RVL) component of the Sentinel-1 Level 2 Ocean (OCN) product
The L2 Doppler and Radial Velocity component is included in the L2 OCN processing and product together with the Ocean Swell Spectra (OSW) and Ocean Wind Field (OWI) information.
This document is the Sentinel-1 IPF delivery item PAL2-1.
Keywords: Sentinel-1, Doppler Frequency, Radial Velocity, Level 2 Product
Notes:
Publisher: Norut, Tromsø
Authorization: Kjell-Arild Høgda
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CHANGE RECORD
ISSUE DATE PAGE(S) DESCRIPTION
0.9
0.91
1.0
1.1
1.2
27/08/2010
17/09/2010
12/10/2010
08/11/2010
09/05/2011
All
All
All
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Draft version
Draft version
Version 1.0
Version 1.1, Addressed the following RIDS:
RIDS-342, RIDS-341, RIDS-340,
RIDS-339, RIDS-338, RIDS-337,
RIDS-336, RIDS-335, RIDS-334, RIDS-333
Updated according to RIDS: S1IPFDPDRL2-3, S1IPFDPDRL2-4, S1IPFDPDRL2-5, S1IPFDPDRL2-7, S1IPFDPDRL2-15 (moved background theory into appendixes), S1IPFDPDRL2-16,
Merged Sec.5 into Sec.8.
1) Moved flowchart figure to Sec 6.
2) Rename Sec 6 to “Doppler and radial velocity estimation algorithm”
3) In Sec 6.2.1 input parameters are defined and explained from where they are extracted
4) The ADF is removed, and error matrix included
5) This is now Sec 6.3.1. Step by step procedure is now described for computing the profiles, removing noise and computing initial Doppler value
6) This is now Sec 6.3.3. A detailed explanation of the method is given and a Figure 4 is included for illustration purposes.
7) Figure 3 and Figure 5 illustrate the two main modules.
S1IPFDPDRL2-17, S1IPFDPDRL2-18 (removed), S1IPFDPDRL2-19 (All Section 7 modified: input/output data for each module included, processing steps/procedure included in
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ISSUE DATE PAGE(S) DESCRIPTION
the beginning of each module with reference to equations computed)
Included clarifying figure in the processing steps for side-band corrections (Figure 4)
Harmonized Section 3 with OSW, OWI documents.
Contents1 Introduction 5
1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Document Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Documents 62.1 Applicable Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Reference Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 OCN Product Overview 83.1 Product Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Processing Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 RVL Component Overview 12
5 Algorithm input data requirements 135.1 Antenna Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 SLC Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Doppler and radial velocity estimation algorithm 156.1 Algorithm overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2.1 Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2.2 Internal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.2.3 Output data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.3 Doppler and radial velocity estimation procedure . . . . . . . . . . . . . . 186.3.1 Estimation of observed azimuth and range Fourier profiles and
initial value of Doppler frequency . . . . . . . . . . . . . . . . . . . 196.3.2 Computation of reference profiles . . . . . . . . . . . . . . . . . . . 236.3.3 Side-band correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3.4 Estimation of precision Doppler values: . . . . . . . . . . . . . . . . 256.3.5 Resampling to output grid and estimation of radial velocity . . . . 26
7 Input Files 287.1 SAR Image Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.1.1 SAR Product Annotations . . . . . . . . . . . . . . . . . . . . . . . . 287.2 Auxiliary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3 Internal Auxiliary Data Files . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.3.1 Coastline and Land Masking Data - LOP CLM . . . . . . . . . . . . 297.3.2 S-1 Antenna Embedded Row Pattern - LOP PAT . . . . . . . . . . . 297.3.3 S-1 Antenna Excitation Coefficients - LOP COE . . . . . . . . . . . 297.3.4 S-1 Antenna LUT - LOP LUT . . . . . . . . . . . . . . . . . . . . . . 307.3.5 Internal Processing Parameter File - PRM LOPIn . . . . . . . . . . . 30
7.4 External Auxiliary Data Files . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.4.1 S-1 Antenna Error Matrix Auxiliary Data - AUX ECE . . . . . . . . 307.4.2 Atmospheric Model Wind Field - AUX WND . . . . . . . . . . . . 307.4.3 Wavewatch III Model - AUX WAV . . . . . . . . . . . . . . . . . . . 307.4.4 L2 Processor Parameter Auxiliary Data - AUX PP2 . . . . . . . . . 31
8 Symbols 32
9 Output Product Content 34
A Background theory 36A.1 SAR imaging process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
A.1.1 SAR raw data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.1.2 SAR raw data Fourier domain . . . . . . . . . . . . . . . . . . . . . 37A.1.3 SAR data focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A.2 Azimuth time-frequency relations . . . . . . . . . . . . . . . . . . . . . . . 40A.2.1 Antenna pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.2.2 Frequency band location . . . . . . . . . . . . . . . . . . . . . . . . . 40A.2.3 Doppler centroid estimation . . . . . . . . . . . . . . . . . . . . . . . 41A.2.4 Phased array antennas . . . . . . . . . . . . . . . . . . . . . . . . . . 42
B Expected Estimation Quality Performance 46
C Regular gridded 2D-antenna 50C.1 1D beam forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51C.2 Doppler offsets caused by error in the antenna form . . . . . . . . . . . . . 52
D Figures 54
2
Acronyms and Abbreviations
ASAR Advanced Synthetic Aperture Radar
CLM Coast-Line and Land masking
CLS Collecte Localisation Satellites
CNR Clutter to Noise Ratio
dB Decibel(s)
DC Doppler Centroid
DCE Doppler Centroid Estimate
ECMWF European Centre for Medium-range Weather Forecasts
ENVISAT ENVIronment SATellite
ESA European Space Agency
EW Extra Wide
HH Horizontal polarisation on transmit and receive
IDL Interactive Data Language
IM Image Mode (ASAR)
IPF Instrument Processing Facility
IW Interferometric Wide
IWS Interferometric Wide Swath
L1 Level 1
L2 Level 2
MDA MacDonald, Dettwiler and Associates Ltd.
MET Meteorological
NESZ Noise Equivalent Sigma Zero
NRCS Normalised Radar Cross Section
Norut Nothern Research Institute
NWP Numerical Weather Prediction
3
OSW Ocean Swell Wave Spectra
OWI Ocean Wind Field
PDF Probability Density Function
PRF Pulse Repetition Frequency
R&D Research and Development
RMS Root Mean Square
RMSe Root Mean Square Error
RVL Radial Velocity Field
S-1 Sentinel-1
S/C Signal-to-Clutter
SAR Synthetic Aperture Radar
ScanSAR Scanning SAR
SLC Single-Look Complex
SM Stripmap
SNR Signal to Noise Ratio
SOPRANO SAR Ocean Products Demonstrat1ion
SRR System Requirements Review
TBC To Be Confirmed
TBD To Be Determined
TOPS Terrain Observation with Progressive Scans
V Vertical
VV Vertical polarisation on transmit, Vertical polarisation on receive
WS Wide Swath
WSM Wide Swath Mode
WV Wave Mode
WVW Wave Mode Ocean Spectra Level 2 Product
4
1 Introduction
1.1 PurposeThe objective of this document is to define and describe the algorithm implemented inthe S1 L2 IPF and the processing steps for the generation of the Radial Velocity (RVL)component of the Sentinel-1 Level 2 Ocean (OCN) product
1.2 ScopeThe OCN product contains three components: the Ocean Swell spectra (OSW) compo-nent, the Ocean Wind Field (OWI) component, and the Radial Surface Velocity (RVL)component. These three components are all merged into a common OCN product forthe Wave Vignette (WV) and Strip Map (SM) modes. For the S-1 TOPS mode, the OCNproduct consists of only the RVL and OWI components. A description on how all thesethree components (OSW, OWI, RVL) are connected into the L2 ocean processing is out-lined in Section 3. This document contains only the RVL algorithm definition. TheOWI and OSW algorithm definitions are provided in separate documents [A-8], [A-9].This document satisfies the PAL2-1 deliverable defined as per the content defined in theSentinel-1 IPF Statement of Work [A-1],[A-2] for review at the Sentinel-1 IPF PreliminaryDesign Review (PDR L2) and Critical Design Review (CDR L1 & L2).
1.3 Document StructureThis document is structured as follows:
Section 1 introduces the purpose, scope, structure and conventions of the document
Section 2 lists the applicable and reference documents
Section 3 gives a contextual overview of the L2 OCN processing and component
Section 4 gives a short L2 RVL component overview
Section 5 describes RVL algorithm input data requirements
Section 6 describes the Doppler and radial velocity estimator
Section 7 lists and describes all the input files
Section 8 lists the main symbols used in the RVL algorithm
Section 9 lists the content of the output netCDF product file
Appendix A gives the theoretical background for the algorithm
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Appendix B describes the processing requirements for the L1 SLC product
Appendix C outlines the best (idealized) theoretical performance of the algorithm
Appendix D outlines background theoretical derivations of idealized 2D-antenna modeland antenna errors, and impacts on Doppler estimate.
Appendix E includes figures related to Appendix B
2 Documents
2.1 Applicable DocumentsA-1 GMES-DFPR-EOPG-SW-07-00006 Sentinel-1 Product Definitions & Instrument Pro-
cessing Facility Development Statement of Work, Issue/Revision 4/1, 23-05- 2008.
A-2 Contract Change Notice N.2, Changes in ESRIN Contract No. 21722/08/ILG, June21, 2010
A-3 S1-RS-MDA-52-7443 Sentinel-1 IPF Auxiliary Product Specification, Issue/Revision2/2, May 6, 2011
A-4 S1-RS-MDA-52-7441, Sentinel-1 Product Specification, Issue/Revision 2/2, May 6,2011
A-5 SEN-RS-52-7440, Sentinel-1 Product Definition, Issue/Revision 2/3, Mar 21, 2011
A-6 Johnsen H., Engen G., Collard F., Chapron B., ”Envisat ASAR Level 2 Wave ModeProduct Algorithm Specification And Software Requirements Document”, NorutReport No. IT650/1-01, v.2.3.0, 24 Oct. 2006
A-7 Collard F., Johnsen H., ”SAR Ocean Wind Waves And Currents - Software Re-quirements Document”, CLS Report No. BO-024-ESA-0408-SRD-2 waves, v.1.2,Nov.2006
A-8 S1-TN-CLS-52-9049 Sentinel 1 Ocean Wind Field (OWI) Algorithm Definition, Is-sue/Revision 1/2, April 27, 2011, CLS
A-9 S1-TN-NRT-52-7450 Sentinel-1 Ocean Swell Spectra (OSW) Algorithm Definition,Issue/Revision 1/2, April 27, 2011, Norut
A-10 S1-DD-ASD-PL-0003, Issue 4, March 18 2011, Sentinel-1 SAR Instrument AntennaModel Description
A-11 http://www.unidata.ucar.edu/software/netcdf/
6
A-12 S1-IC-MDA-52-7454 Sentinel-1 Instrument Processing Facility Interface Control Doc-ument, Issue/Revision 1/3, Apr. 9, 2011, MDA
2.2 Reference DocumentsThe following documents provide useful reference information associated with thisdocument. These documents are to be used for information only. Changes to thedate/revision number (if provided) do not make this document out of date.
R-1 ES-RS-ESA-SY-0007 Mission Requirements Document for the European Radar Ob-servatory Sentinel-1, EUROS, Issue 1/4, ESA, July 11, 2005
R-2 Kerbaol V., et al., ”SAR Ocean Wind, Waves and Currents : Final Report and Ex-ecutive Summary”, Esrin/Conract NO 18709/05/I-LG, Ref. BO-024-ESA-0408-RF,v.2.0, 15 November 2006
R-3 BOOST Technologies, SAR WINDS WAVES CURRENTS Validation technical notes,Technical note (WP6), BO-024-ESA-0408-VTN, version 1.0, 09/08/2006
R-4 Chapron B., Collard F., Ardhuin F., ”Direct measurements of ocean surface veloc-ity from space: Interpretation and validation”, Journal of Geophysical Res., 110,C07008, 2005
R-5 Mouche A.A., Chapron B., Reul N., Collard F., ”Predicted Doppler shifts inducedby ocean surface wave displacements using asymptotic electromagnetic wave scat-tering theories”, Waves in Random and Complex Media, Volume 18, Issue 1, Febru-ary 2008, pp. 185
R-6 Collard F., Chapron B., Johnsen H., ”Sentinel-1 SAR Wave Mode”, Technical Note,Boost Technologies, v1.0, Dec 2007.
R-7 Johnsen H., Engen G., Guittion G., ”Sea-Surface Polarisation Ratio from EnvisatASAR AP Data”, IEEE Trans. on Geo. Rem. Sensing, Vol.46, No.11, Nov. 2008.
R-8 Ardhuin F., A. D. Jenkins, D. Hauser, A. Reiers, B. Chapron, ”Waves and Opera-tional Oceanography: Toward a Coherent Description of the Upper Ocean”, Eos,Vol.86, No.4, 25 January 2005
R-9 Engen G., Pedersen I. F., Johnsen H., Elfouhaily T., ”Curvature Effects in OceanSurface Scattering”, IEEE Trans. on Antennas and Propagation, Vol.54, No.5, May,2006.
R-10 Antenna Doppler Contribution Predictor Preliminary Validation,Doc. No.: SAR-EQWG-132-TEN, Aresys, Sept. 2010
7
3 OCN Product OverviewThe level-2 (L2) ocean product (OCN) has been designed to deliver geophysical param-eters related to the wind, waves and surface velocity to a large panel of end-users. TheL2 OCN products are estimated from Sentinel-1 (S-1) Synthetic Aperture Radar (SAR)level-1 (L1) products. L2 OCN products are processed by the Level 2 IPF processor andbenefit from robust and validated algorithms [R-3]. A diagram of the L2 Ocean pro-cessing unit context is presented in Figure 1. In this figure, external IPF interfaces havea white background, internal IPF interfaces are identified by a grey background, andinterfaces with a yellow background are only applicable when the L2 processor is usedin test mode outside of the normal IPF environment.
The processor can be used in PDGS environment or in a stand-alone HMI mode. Inboth cases, a job order is read by the processor to get all high level information requiredfor processing a particular product (e.g. names and directories of input L1 files, namesand directories of auxiliary data files, directories of outputs files, etc). Processing thenstarts from L1 products using the auxiliary data files provided (e.g. the L2 processorparameter file). During the processing, a log file is generated to monitor the status ofeach processing step. The final step of the processing is the creation of the productincluding writing of all the geophysical information into netCDF files.
L2 Ocean Processing
AUX_PP2AUX_ECEAUX_ICEAUX_SCS
AUX_WAVAUX_WND
Logging
Start/Stop Exit Code
Job Order
Product Report
Product List
L2 OCN Product
PDGS ML or Stand-Alone HMI
PRM_LOPInLOP_CLMLOP_GEBLOP_ANTLOP_FOU
PRM_LOPOutProcessor Configuration File
Internal S1 L1 Products(SLC / GRD)
ASAR L1 Products (IMS / IMP / WVI / WSM)
Figure 1: Sentinel-1 L2 Ocean Processing Context Diagram
8
3.1 Product OrganisationEach L2 OCN product contains up to three geophysical components: the radial velocity(RVL), the ocean surface wind field (OWI) and the ocean swell wave spectra (OSW)components. These components are formatted into one output netCDF file. For SM andWV modes, the L2 product contains all three components. For TOPS mode, the productcontains only RVL and OWI components. The detailed algorithm definition of eachcomponent is described in a dedicated document (this document is for the RVL and [A-8], [A-9] are for the OWI and OSW). The outputs variables related to each componentare listed and defined in the product definition documentl [A-5] .
For the SM and TOPS modes, the information related to each component is esti-mated onto a specific grid cell (ground range) whose properties are chosen to optimizethe inversion schemes. As a consequence, the SM mode output netCDF file has threecomponents and the TOPS mode output netCDF file has two components, each set hav-ing its own resolution. In addition, the most pertinent geophysical parameters fromRVL and OSW components are interpolated onto the OWI grid to get a set of variablesdefined at the same resolution. The default value for the resolution of this common gridis 1 km for SM and TOPS modes. The set of variables from RVL and OSW interpolatedonto OWI grid is listed in [A-5], [A-8] Section 8. RVL and OSW are estimated from L1SLC internal product. OWI is estimated from L1 GRD internal product. For WV mode,there is no grid. In this case, the resolution of the components is simply the size of theimagette: 20 km. The three components are estimated from L1 SLC internal products.
3.2 Processing WorkflowFor SM and TOPS modes, the components are estimated independently. This meansthat for a given acquired scene, the steps for each component are:
• the appropriate L1 internal product is read,
• the variables corresponding to the considered component are estimated
• a temporary file containing the results is saved locally.
For each component, these three latter steps are executed by different IDL scripts basedon the same library of IDL functions. These 3 scripts are coordinated by a Python scriptwhich collects all information mandatory for L1 processing of each component. Then,when it is completed for all components, the components outputs and logging files aremerged into a single netCDF file and a single logging file by another script. For WVmode, the three components are estimated sequentially from the same L1 SLC internalproduct with the same IDL script. The SM and TOPS modes have the dual-polarizationoption. However, the L2 OCN components are always estimated only using the infor-mation from the co-polarized signal. Thus, the algorithms for each component as wellas the workflow for the L2 OCN product generation are not different from that of sin-gle polarization product. The OCN Product consists of three components (OWI, OSW
9
and RVL), and these three components are derived through three different processingalgorithms:
• The Radial Velocity algorithm, as described by this document
• The Ocean Swell Spectral algorithm, as described [A-9]
• The Ocean Wind Field algorithm, as described in [A-8]
Table 1 presents a summary of which components are included in the OCN Productper acquisition mode, the L2 processing algorithm used to calculate the values for thatcomponent, and the type of input L1 product (either SLC or GRD) is required by eachalgorithm. Further details about these L1 GRD and SLC L1 products can be found inthe respective algorithm documents
Table 1 : L2 OCN Product Content and Processing Algorithm per Acquisition Mode
L2 Processing Algorithm Acquisition
Mode
L2 OCN Product
Component
Input L1 Product
Ocean Swell Spectra
Algorithm
Ocean Wind Field Algorithm
L2 Doppler Grid Algorithm
OSW SLC !
OWI GRD ! SM
RVL SLC !
OWI GRD ! IW
RVL SLC !
OWI GRD ! EW
RVL SLC !
OSW SLC !
OWI SLC ! WV
RVL SLC !
The L2 processing algorithms support the processing of both Sentinel-1 and ASARL1 products that have been produced by the S1 IPF in the Stand-Alone environment.In the Test Mode of the L2 OCN Processor, they can also use L1 products that havebeen produced by the PF-ASAR processor, and therefore follow the ENVISAT/ASARproduct format. Table 2 shows the acquisition modes, sensors and input product typessupported by each of the three L2 processing algorithms.
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Table 2: Input L1 Products Supported by L2 Processing Algorithms
L2 Processing Algorithm
Acquisition Mode Sensor L1 Processor L1 Product Type
Sentinel-1 S1 IPF SLC
ASAR S1 IPF SLC SM
ASAR PF-ASAR IMS
Sentinel-1 S1 IPF SLC
ASAR S1 IPF SLC
Ocean Swell Spectra
WV
ASAR PF-ASAR WVI
Sentinel-1 S1 IPF GRD-MR
ASAR S1 IPF GRD-MR SM
ASAR PF-ASAR IMP
IW Sentinel-1 S1 IPF GRD-MR
EW Sentinel-1 S1 IPF GRD-MR
Ocean Wind Field
WS ASAR PF-ASAR WSM
Sentinel-1 S1 IPF SLC
ASAR S1 IPF SLC SM
ASAR PF-ASAR IMS
Sentinel-1 S1 IPF SLC
ASAR S1 IPF SLC WV
ASAR PF-ASAR WVI
IW Sentinel-1 S1 IPF SLC
L2 Doppler Grid
EW Sentinel-1 S1 IPF SLC
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4 RVL Component OverviewThe Sentinel-1 SAR can be operated in one of four nominal acquisition modes:
• Stripmap Mode (SM)
• Interferometric Wide-swath Mode (IW)
• Extra-Wide-swath Mode (EW)
• Wave Mode (WV)
The Sentinel-1 ocean Doppler and radial velocity processing supports all the modeslisted above. The Sentinel-1 Level 2 RVL component consists of an estimate of the totalDoppler frequency [Hz] and the corresponding radial velocity [m/s] estimated from aSentinel-1 Level 1 Single-Look Complex (SLC) SAR image. The RVL component con-tains an estimate of the width of the ocean Doppler spectra. The Doppler width is a newgeophysical parameter that has never been estimated from SAR data before. For each ofthese parameters the RVL component contains the corresponding standard deviation ofthe estimates. The image from which a single RVL is computed can be a SLC imagettefrom the WV mode, or sub-image extracted from a SM SLC image, IW SLC image, oran EW SLC image. The RVL product is given on a grid similar as to the OSW or OWIcomponents.
The Doppler frequency is estimated from the SLC data by fitting (least square mini-mization) the antenna model to the observed azimuth spectra taking into account effectsfrom additive noise and side band effects. The Doppler frequency is the total estimatedDoppler frequency offset without any geometric or mispointing corrections. The cor-responding radial velocity is retrieved from the Doppler frequency after correcting theestimated total Doppler frequency for antenna mispointing as function of elevation an-gle and compensating for the attitude/orbit Doppler signal error. As for the OSW andOWI component the RVL component contains also information on the block size usedfor estimation. The spatial coverage of the RVL product is equal to the spatial coverageof the corresponding L1 WV-SLC or sub-images extracted from the L1 SM/IW/EW-SLCproducts, limited to ocean areas.
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5 Algorithm input data requirementsThe RVL processing algorithm requires access to S-1 antenna error matrix in order togenerate internally the radiation beam patterns [A-10]. The error matrix data are pro-vided as an IPF auxiliary file. The RVL algorithm also set some specific requirements tothe internal SLC that will be generated for the L2 processing. These are specified belowand the reason for needing them is justified.
5.1 Antenna InformationThe RVL algorithm models the contribution from alliasing of energy from neighbour-ing areas into the estimation area (side band effects). In case of intensity gradients inthe image (which is very likely to happens over ocean areas) this will cause slightlyasymmetric azimuthal profile. Although small, this asymmetry will cause a significantbias in the Doppler frequency estimate. In order to model this effect the RVL algorithmmodels the behaviour of the antenna during acquisition including the side bands. Andfor TOPS mode the knowledge of antenna information for the various steering angles isrequired.
One option to provide the antenna information to the RVL algorithm is to providethe full 2D antenna pattern. However, for TOPS mode this means that the patterns forall steering angles must be provided, resulting in a large amount of auxiliary input data.The solution is to synthesize the 2D antenna pattern in the L2 RVL processing for vari-ous swaths and steering angles (TOPS) including all the side bands following the idealantenna model approach described in [A-10],[R-10]. This require access to the errormatrix, the embedded row pattern, the excitation coefficients, and the correspondingassociation of the excitation coefficients with the PRI within a burst (LUT). The excita-tion coefficients, the error matrix and the embedded row patterns can be provided asInternal Auxiliary Data, and also the LUT indexes since they will also be kept mostlyconstant from product to product of same swath/mode.
The requirements are summarized :
• access to antenna error matrix
• access to the excitation coefficients
• access to embedded row patterns
• access to LUT indexes (for TOPS for each burst)
5.2 SLC DataThe RVL algorithm requires SLC data where the data is focused in range such that theadditive noise is kept white. This means no weighting or windowing applied during
13
data focusing. An internal SLC product, that differs from the external SLC product,is developed to meet these requirements. The reason for this requirement is that thealgorithm is estimating the additive noise from the range profile, independently fromthe azimuth profile. In this way the RVL algorithm derives both the Doppler frequencyand the geophysical Doppler width from the azimuth profile fitting procedure. Thisprocess makes also the fitting procedure more robust.
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6 Doppler and radial velocity estimation algorithmIn this section the Doppler and radial velocity estimation algorithm is described step bystep. The steps outlined below are described in detail in sub-section 6.3.
6.1 Algorithm overviewThe procedure consists of five main parts:
I : Estimation of Observed azimuth Fourier Profile P initial values of signal energy a
and Doppler offset dc. This process is done for each burst for IW/EW and forSM/WV in SLC products, on a equidistant grid in zero Doppler SAR coordinates.From the range direction Fourier profiles (observed and model) is the additivenoise level estimated outside the bandwidth of the signal (Figure 3). The outputof this procedure is stored in the array (EST) of estimation point structures.
II : For each estimation point in one burst, the reference Fourier profile P is computed.The resulting profiles are stored in the array (REF) of reference functions. Thisarray is assumed to be fixed and will be used in the precise parameter estimationfrom several bursts.
III: The weighted sum of the aliased side-bands is computed using the signal energyof the corresponding neighbor areas, and subtracted from the observed profileP (all based on previous estimated values from the EST array). The side-bandcorrected profile is saved in the EST array.
IV: From the side-band corrected observed profile (in the EST array) combined withthe reference profile (from the REF array), the precise estimates of a and dc arecomputed and stored in the structure of the estimation point array (EST) .
V : From the values stored in the estimation point array (EST) the output product isgenerated on a given output grid and the radial velocity is computed, combiningall swaths (for TOPS) and stored in the output array (RET).
The procedure is outlined in Figure 2.
6.2 DataIn this section the input data, the main internal data and the final output data are de-scribed.
6.2.1 Input data
SLC : Data product where the following additional information is included:
15
Level 1 SLC Products
Read Processing Set-Up File Read Level 1 SLC Data AUX_PP2
Step-I: Estimation of Range and Azimuth Fourier Profiles Estimation of Initial Doppler and Additive Noise
Step-II: Computation of Reference Azimuth Profiles from
Antenna Data
Step-IV: Precise Estimation of Doppler and Doppler Width by
Minimization of Cost-Function
Step-V: Computation of Radial Velocity on Output Grid
RVL Component of OCN Product
AUX_WND
OWI unit OSW unit
AUX_ECE
Step-III: Estimation and Compensation for Side-Band Effects in
Observed Profiles
Generation of RVL Output Data Structure
(optional)
Figure 2: Flowchart of RVL Algorithm
16
– Number of bursts (Nbursts) in input SLC product (TOPS)
– Size (Mraw, Nraw) of each raw-data burst
– Start-time (t0, τ 0) of each raw-data burst
– Sampling-rate (ω∆, ∆) of each raw-data burst
– Azimuth steering-rate (γ) of antenna.
– Array[Nraw] containing the reference between line-number in raw-data burstand antenna T/R coefficient LUT index.
– Array[∗, Nrep] containing the replica pulses (only a small subset is needed).
The SLC data must be processed with no weighting functions in the Fourier-domain. This also includes the range compression transfer function, whose ab-solute value has to be constant for the whole Fourier-domain (also outside thesignal bandwidth).The following parameters are extracted from the auxiliary antenna data products:
Na : number of antenna elements in azimuth.
Ne : number of antenna elements in elevation.
Zt : Array[Ne , Na , *] containing the antenna transmission (exitation) coefficients.
Zr : Array[Ne , Na , *] containing the antenna receive (exitation) coefficients.
δZt : Array [Ne , Na] containing the multiplicative weighting (error) coefficients to the
antenna transmission coefficients.
δZr : Array [Ne , Na] containing the multiplicative weighting (error) coefficients to the
antenna receiveing coefficients.
f : Array[Ne] containing the radiation pattern of the T/R modules (embedded rowpattern).
L2 AUX PP2 : L2 processor parameter auxiliary data:
– RaEst : Size of Doppler estimation block in range [m]
– AzEst : Size of Doppler esimation block in azimuth [m]
– RaRes : Size of output grid cell interval in range direction [m]
– AzRes : Size of output grid cell interval in azimuth direction [m]
17
6.2.2 Internal data
EST : Array[Mest, NestNbursts] containing the following structure:
Mest : Number of estimation blocks in rangeNest : Number of estimation blocks in azimuth
a : Scalar containing the estimated energy of the signal.b : Scalar containing the estimated energy of the additive noise in the signal.
dc : Scalar containing the estimated angular Doppler frequency offset.ϑ : Scalar containing the estimated width of the Doppler frequency spread.P : Array[N ] containing the observed azimuth Fourier profile.R : Array[M ] containing the observed range Fourier profile.
REF : Array[Mest, Nest] containing the following structure:
P : Array[N ] containing the reference azimuth Fourier profiles.R : Array[M ] containing the reference range Fourier profiles.
Here (Mest, Nest) is the number of estimation blocks (range, azimuth) (per burst forTOPS). Note that the number of estimation points is much larger than the number ofoutput grid cells. These numbers will depend on the input parameters (block sizes)and imaging mode and must be computed for each cases. N is the number of pixels inthe reference and observed azimuth Fourier profile. M is the number of pixels in thereference and observed range Fourier profile.
6.2.3 Output data
RET : Array[*,*] containing the following structure:
dc : Scalar containing the estimated angular Doppler frequency.ϑ : Scalar containing the estimated width of the Doppler spread.
Ur : Scalar containing the estimate of radial velocity.σ
2dc
: Scalar containing the variance of the estimate of dc.σ
2ϑ : Scalar containing the variance of the estimate of ϑ.
σ2Ur
: Scalar containing the variance of the estimate of Ur.
6.3 Doppler and radial velocity estimation procedureIn this section the five main steps of the algorithm are described. The background theoryfor the formulas used in the algorithm description can be found in Appendix A. Theformulas of the algorithm and the procedure described in the following sections areapplicable to TOPS, SM and WV modes. For SM and WV the only difference is that theangular steering rate, γ is set to zero.
18
6.3.1 Estimation of observed azimuth and range Fourier profiles and initial value ofDoppler frequency
This subsection refers to Step I of the procedure outlined in Section 6.1.
Input parameters: Observed complex image product: Ic(t, τ)
Output parameters: Observed azimuth spectral profile: P (), Variance of azimuthspectral profile: VarP, signal energy: a, additive noise: b , initial value of Dopplerfrequency: ς , weight factor in range: χra, weight factor in azimuth: χaz.
Procedure: The procedure below is done for all estimation blocks within the product.The procedure for block extraction from input SLC product is part of the reader, andsimilar to the one used for the OSW stripmap processing. The reader provides also allthe SLC header informations, as well as all the necessary geometry/satellite informa-tion (incidence angle, latitude, longitude, radar velocity, heading, etc) for each of theestimation blocks. These parameters are thus not specifically mentioned in the InputParameter descriptions for the various procedures. The equations refered in the pro-cedure below (and implemented in the code) are given in the Background Equationssection below. Initially all weight functions are set to one (WP = 1,W = 1,WR = 1), andthen the following steps are done:
• compute the windowing functions in Fourier domain, haz and hra
• compute the weight factors, χra, χaz, using eq.(21)
• compute model of range fourier profile, R, using eq.(7)
• compute estimates of energy, a, and noise, b using eqs.(16), (17)
• compute estimate of range, R(ω) , and azimuth, P (ω) fourier profiles and corre-sponding variances, VarR, VarP using eqs.(9), (10), (22), (23)
• compute model of azimuth fourier profile, P , using P (ω)
• recompute weight functions W and WP using eqs.(19) and (24), and repeat the
last four steps twice
The initial value of the frequency shift ς of P is estimated using a standard cyclicmethod:
ς =1
∆τArg
d P () e
i∆τ
. (1)
The initial Doppler offset is related to this shift as described in eq (28) and together withthe estimated values of a, b and P they all are stored in the structure of the estimationpoint array EST. The initial values of Doppler offset and signal energy are used in theside-band correction procedure described in sub-section 6.3.3.
19
Background Equations: First the 2D-Fourier-profile is estimated from the compleximage Ic multiplied with a windowing function and with the dechirping function (forTOPS) as follows:
I(ω, ) =
dt dτ e−i(ωt+τ)
Ic(t, τ) ei2
βγγ+β τ2
h(t− t, τ − τ)2 (2)
where Ic is the complex image, γ is the angular azimuth steering rate of the antenna(given as input data), h is the dyadic constructed spatial window function
h(t, τ) = hra(t) haz(τ)
(TBD) centered at range and azimuth position t = (t, τ), and β is the angular signalDoppler rate (in rad/s
2) computed using the relation:
β =ν
2Ω
t(3)
where Ω(ω, t) ≡
(ω0 + ω)2 −2/ν2(t) and ν(t) = 2 vrc . Here vr is the effective radar
velocity, and ω0 is the carrier frequency (rad/s). Eq. 3 needs only be computed at theestimation center position, t = t. We assume the following model for the mean andvariance of the 2D-model:
EI(ω, ) = a R(ω) P () + b , (4)VarI(ω, ) = (a R(ω) P () + b)2
, (5)
where the model range and azimuth Fourier profiles satisfies:
dω R(ω) =
d P () ≡ 1 . (6)
Let s(1)rep(t) . . . s
(Nrep)rep (t),, represent the array of pulse replicas, then the model range
Fourier profile is computed by:
R(ω) =
|hra|2 ⊗R0
(ω)
dω
|hra|2 ⊗R0
(ω)
, (7)
where s(n)rep and hra are the Fourier transforms of s
(n)rep and hra, respectively, and
R0(ω) =
Nrep
n=1
s(n)rep(ω)
2 . (8)
We define the observed range and azimuth Fourier profiles as
R(ω) =
d I(ω, ) WR(ω, ), (9)
P () =
dω
I(ω, )− b
WP (ω, ). (10)
20
where b is an estimator of b, and WR and WP are positive weight functions with con-straints
d WR(ω, ) P () = 1, (11)
dω WP (ω, ) R() = 1. (12)
The means of the profiles are
ER(ω) = a R(ω) + b, (13)
EP () = a P () + (b− Eb)
dω WP (ω, ) , (14)
where b is related to b by
b ≡ b
d WR(ω, ) . (15)
we observe that if b is unbiased the last term of of equation (14) is zero. Unbiasedestimators for both a and b
are
a =
dω R(ω) ga(ω) W
(ω) , ga(ω) ≡ γ0R(ω)− γ1
γ0γ2 − γ21
, (16)
b =
dω R(ω) gb(ω) W
(ω) , gb(ω) ≡ γ2 − γ1R(ω)
γ0γ2 − γ21
, (17)
whereγ ≡
dω R
(ω) W(ω) (18)
for all positive choice of W. However choosing
W(ω) =
χaz
VarR(ω)(19)
yields the minimum variance unbiased estimators (for a given R) and the estimatorvariances become
Vara =γ0 χraχaz
γ0γ2 − γ21
, Varb =γ2 χraχaz
γ0γ2 − γ21
, (20)
where
χra =
n
(hra ⊗ h∗ra)n
2
|(hra ⊗ h∗ra)0
2 , χaz =
n
(haz ⊗ h∗az)n
2
|(haz ⊗ h∗az)0
2 . (21)
The variance of the Fourier profiles are
VarR(ω) = χaz
d(a R(ω) P () + b)2
W2R(ω, ) , (22)
VarP () = χra
dω(a R(ω) P () + b)2
W2P (ω, ) . (23)
21
-0.4 -0.2 0.0 0.2 0.4Normalized range frequency
0.0
0.5
1.0
1.5
Figure 3: Range Fourier profile. The green line represents the additive noise level.
We can use the freedom of choice of weight functions (with the given constraints ofeq. (11 and (12) to minimize the profile variance. However, since we at this stage (theinitial stage), does not have a good model for P , and we need to satisfy the constrain ofeq. (11) to obtain an unbiased estimate of the profile R, we choose WR ≡ 1. Minimizingthe variance of P , yields
WP (ω, ) =λ() R(ω)
(aR(ω)P () + b)2, λ() ≡ 1
dω
R2(ω)(aR(ω)P ()+b)2
(24)
and with this choice of weight function: VarP = λ χaz.Since we need a, b and P to compute the variance used in the weight functions WP
and W, the procedure has to be done twice. The first time, flat weighting functions are
used to get initial estimates a and b for a and b, and as a model for P we will use thelowest Fourier coefficients of P . The procedure is then repeated with updated weightfunctions W
and WP based on the latest estimated parameters.
Remark I: The size of the sub-images, the spacing between each of the estimationpoints (t, τ) and the form of the windowing function h, used to compute the Fourierprofiles, are defined such that the estimated values of dc and a can be resampled toany grid later (they need to satisfy the Nyquist sampling criteria). This means that theestimation is done with at least 50% overlap in both range and azimuth direction. Atthe same time the estimation area in azimuth for TOPS mode must be sufficiently smallto provide a continuous grid over the burst borders (restricted by the azimuth size ofthe overlap areas).
22
Remark II: The main contributor to the total time consumed by computing the Dopplercentroid values is the computation of the 2D-Fourier profile represented by equation (2).A good estimate is that the algorithm will use slightly more than the time it takes per-forming 3 multiplications (window function, dechirping, complex conjugate) and oneFFT of size (256×64) (range,azimuth) with 2.5 times overlap in both directions, coveringthe total data-set.
6.3.2 Computation of reference profiles
This subsection refers to Step II of the procedure outlined in Section 6.1.
Input parameters: Number of antenna elements in azimuth: Na, number of antennaelements in elevation: Ne, antenna transmission (excitation) coefficients: Z
t, antennareceive (excitation) coefficients: Z
r, AUX ECE file: multiplicative weighting (error) co-efficients to the antenna transmission coefficients: δZ
t, multiplicative weighting (er-ror) coefficients to the antenna receiveing coefficients: δZ
r, radiation pattern of the T/Rmodules (embedded row pattern), f , and LUT index.
Output parameters: Reference azimuth spectral profiles: P (; t)
Procedure: The one-dimensional (azimuthal) reference function in the two-dimensionaldata set space is computed as follows:
P (; t) =A( + βγ
γ+β τ ; βγ+β τ − β
−1)
2 (25)
where A(; τ ) is the two-way antenna diagram in the Fourier domain of the raw-datacomputed using equations (85) and (86). Here t = (t, τ) is the center position of the es-timation area, γ is the angular azimuth steering rate of the antenna and β is the angularsignal Doppler rate given by equation (3). The reference profile is computed for eachestimation position inside one burst and put into the REF array. The same REF arrayis to be used for several subsequent bursts. To speed up the computation, only a sub-set of the reference profile is computed in the range direction, and the profiles for theintermediate range positions are computed by interpolation.
6.3.3 Side-band correction
This subsection refers to Step III of the procedure outlined in Section 6.1.
Input parameters: 2D antenna model: A. Initial Doppler frequency: dc. Estimatedsignal energy: a
Output parameters: Side-band corrected estimated azimuth spectral profiles: P (; t)
23
Figure 4: Illustration of aliasing of energy from side-bands ( = ±1) into base-band. Thevertical red lines define the base-band. The blue dotted lines are the doppler frequenciesaliased into the base-band i.e. energy from nearest left and right neighboring areasmapped into estimation area. Here equal signal energy is assumed for the neighborareas.
Procedure: The side-band correction is basically to correct the estimated azimuth Fourierprofile for aliased energy arising from doppler frequencies outside the processed band-width or equivalently from neigbour areas left and right to the estimation area. If thesignal energy is different in the neighbor areas, the uncorrected estimated profile will beskewed due to aliasing causing a bias in the estimated Doppler frequency. The correc-tion of the estimated profile is done by combining the reference profile with the signalenergy of the left and right neigbor areas. This is illustrated in Figure 4 assuming (forsimplicity) equal signal energy in the left and right neighbor areas.
For the side-bands, we use the following approximation for the reference profile
P (; t) ≈A((1 + γ
β ) + βγγ+β τ ; β
γ+β τ)2 (26)
which only involves computing the azimuth raw-data Fourier diagram of the two-wayantenna for only one time instance
τ =
β
γ + βτ
In order to avoid errors in the side-band correction, the reference function is centeredaround the initial Doppler offset. The energy of all critical sidebands (here indexed with) is then computed as:
a()
P ( − ς() + ∆; t) (27)
This energy is subsequently removed from the observed Fourier profiles P (; t) and thecorrected profile is put back into the EST array. Here 2∆ is the width of the base-band
24
(bandwidth processed) given in rad/s. The corresponding energy a() for the side-bands
and profile offsets
ς() =
β
γ + β
()dc (28)
are computed at azimuth time position
τ() = τ + β
−1∆ (29)
by combining the values of a and dc in the EST array. By critical side-bands, is meantthe first side-bands ( = ±1) and the bands containing the first left and right side gratinglobes. The grating-lobe correction is only needed for the estimation points having largesteering angles.
6.3.4 Estimation of precision Doppler values:
This subsection refers to Step IV of the procedure outlined in Section 6.1.
Input parameters: Initial values of Doppler frequency: ς , Doppler spread: ϑ and the
corresponding variances on the estimation grid. Estimate of signal energy: a. Weightfactors: χra χaz. Estimated azimuth antenna pattern: P () . Modelled azimuth antennapattern: P ()
Output parameters: Doppler frequency: dc, Doppler spread: ϑ and the correspond-ing variances on the estimation grid.
Procedure: The precision values of a, dc and ϑ are computed by minimizing the fol-lowing cost-function using standard minimization routine:
J(a, ς, ϑ) =
d W ()
a Pϑ( − ς)− P ()2 (30)
where Pϑ is connected to the azimuth reference profile P by
Pϑ( − ς) ≡
dςP (ς ) 1√
2π ϑe− 1
2 (−ς−ςϑ )2 (31)
and is introduced because there is a spread in the geophysical Doppler, mainly due tothe orbital velocity of the ocean waves. The width ϑ
of the Doppler spread shouldbe zero for land measurement and the Gaussian probability function collapses to a δ-function. The minimizing of the cost function of eq. (30) is done iterativly by defining:
P() ≡1,
∂
∂ς,
∂
∂ϑ
Pς( − ς) , α ≡ (a, ∆ς, ∆ϑ
) (32)
25
where ∆ς and ∆ϑ represent the updates to the values of ς and ϑ
from the previousiteration. The solution of one iteration is given by
α =
d Γ−1 ·P() P () W () (33)
where Γ is a 3× 3-matrix with elements
γm,n =
d P
(m)() P(n)() W () (34)
Here P() represents the -th element of P. The covariance matrix of this estimator is
given by
Covara = χaz
d
Γ−1 ·P
))T
Γ−1 ·P
)) VarP ()W
2() (35)
The variance of the estimator α is minimized by choosing
W () =χra
VarP (), (36)
and the covariance matrix simply becomes
Covara = χra χaz Γ−1
. (37)
where χra and χaz are computed in Step I. After the final minimization the updatedvalues of a and
dc = (1 + γβ ) (ς − ς0) (38)
ϑ = (1 + γβ ) ϑ
(39)
are stored in the EST array, where ς0 is the offset from zero of the reference profile cor-responding to zero azimuth steering angle of the antenna. The algorithm will not becorrecting for absolute errors predicted by the computed antenna diagrams, but onlyrelative errors. This means that the estimated Doppler centroid is still going to be thesum of the geophysical Doppler and the effects due to errors in the antenna model.
6.3.5 Resampling to output grid and estimation of radial velocity
This subsection refers to Step V of the procedure outlined in Section 6.1.
Input parameters: Doppler frequency: dc, Doppler spread: ϑ, Radial velocity: Ur
and the corresponding variances on the input grid. Geometric Doppler offset: dc. L2Aux PP2 file. AUX WND: ECWMF wind field. AUX WAV: WaveWatch III. AUX CLM:Coastline data
26
Output parameters: Doppler frequency: dc, Doppler spread: ϑ, Radial velocity: Ur
and the corresponding variances on the output grid.
Procedure: By combining the EST-array (for TOPS from all swaths), the Doppler cen-troid dc and width of the Doppler spread ϑ are resampled to the output grid, and theradial velocity is computed from the relation:
Ur =dc − dc
2kr(40)
where dc is the known (from attitude data) geometric Doppler offset. Those three val-ues (radial velocity, Doppler offset, Doppler spread) are in combination with the vari-ance of the estimates, stored in the structure RET array and returned to the caller. Thesize of the output grid is computed from the size of the EST-array (Mest,Nest), the inputSLC pixel sizes, the percentage of estimation block overlap (50%), and the output gridresolutions (RaRes,AzRes). Internal IDL resampling routine will be used for the resam-pling operation. Then the percentage land overlap is computed for for each output gridusing the lat/lon of the grid point and the AUX CLM file (similar as for OSW)
Finally, the RVL output data structure is generated including resampling of auxiliarydata (ECMWF, Wavewatch III (optional) into the output grid, and stored internally forlater input to the OCN merging process.
27
7 Input FilesThe Sentinel-1 RVL processing will access at input both SAR products and external data.These are specified in the following subsections, and a detailed description of theseproducts and data can be found in [A-4] and [A-3], respectively.
7.1 SAR Image ProductsThe RVL processing system can access and process the SAR SLC data products listedbelow. The SLC products supported are the internal L1 SLC product suitable for L2processing.
• Sentinel-1 Stripmap Mode (SM) on S-1 format - SM SLC (internal)
• Sentinel-1 Wave Mode (WV) on S-1 format - WV SLC (internal)
• Sentinel-1 Interferometric Wide-swath Mode (IW) on S-1 format - IW SLC (inter-nal)
• Extra-Wide-swath Mode (EW) - EW SLC (internal)
• ASAR L1 IMS and WVI products, supported in test mode only
7.1.1 SAR Product Annotations
The RVL processing algorithm will extract the following key data (parameters) from theSLC (internal) product:
• Range and azimuth dimensions of input SLC image
• Range and azimuth pixel size of input SLC image
• Time of first line in SLC data
• Two way slant range time to first pixel in SLC data
• State vector (at least one)
• Number of bursts in input SLC (for TOPS)
• Raw data azimuth time (for each burst for TOPS)
• Raw data two way slant range time (for each burst for TOPS)
• Rawdata PRF and sampling rate (for each burst for TOPS)
• Size of raw data (for each burst for TOPS)
28
• Steering rate (for TOPS data)
• Yaw, pitch and roll of satellite platform at zero doppler time
• Doppler centroid frequency as predicted from orbit and attitude data
• Chirp replica
7.2 Auxiliary DataThis section is an overview of the auxiliary files used by RVL processing, and moreinformation is provided in the Auxiliary Specification document [A-3], the ProductSpecification document [A-4], and Interface Control document [A-12]. The auxiliaryfile description is here separated into internal and external files. The internal files aremeant not to be changed during mission, while the external files may change. The aux-iliary data from ECMWF and Wavewatch III are not explicitly used by the algorithmbut provided as part of the RVL component of the OCN product for the benefit of users.Detailed specifications of the auxiliary data can be found in [A-3].
7.3 Internal Auxiliary Data Files7.3.1 Coastline and Land Masking Data - LOP CLM
Doppler and RVL processing is not performed if land coverage is greater that 10% in theimagette considered. The land coverage is estimated as the ratio between the surfacearea of imagette and the surface area of a local land mask that covers the imagette. Thecoastline to be used should be quite accurate; an accurate shoreline polyline from theGSHHS is required. This shoreline database is available at
http : //www.ngdc.noaa.gov/mgg/shorelines/gshhs.html.
7.3.2 S-1 Antenna Embedded Row Pattern - LOP PAT
The embedded row patterns are required by the S-1 IPF as part of the L2 RVL compo-nent. The data is used to derive accurate Doppler estimation by synthesizing a radiationbeam pattern using an idealized antenna model.
7.3.3 S-1 Antenna Excitation Coefficients - LOP COE
The excitation coefficients are required by the S-1 IPF as part of the L2 RVL component.The data is used to derive accurate Doppler estimation by synthesizing a radiation beampattern using an idealized antenna model.
29
7.3.4 S-1 Antenna LUT - LOP LUT
LUT index - the association of the excitation coefficients with the PRI within a burst.The data is used in synthesizing a radiation beam pattern using an idealized antennamodel.
7.3.5 Internal Processing Parameter File - PRM LOPIn
When the L2 processor receives a job order request, the L2 processor will access theprocessing parameters via the L2 Processor Parameter Auxiliary File (AUX PP2 file).There is also an option to include an IPF internal parameter file for the L2 RVL processorin case more parameters than those defined in the AUX PP2 file. These parameterswill be provided in the Internal Processing Parameter File, and will be used during thealgorithm development phase for tuning the L2 RVL processor performance.
7.4 External Auxiliary Data Files7.4.1 S-1 Antenna Error Matrix Auxiliary Data - AUX ECE
The error maxtrix (EM) is required by the S-1 IPF as part of the L2 RVL component.The EM is used to derive accurate Doppler estimation by synthesizing a radiation beampattern using an idealized antenna model. The error matrix is used to correct the t/rexcitation coefficient with the latest available correction data. This file is optional.
7.4.2 Atmospheric Model Wind Field - AUX WND
Wind speed and direction at 10 m above the sea surface from the ECMWF atmosphericmodel is required with spatial and temporal resolution of at 0.125x0.125 degrees every3 hours. The model wind field will be resampled to RVL grid and provided as part ofthe RVL product output for the benefit of users.
7.4.3 Wavewatch III Model - AUX WAV
The Wavewatch III Stokes drift is a vector calculated as the third order momemt ofthe wave spectra. The Stokes drift is mostly dependent on the wind sea which is notimaged spectrally by the SAR. Stokes drift forecast files in NetCDF format generated bythe operational Wavewatch III model run at IFREMER will be used. The Wavewatch IIImodel provides surface Stokes drift velocity and direction for frequency range fmin =0.032 to fmax = 0.032 Hz with a spatial resolution of at least 0.5 degrees every 6 hours.The Stoke drift field will be resampled in to RVL grid and provided as part of the RVLproduct output for the benefit of users.
30
7.4.4 L2 Processor Parameter Auxiliary Data - AUX PP2
The RVL processing is flexible, and the processing can be configured with the key pa-rameters listed below.
• Size of Doppler estimation block in range [m]
• Size of Doppler estimation block in azimuth [m]
• Size of grid cell interval in range direction [m]
• Size of grid cell interval in azimuth direction [m]
31
8 SymbolsThe following list provides the definitions of most important symbols used in the algo-rithm description.
t = (t, τ): 2D time variable of SLC image (range, azimuth)
T : Scatter two-way time
A: 2D angular antenna pattern for stiff antenna
A : 2D angular pattern for phase array antenna
Θ0: Antenna elevation angle
Θ: Local elevation angle to scatterer
Φ0: Antenna squint angle
Φ0 : Antenna squint rate
β : Signal Doppler rate
γ : Angular frequency squint rate
vr : Effective radar velocity
va : Azimuth velocity of zero-Doppler plane
vp : Platform velocity
ω = (ω, ) : 2D-Fourier SLC image variable (range, azimuth)
Zt : Array containing the corrected antenna transmission coefficients.
Zr : Array containing the corrected antenna receive coefficients.
f : Array containing the radiation pattern of the T/R modules (embedded sub-arraypattern).
dc : Array containing the error in Doppler centroid values as function of elevationangle
Ic : Complex SAR image (internal L1 SLC product)
kr : Radar wavenumber
ωo : Radar carrier frequency (angular)
∆: Bandwidth of SLC in azimuth
32
P : Array containing the observed azimuth Fourier profile
P : Array containing the reference azimuth Fourier profiles.
R : Array containing the reference range Fourier profiles.
Mest : Number of estimation points in range direction (per burst for TOPS)
Nest : Number of estimation points in azimuth direction (per burst for TOPS)
N : Number of points in azimuth Fourier profiles
W : Weight function used in the cost function
h : Weight function used in profile estimation
(t, τ): 2D time position of estimation points (range, azimuth)
a : Scalar containing the estimated energy of the signal.
b : Scalar containing the estimated energy of the additive noise in the signal.
dc : Scalar containing the estimated angular Doppler frequency offset.
ϑ : Scalar containing the estimated width of the Doppler frequency spread.
Ur : Scalar containing the estimate of radial velocity.
σ2dc
: Scalar containing the variance of the estimate of dc.
σ2ϑ : Scalar containing the variance of the estimate of ϑ.
σ2Ur
: Scalar containing the variance of the estimate of Ur.
srep(t): Pulse replica
La : Size of antenna in azimuth direction
Le : Size of antenna in elevation direction
33
9 Output Product ContentThe following is a brief description/content of the numerical NetCDF file for the RVLcomponent of the OCN product. When the OSW, RVL and OWI components are gener-ated on internal files the final OCN product will be generated by merging the informa-tion from these files into a common NetCDF file.
The L2 RVL annotations data set includes the following dimensions:
• Number of RVL cells in the range direction
• Number of RVL cells in the azimuth direction
The L2 RVL data set also includes the following variables per RVL cell:
• Geodetic latitude at cell centre [deg]
• Geodetic longitude at cell centre [deg]
• Zero Doppler time at cell centre
• Two way slant range time at cell centre
• Ground range size of estimation area [m]
• Azimuth size of estimation area [m]
• Estimated Doppler frequency at cell centre [Hz]
• Estimated standard deviation of Doppler frequency at cell centre [Hz]
• Estimated width of Doppler frequency spectra at cell centre [Hz]
• Estimated standard deviation of width of Doppler frequency spectra at cell centre[Hz]
• Estimated radial velocity at cell centre [m/s]
• Estimated standard deviation of radial velocity at cell centre [m/s]
• Predicted geometric Doppler frequency (from Level 1 resampled to L2 dopplergrid) [Hz]
• Yaw of satellite platform at zero Doppler time (from Level 1)
• Pitch of satellite platform at zero Doppler time (from Level 1)
• Roll of satellite platform at zero Doppler time (from Level 1)
• Local northing angle [degN]
34
• Incidence angle at cell centre [deg]
• Signal-to-noise ratio [dB]
• Confidence of Doppler frequency estimate
• Auxiliary Data derived statistics
– Percentage of land coverage for each cell
– Land flag
– Ice flag
– Doppler frequency from antenna mispointing at cell centre [Hz]
– ECMWF wind speed [m/s]
– ECMWF wind direction [degN]
– Radial component of surface stokes drift from Wavewatch III model [m/s](optional)
– Azimuthal component of surface stokes drift from Wavewatch III model [m/s](optional)
– Sea state induced radial velocity from model [m/s] (optional)
The L2 RVL component of the OCN product includes the global attributes:
• Mission name
• Level1 source filename
• Processing time
• Polarization
• State vector
• Doppler algorithm version
• Grid cell size [m2]
35
A Background theoryThis section outlines some background theory for the Doppler estimation algorithmdescribed in the subsequent section. In sub-section A.1 basic notations, variables andcoordinate system (Figure 5) are defined, and relation between complex SAR image andrawdata is derived in frequecy domain. In sub-section A.2 the azimuth time-frequencyrelations for phase array antenna are derived.
. ............................................................................................................................................................................... ...................................................................................... ...................................................................................... ....................................................................................... ........................................................................................
.
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
. .............................. .
............................... .............................. .
..............................
. .......................... .
..........
..........
.......... . .......................... .
..........
..........
..........
..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..............................................................................................................................................................
p(0, 0, τ) p(0, 0, τ )
p(t, Θ, τ) p(t, Θ, τ)
ct2
ct2
cT2
. ............ ............
Φ
Figure 5: Curved track radar-to-target geometry. p is the function mapping from 3Dzero-Doppler system to xyz coordinate system, t is the fast time, τ is the slow time, Θ isthe elevation angle, Φ is the squint angle.
A.1 SAR imaging processA.1.1 SAR raw data
The base-band sampled signal reflected from a frozen surface, can be written as a two-dimensional integral over the scatter location t = (t, τ) in the 2D zero-Doppler coordi-nate system:
Iraw(t) =
dt e
−iω0T (τ −τ ;t)γc(t) s(t− T (τ − τ ; t)) V0(τ
− τ ; t) (41)
where γc is the complex back-scatter coefficient of the imaged surface, s is the modula-tion of the transmitted signal, V0 is the antenna ground pattern, and T is the two-waytime given as
T (τ − τ ; t) =
t2 + ν2(t)(τ − τ)2, ν(t) = 2 vrc , (42)
36
where vr is the effective radar velocity.The range direction Fourier transform of the raw data is
Iraw(ω, τ) = s(ω)
dt e
−i(ω0+ω)T (τ −τ ;t)γc(t) V0(τ
− τ ; t) , (43)
where s is the Fourier transform of s. In this domain we observe that SAR imaging pro-cess is independent of the form of the modulated signal. Strictly speaking, this equationis the correct starting point for definition of the raw data, since the antenna patternis also a function of range frequency ω, which can only be neglected for narrow-bandsignals.
A.1.2 SAR raw data Fourier domain
The 2D Fourier transform of the raw data can be written as
Iraw(ω) = s(ω)
dt γc(t)
12π
dτ
e−iφ(τ )
V0(τ − τ ; t) , (44)
where ω = (ω, ) is the 2D-Fourier variable and the phase function in the last integralis defined by
φ(τ ) = τ + (ω0 + ω) T (τ − τ ; t) . (45)
To find an explicit expression for this integral, we will use the method of stationaryphase. By expanding the phase to order two around the stationary phase value τ
s
φ(τ ) ≈ φ(τ s) + φ(τ s) (τ − τs) + 1
2 φ(τ s) (τ − τs)
2, (46)
and demanding that φ(τ s) = 0 , gives the following solution for the stationary phasevalue
τs = τ − t
ν2(t) Ω(ω,t) , (47)
whereΩ(ω, t) ≡
(ω0 + ω)2 −2/ν2(t). (48)
By inserting the expression for the stationary phase value τs, and computing the inte-
gral, the 2D Fourier transform of the raw data can be approximated by:
Iraw(ω) ≈ s(ω)
dt e
−iτ−iΩ(ω,t)tγc(t) V (ω, t) (49)
whereV (ω; t) = V0(− t
ν2(t) Ω(ω,t) ; t)
(ω0+ω)2t2πi ν2(t) Ω3(ω,t) (50)
is the azimuth direction envelope function of the Fourier domain. This is a slowly vary-ing function of the scatter location t. The azimuth direction envelope function is directlyrelated to the two-way antenna pattern (see Eq. 61).
37
A.1.3 SAR data focusing
Without loss of generality, we may write the Fourier transform of the transmitted signalas
s(ω) = U(ω) e−iψ(ω) (51)
where U is the magnitude of s. Further we define:
Ψ(ω; t) ≡ (Ω(ω, t)− (ω0 + ω)) t + ψ(ω) + ω · t (52)
where t is some reference time position. With those definitions, the Fourier transformedSAR raw data can we written as
Iraw(ω) = U(ω)
dt e
−iω·(t−t)−iΨ(ω;t)γc(t) e
−iω0tV (ω; t) (53)
Here, e−iΨ(ω;t) represents the transfer function between the complex focused SAR data
and the raw data. Since there exists an area around t = t where |Ψ(ω; t)−Ψ(ω; t)| < Ψ
for any real positive Ψ > 0, the phrase Exact Transfer Function is meaningful, and theSAR data may be focused with e
iΨ(ω;t) as a reconstruction transfer function
∀ > 0 ∃ δ > 0 :
Ic(t)−
dω e
iω·(t−t)e
iΨ(ω;t)Iraw(ω)
< for |t − t| < δ , (54)
where Ic is the theoretical perfectly focused complex SAR data. The practical problemof using Ψ(ω; t) as the phase function of the reconstruction transfer function is that thearea around t = t where satisfying results are achieved is small.
A better approach is to use the following expansion for the weak two-way time de-pendence in the phase function
Ψ(ω; t) ≈ Ψ(ω; t) Ψ0(ω)
+ Ψ(0, , t, τ)−Ψ(0, , t, τ) Ψ1(t,)
+ (t− t) ω∂2Ψ∂t∂ω (0, , t, τ)
α()−1
. (55)
The focusing scheme then becomes
Ic(t) =
d e
i(τ −τ)e
iΨ1(t,)
dω e
iα()ω(t−t)e
iΨ0(ω)Iraw(ω) , (56)
yielding the following relation between the focused SAR data and the backscatter coef-ficient
Ic(t) =
dtH(t − t; t) γc(t) e
−iω0t, (57)
where the impulse response function is given by:
H(t; t) ≈
d eiτ
dω e
iα()ωtU(ω) V (; t) . (58)
38
To illustrate the effects of the different terms of the phase function, we approximateΨ1 to first order in t:
Ψ1(t, ω) ≈ (t− t) ∂Ψ∂t (0, , t, τ) ≡ (t− t) µ() . (59)
With this approximation, the relation between the Fourier domains of the raw and thecompressed SAR data can be expressed as:
Ic(ω, ) ≈ eiΨ0(ω,)
Iraw(α()ω + µ(), ) . (60)
This means that the Ψ1-term in (55) is causing a range frequency shift (through µ), whilethe term containing α is causing a scaling of the the range frequency domain. This isoften referred to in the literature as Stolt migration.
39
A.2 Azimuth time-frequency relationsIn this section we start describing how the antenna ground pattern is related to the 2Dangular antenna diagram of the radar, then how the azimuth direction frequency bandis limited by the antenna diagram, and how the antenna squint angle is shifting thelocation of the frequency band. In sub-section A.2.3, estimation of the Doppler centroidis discussed, for the raw and focused data cases, respectively. In sub-section A.2.1 toA.2.3 is the theory given for a stiff rotated antenna, whereas in the last sub-section A.2.4,the same properties are discussed, using a phased array antenna.
A.2.1 Antenna pattern
The antenna ground pattern can be written as
V0(τ − τ ; t) = C
T 2(τ −τ ;t) A(Φ(τ − τ ; t)− Φ0(τ), Θ(t)−Θ0) , (61)
where A represents the 2D angular antenna pattern, where Θ0 is the antenna elevationangle, Θ is the local elevation angle to the target, Φ0 is the antenna squint angle and
Φ(τ − τ ; t) ≈ 2va(t) (τ −τ)c t (62)
represents the angle between the zero-Doppler direction at time τ and the direction to
a point on the ground in time position t, see Figure 5. The angular antenna pattern isgenerally also a function of the signal frequency and antenna pitch angle, but explicitnotation of those dependencies are omitted for simplicity. va is the azimuth velocity ofzero-Doppler plane.
A.2.2 Frequency band location
Assume that the antenna squint angle is changing through the imaging burst. If thechange is linear in time, we may write the antenna squint angle as
Φ0(τ) = Φ0(τ
0) + Φ0(τ
0)(τ
− τ0) (63)
where Φ0 is the antenna squint rate. In the following, we will, without loss of generality,assume that the azimuth time τ
of the raw data is zero for the center position of the rawdata burst (τ0 = 0). By combining this expression with the time-frequency relation
τ − τ = −β
−1 , (64)
whereβ =
ν2(ω0 + ω)
T=
ν2Ω
t(65)
is the signal Doppler rate (in rad/s2). Substituting for τ or τ
, respectively, we get thefollowing time-frequency relations for raw and focused data:
40
Raw data time-frequency relation: Let τ → τ + β
−1, the expression for the antenna
diagram becomes
A(Φ− Φ0, ·) −→ A
− c
2vpΩ ( −dc − γτ) , ·
(66)
whereγ = −2vpΩ
cΦ0 (67)
is the angular frequency squint rate of the antenna and
dc = −2vpΩ
cΦ0 (68)
is the angular Doppler centroid offset frequency. Here vp is the platform velocity andΩ is given by (48). For the center of the burst, where τ
is zero, we get the stripmap
mode (γ =0) azimuth frequency envelope function centered around dc, but generally,the expression is now centered around dc + γτ
, meaning that the azimuth frequencyband is moving as function of azimuth time. This effect is illustrated in time-frequencydiagram of Figure 6-a. The anti-diagonal lines, with slope equal to −β, represents theinfluence of equally spaced targets. Note that we have assumed a Doppler rate equal tothe pulse repetition frequency (PRF) in this plot.
Focused data time-frequency relation: By letting τ → τ − β
−1, the expression for
the antenna diagram becomes
A(Φ− Φ0, ·) −→ A
− c
2vpΩ
1 + γ
β
− β
β+γ (dc + γτ)
, ·
(69)
This expression is different from the usual Fourier domain envelope function for stripmap
mode focused data (γ = 0) in two ways: First, the bandwidth is changed with a factorβ/(β + γ), and second, it is not centered around the angular Doppler centroid offsetfrequency (not even for τ = 0), but around β/(β +γ)(dc +γτ). This means that the esti-mation procedure for the Doppler centroid offset anomaly is slightly more complicated,since β is a function of the estimation position. The time-frequency diagram of the fo-cused data is illustrated in Figure 6-b. Here we observe that the lines separating thecolors in the time-frequency diagram are vertical, meaning that the targets are sharplylocated in time.
A.2.3 Doppler centroid estimation
By deramping the raw data or the focused data with chirps aligning the frequencybands, the azimuth direction Fourier domain envelope functions becomes proportionalto
Raw: A
− c
2vpΩ ( −DC), ·
(deramped with ei2γτ 2) (70)
Focused: A
− c
2vpΩ (1 + γβ )( − βDC
β+γ ), ·
(deramped with ei2
βγγ+β τ2
) (71)
41
a) Raw signal (unfolded)
-2 -1 0 1 2Normalized azimuth time
-2
-1
0
1
2
Azi
mut
h fr
eque
ncy/
PRF
b) Focused signal (unfolded)
-2 -1 0 1 2Normalized azimuth time
-2
-1
0
1
2
Azi
mut
h fr
eque
ncy/
PRF
Figure 6: Time-frequency diagrams for an azimuth line (range gate) of TOPS mode SARdata. Each color represents the energy belonging to a collection of nearby targets.
The Doppler centroid frequency offset DC can now be found, by estimating the locationof the envelope maximum. The value is equal the location of the envelope maximumfor the raw data case and equal to 1+ γ/β times the envelope maximum for the focuseddata case. Here, ordinary stripmap mode centroid estimators may be used.
A.2.4 Phased array antennas
So far, only the ideal stiff rotated antenna case has been discussed, in this sub-sectionwe will introduce the phased array antenna. The general form of the transmission orreceive pattern of a regular gridded 2D-antenna with N ×M elements, can be writtenas
At/r(ζ; ζ0) =
n,m
Zr/tn,m(ζ0) e
iζ·( nN , m
M )fn,m(ζ) (72)
where fn,m represents the pattern of each individual module, Zr/tn,m is the receive/transmit
excitation coefficients, and ζ and ζ0 is the signal- and antenna steering direction, relatedto the azimuth and elevation angles in the following way
ζ = (ζ, ξ) ≡ (krLa sin Φ, krL
e sin Θ) . (73)
Here is La and L
e the size of the total antenna in azimuth and elevation direction, re-spectively. The two-way pattern is given by
A(ζ; ζ0) = At(ζ; ζ0) A
r(ζ; ζ0) . (74)
Beam formation: The big advantage of the phased antenna, is not only the ability tosteer the antenna by adding a phase shift to the T/R coefficients
Zt/rn,m(ζ0) ≡ Z
t/rn,m(ζ0) e
−iζ0·( nN , m
M ) (75)
42
-3 -2 -1 0 1 2 310-8
10-6
10-4
10-2
100
Intensity
Figure 7: Antenna pattern showing the difference between a classic antenna (red) whereall T/R coefficients Z are equal to one, and an antenna where the receive coefficients areformed to reduce the energy in the side-lobes (black) by steering the zeros away fromthe zeros of the transmitted pattern.
but also have the freedom to form the transmitting and receive antenna diagrams indi-vidually. By shaping the two diagrams such that the zero-crossing is not co-located, theenergy of the side-lobes of the two-way antenna diagram is dramatically reduced. Inthe following discussion, we drop the explicit notation for the elevation directions onthe left side, and rewrite equation (72) as
At/r(ζ; ζ0) =
n,m
Zr/tn,m(ζ0) e
i(ζ−ζ0)·( nN , m
M )fn,m(ζ) (76)
The big challenge, in the beam formation for TOPS systems, is to be able to keep theshape of the main lobe of the antenna constant for different azimuth steering directions:
A(ζ; ζ0) ≈ A(ζ − ζ0; 0) , ζ ∈ (ζ0 − ζ∆2 , ζ0 + ζ∆
2 ) (77)
where ζ∆ is the width of the main-lobe, and at the same time keep the energy in theside-lobes low. The disadvantage of using a phased array is the appearance of the grat-ing lobes. All those effects are visualized easily with some small simplifications of themodel.
Simplifications: By assuming all fn,m to be the radiation diagrams of perfect rectan-gular sub-antennas with the same size (La
/N × Le/M ):
fn,m(ζ) ≈ fa(ζ) f
e(ξ) = sinc ζ2N sinc ξ
2M , (78)
43
-15 -10 -5 0 5 10 1510-8
10-6
10-4
10-2
100
Intensity
-15 -10 -5 0 5 10 1510-8
10-6
10-4
10-2
100
Intensity
-15 -10 -5 0 5 10 1510-8
10-6
10-4
10-2
100
Intensity
Figure 8: Antenna pattern, showing how the grating lobes are growing when the an-tenna steering angle is increasing. The red lines are the sub-antenna pattern.
then the antenna diagrams may be written as
At/r(ζ; ζ0) ≈ f
a(ζ)
n
zt/rn (ζ0) e
i(ζ−ζ0)n/N, (79)
wherez
t/rn (ζ0) ≡ f
e(ξ)
m
Zt/r(ζ0, ξ0) e
i(ξ−ξ0)m/M. (80)
The two-way antenna diagram can with this simplification be written as
A(ζ; ζ0) ≈ f a(ζ)2
c(ζ0) ei(ζ−ζ0)/N
, (81)
wherec(ζ0) ≡
n
zt−n(ζ0)z
rn(ζ0) (82)
is different from zero for (2N − 1) values of . This means there exist 2N − 1 degree offreedom to keep the main-lobe of the diagram close to constant and at the same timeminimizing the energy of the side-lobes. To keep the beam-form close to constant, fordifferent steering direction, is the same as the integrated effects of c mimics
c(ζ0) ∼ c(0)fa(ζ−ζ0)
fa(ζ)
2.
44
In Figure 7 is the steering angle equal to zero, and we observe how, the side-lobes arereduced by the effect of not having the same locations of the zero-crossing in the receivediagram as in the transmitted one. In Figure 8 the steering angle is not zero, and weobserve how the grating-lobes grow as the main lobes of the periodic beam-train areshifted away from the zeros of f
a(ζ).
Time-frequency relations: Given a raw-data time τ the azimuth direction component
ζ of the directional vector is related to a target in zero Doppler time position t as follows
ζ = krLa 2va(t)
c t (τ − τ) = − (ω+ω0)La
2Ω vp ≡ (83)
where the second relation is coming from using the time-frequency relation of equa-tion (64). L
a is length of antenna in azimuth. For the steering direction we have
ζ0 = γτ. (84)
Combining those two equations, the antenna diagrams becomes
A(; τ ) = At(; τ ) A
t(; τ ) (85)
At/r(; τ ) ≡ A
t/r; γτ
=
n,m
Zt/rn,m(τ ) δZ
t/rn,m e
i(−γτ ) nN +i(ξ−ξ0) m
M fn,m(, ξ) , (86)
where we have chosen to annotate the coefficients as function of azimuth time τ. Here
δZt/r are the multiplicative weigthing (error) coefficients to the t/r modules provided as
auxiliary data. In the following we will use the following substitution τ → τ − β
−1 to
express the focused time-frequency relations. Assume we have a frequency offset dc
the two-way antenna diagram becomes:
Raw: A −dc; τ
(87)Focused: A
−dc; τ − β
−1
(88)
and if deramped, we get:
Raw: A −dc + γτ
; τ (deramped with e
i2γτ 2) (89)
Focused: A −dc + βγ
γ+β τ ; ββ+γ τ − β
−1
(deramped with e
i2
βγγ+β τ2
) (90)
If (and only if) the form of the main-lobe is invariant of the azimuth steering direction,we then get the following results:
Raw: A −dc; 0) (deramped with e
i2γτ 2) (91)
Focused: A(1 + γ
β )( − βdc
β+γ ); 0) (deramped with ei2
βγγ+β τ2
) (92)
This result is similar to the result of the previous sub-section. The requirements of hav-ing a time invariant shape of the main-lobe is a strong requirement. Even, if the antennaform is created such that the energy and the mean location (Doppler offset) is invariantof time τ
, the antenna diagram (in the focused Fourier-domain) will not necessarily beradiometric or mean location invariant of time τ .
45
B Expected Estimation Quality PerformanceThis section derives the expected estimation quality performance for the ideal case ofno errors in the antenna.
Assuming linear statistics:
P(j)n − P
(j)n ≈
(a(j+)n − a
(j+)) P (n − ς(j+) − ∆) + (b(j)
n − b(j))
−
(a(j+) − a(j+)) P (n − ς
(j+) − ∆) − (b(j) − b(j))
+
(ς(j+) − ς(j+)) a
(j+)P (n − ς
(j+) − ∆)
(93)
Demanding∂J
(j)
∂(ς(j), a(j), b(j))= 0 (94)
yield the following set of equations:
0 =
n
Wn
P
(j)n − P
(j)n
P (n − ς
(j)) (95)
0 =
n
Wn
P
(j)n − P
(j)n
P (n − ς
(j)) (96)
0 =
n
Wn
P
(j)n − P
(j)n
. (97)
By introducing the following short-hand notation:
P()n ≡ P (n − ς
() − ∆) . (98)
and letting j = 0, we get:
∆ς()
a()
n
WnP(0)n P
()n −
∆a()
n
WnP(0)n P
()n −∆b
(0)
n
WnP(0)n
= −
n
WnP(0)n
∆a()n P
()n + ∆b
(0)n
(99)
∆ς()
a()
n
WnP(0)n P
()n −
∆a()
n
WnP(0)n P
()n −∆b
(0)
n
WnP(0)n
= −
n
WnP(0)n
∆a()n P
()n + ∆b
(0)n
(100)
∆ς()
a()
n
WnP()n −
∆a()
n
WnP()n −∆b
(0)
n
Wn
= −
n
Wn
∆a()n P
()n + ∆b
(0)n
. (101)
46
Mean value: If the mean value is taken on both sides of the equation set above, we getthree equations for each j. Since all the equations are zero on the right hand side, andwe have the same number of unknowns as equations, we get
Eς(j) = ς(j)
, Ea(j) = a(j)
, Eb(j) = b(j)
. (102)
for all j.
Estimator variance: If we drop the superscript when = 0, and impose the followingtwo constrains:
n
WnPnPn = 0 ,
n
WnPn = 0 , (103)
the equation set can be rewritten as
−∆ς a
n
WnP2n =
n
WnPnun + qn (104)
∆a
n
WnP2n + ∆b
n
WnPn =
n
WnPnun + qn (105)
∆a
n
WnPn + ∆b
n
Wn =
n
Wnun + qn (106)
where
un =
∆a()n P
()n + ∆bn , qn =
=0
∆ς
()a
()P
()n −∆a
()P
()n
. (107)
Solving the set of equations, yield
∆ς = −
n WnPnun + qna
n WnP
2n
(108)
∆a =
n Wn
M22Pn −M12
un + qn
detM (109)
∆b =
n Wn
M11 −M21Pn
un + qn
detM (110)
where Mjk are the elements of the following matrix:
M =
n WnP
2n
n WnPn
n WnPn
n Wn
. (111)
In the computation of the of the variance we will treat (∆ς()
, ∆a()
, ∆b()), for = 0,
deterministic (all in q). That means that we will under-estimate the variances. Later we
47
will make a conservative estimate for the errors made. The variance of ς way be writtenas
σ2ς =
1
NMr, (112)
where N is the number of samples of the profiles, M is the number of averages donecreating the profile. The estimator performance number:
r =
1N
n Wn P
2n
2
1N
n W 2
n P 2n χ2
n
(113)
and the normalized noise variance:
χ2n =
µ2P ()
n 2 + ρ2, ρ =
b
a, µ =
a()
a, (114)
are numbers, independent of M and N (asymptotic). The variance of un is related to χ2n
in the following wayσ
2un
=a
Mχ
2n . (115)
Optimal weight function: Finding the optimal weight function is the same as maxi-mizing r with the constrains given by equation (103). By defining the following func-tion:
r ≡ r +2λ1N
n WnPnPn + 2λ2
N
n WnPn
1N
n W 2
n P 2n χ2
n
. (116)
where the last terms contains products between two Lagrange multipliers (λ1, λ2) andthe constrain conditions. Solving the minimizing problem is the same as solving:
∂r
∂Wm= 0 ,
∂r
∂(λ1, λ2)= 0 . (117)
First we rewrite equation (116) as
r1
N
n
W2nCn =
1
N
n
WnB(0)n
2+
2λ1
N
n
WnB(1)n +
2λ2
N
n
WnB(2)n (118)
where
Cm ≡ P2mχ
2m B
(1)m ≡ PmPm (119)
B(0)m ≡ P
2m B
(2)m ≡ Pm . (120)
Then perform the operation ∂/∂Wm on both sides, yielding the following expression forthe weight function:
Wm =1
r
2
k=0
λkB
(k)m
Cm, (121)
48
where the definitionλ0 ≡
1
N
n
WnB(0)n (122)
is used. If we now multiplies equation (121) with B(j)m (j = 0, 1, 2), followed by a sum-
mation over m divided by N we get
B(0,0) + λ1 B(1,0) + λ2 B(2,0) = r (123)
B(0,1) + λ1 B(1,1) + λ2 B(2,1) = 0 (124)
B(0,2) + λ1 B(1,2) + λ2 B(2,2) = 0 (125)
where λk = λk/λ0 and
B(j,) =1
N
m
B(j)m B
()m
Cm. (126)
From the two last lines (eq. (124) and (125)) (λ1, λ2) can be found, and from the first linewe get
r =1
N
n
P2n + λ1PnPn + λ2Pn
χ2n
(127)
Special case: One special case is when χ2m is symmetrical, then both B(0,1) and B(0,2) is
zero, and from equation (124) and (125) we have the solution: λ1 = λ2 = 0. This specialcase occurs when the intensity in the side-bands are symmetrical: µ = µ−, yielding
Wm ∝ 1
χ2m
r =1
N
n
P2n
χ2n
. (128)
This is the optimal solution of the problem and the value of r represents the maximumpossible performance number.
49
C Regular gridded 2D-antennaIdeal form: Let L
a and Le be the distance between the (Na ×N
e) elements in azimuthand elevation direction, respectivly, and let the size of the elements be equal to thedistance between the elements. If the elements are radiationg uniformly the patternfrom one element is given by
d
eikr
√x2+y−2
4πi
x2 + y − 2≈ e
ikrr
4πir
d e
−kry·r
=e
ikrr
4πir
LaL
e
4sinc ζa
2 sinc ζe
2 ,
(129)
wherer =
x2 + y2 (130)
is the distance between target and antenna center, and
ζa ≡ krLa
r ya = krL
a sin Φ , ζe ≡ krLe
r ye = krL
e sin Θ . (131)
Here is Φ the azimuth angle and Θ the elevation-angle. The full two-way antenna pat-tern can be written as
A(ζ; ζ0) = At(ζ; ζ0) A
r(ζ; ζ0) (132)
where the transmision and receive pattern are given by
At/r(ζ; ζ0) = f(ζ)
m,n
Zt/rm,n(ζ0) e
i(ζ−ζ0)·(m,n). (133)
and ζ = (ζa, ζ
e). The antenna steering direction is given by ζ0 and
f(ζ) ≡ sinc ζa
2 sinc ζe
2 , (134)
represents the 2D sub-antenna pattern. Since f is dyadic in form (azimuth and eleva-tion direction are separable), the full antenna can also be dyadic if the reflection andtransmission coefficients are chosen dyadic: Z
t/rm,n(ζ) = z
at/rm (ζa) z
et/rn (ζe).
General form: In the real world, the sub-antenna pattern are not necessarily dyadicand the individual patterns are not necessarily equal, neither as function of position,polarization or transmission/receive direction. The transmission and receive patternfor the full antenna, then takes the following general form:
At/r(ζ; ζ0) =
m,n
Zt/rm,n(ζ0) e
i(ζ−ζ0)·(m,n)f
t/rm,n(ζ) . (135)
50
C.1 1D beam formingIn this section it is assumed that the transmitted antenna pattern is given, and optimiza-tion is only done for the receive pattern. Minimizing the following cost-function:
J =
IR\L(ζ0)
dζ
f(ζ)
n
zrn e
i(ζ−ζ0)n2 At(ζ; ζ0)
2
+ λ
L(ζ0)
dζ
f(ζ)
n
zrn e
i(ζ−ζ0)n2 At(ζ; ζ0)
2 − 1
,
(136)
where L(ζ0) is the interval (ζ0 − ζ∆/2, ζ0 + ζ∆/2), yield
n
pn−ma
rn − qn−mb
rn
+ λ
n
pn−ma
rn − qn−mb
rn
= 0 (137)
n
qn−ma
rn + pn−mb
rn
+ λ
n
qn−ma
rn + pn−mb
rn
= 0 (138)
where arn, b
rn are the real and imaginary part of the transmission coeffients:
zrn ≡ a
rn + ib
rn , (139)
and
pn−m =
L(ζ0)
dζAt(ζ; ζ0) f(ζ)
2 cos((ζ − ζ0)(n−m)) ,
qn−m =
L(ζ0)
dζAt(ζ; ζ0) f(ζ)
2 sin((ζ − ζ0)(n−m)) ,
pn−m =
IR\L(ζ0)
dζAt(ζ; ζ0) f(ζ)
2 cos((ζ − ζ0)(n−m)) ,
qn−m =
IR\L(ζ0)
dζAt(ζ; ζ0) f(ζ)
2 sin((ζ − ζ0)(n−m)) .
(140)
By defining the following matrix
P =
p0 p1 p2 · · · pN−1
p−1 p0 p1 · · · pN−2
p−2 p−1 p0 · · · pN−3...
...... . . . ...
p1−N p2−N p3−N · · · p0
(141)
and using the same convention for defining P , Q and Q, from the coefficients pn−m, qn−m
and qn−m, respectively, we may write the system asA− λA
xT = 0 , (142)
where
A ≡P −QQ P
, A ≡
P −QQ P
, x ≡ [ar
0, . . . , arN−1, b
r0, . . . , b
rN−1] . (143)
51
C.2 Doppler offsets caused by error in the antenna formThe azimuth direction 1D-problem is firts considered. The Fourier-profile can be writtenas:
P (ζ − ς;x) = |A(ζ − ς; ζ0)|2 . (144)
Here is x given as the collection of the real and imaginary part of the transmission andreceive coefficients:
x = (at0, . . . , a
tNa−1, b
t0, . . . , b
tNa−1, a
r0, . . . , a
rNa−1, b
r0, . . . , b
rNa−1) (145)
The goal is to find an estimate ς of the true offset ς , given only a estimate x of the truevector x. This yields the following minimizing problem:
0 =∂
∂ζ0
dζ W (ζ)
P (ζ − ς;x)− P (ζ − ς; x)
2(146)
where in the computations, without loss of generality, is assumed ς to be zero. By writ-ing
P (ζ − ς; x) ≈ P (ζ;x) + (x− x) ·∇P (ζ;x)− ςP (ζ;x) , (147)
the minimization problem becomes
0 =
dζ W (ζ)
(x− x) ·∇P (ζ;x)− ςP (ζ; x)
P (ζ;x) (148)
Solved with respect to ς , we get
ς = (x− x) ·
dζ W (ζ)∇P (ζ;x) P (ζ;x)
dζ W (ζ) P 2(ζ;x)(149)
This gives the following expression for the variance of ς as function of the variance ofthe estimated transmission and reflection coefficients:
σ2ς =
σ2x
dζ W (ζ)∇P (ζ;x) P (ζ;x)
dζ W (ζ) P 2(ζ;x)
2
. (150)
The response of an error in the coefficients on the profile is given as
∂P
∂at/rn
=B
t/rn (ζ; ζ0) + B
t/rn
∗(ζ; ζ0)
2= Bt/r
n (ζ; ζ0) , (151)
∂P
∂bt/rn
=B
t/rn (ζ; ζ0)−B
t/rn
∗(ζ; ζ0)
2i= Bt/r
n (ζ; ζ0) , (152)
whereB
t/rn (ζ; ζ0) ≡ 2At/r(ζ; ζ0) f
t/r∗(ζ) e−i(ζ−ζ0)n
Ar/t(ζ; ζ0)2 . (153)
52
By the following definition
pt/rn ≡
dζ W (ζ) B
t/rn (ζ; ζ0) P (ζ;x)
dζ W (ζ) P 2(ζ;x)
(154)
can ς be written as
ς =
n
(at
n − atn)pt
n+ (btn − b
tn)pt
n+ (arn − a
rn)pr
n+ (brn − b
rn)pr
n
(155)
or even more compact as
ς =
n
(zt
n − ztn) p
tn∗+ (zr
n − zrn) p
rn∗
, (156)
and if the variance of the real and imaginary part of each t/r-coefficient are equal, theexpression for the variance simplifies to
σ2ς =
n
σ
2at
n
ptn
2 + σ2ar
n
prn
2 . (157)
53
D FiguresThis section shows figures related to Appendix B on the expected performances.
In Figure 9 is shown the effect of aliasing on the baseband azimuth frequency profilefor different fraction (κ) of the bandwidth processed.
In Figure 10 is shown the estimated Doppler bias as function of fraction of band-width for different SNR, where for the last three plots weight functions scaled to theSNR are used.
In Figure 11 is the estimator performance (eq. (127)) versus SNR plotted for differentfraction of bandwidth and weight functions.
In Figure 12 is the Doppler offset standard devation (Hz) (eq. (112 and eq. (127))estimated over a given number of pixels shown versus SNR for different fractions ofbandwidth and weight functions.
In Figure 13 is shown the Doppler width standard deviation (Hz) (eq. (115)) esti-mated over a given number of pixels shown versus Doppler width for different SNR.
The figures show that the requirement for the Doppler offset accuracy (5Hz) can beachieved.
54
(a)
-0.4 -0.2 0.0 0.2 0.4Normalized frequency [1/PRF]
0.0
0.2
0.4
0.6
0.8
1.0
1.2Kappa = 0.6
(b)
-0.4 -0.2 0.0 0.2 0.4Normalized frequency [1/PRF]
0.0
0.2
0.4
0.6
0.8
1.0
1.2Kappa = 0.7
(c)
-0.4 -0.2 0.0 0.2 0.4Normalized frequency [1/PRF]
0.0
0.2
0.4
0.6
0.8
1.0
1.2Kappa = 0.8
(d)
-0.4 -0.2 0.0 0.2 0.4Normalized frequency [1/PRF]
0.0
0.2
0.4
0.6
0.8
1.0
1.2Kappa = 0.9
Figure 9: Azimuth Fourier profiles, where the aliased profile is in black, the zero-bandprofile in red and the additive noise level in green. The different plots are for κ =(0.6, 0.7, 0.8, 0.9), all with a 3 dB increase in energy of first side-band.
55
(a)
0.6 0.7 0.8 0.9 1.0kappa
-0.5
0.0
0.5
1.0
1.5
2.0
Do
pp
ler
Bia
s [H
z]
(Flat weight case)
(b)
0.6 0.7 0.8 0.9 1.0Kappa
-0.5
0.0
0.5
1.0
1.5
2.0
Do
pp
ler
Bia
s [H
z]
(SNR = 0 dB)
(c)
0.6 0.7 0.8 0.9 1.0Kappa
-0.5
0.0
0.5
1.0
1.5
2.0
Do
pp
ler
Bia
s [H
z]
(SNR = 12 dB)
(d)
0.6 0.7 0.8 0.9 1.0Kappa
-0.5
0.0
0.5
1.0
1.5
2.0
Do
pp
ler
Bia
s [H
z]
(SNR = 24 dB)
Figure 10: Doppler bias as function of κ, caused by a 3 dB non-corrected increase in firstside-band (solid line), second side-band (dashed line) and third side-band (dotted line).Plot (a) represents the flat weight function and plot (b), (c) and (d) are for different SNRwith the Wn = 1/χ2
n weight-function (PRF = 1650 Hz in all plots).
56
0 5 10 15 20SNR [dB]
0
5
10
15
Est
imat
or
per
form
ance
SQ
RT
(r) Kappa = 0.7, 0.8, 0.9
Figure 11: Estimator performance as function of additive noise level. The differentlines represents
√r for different values of κ (the upper lines represents κ = 0.9). The
solid lines represents the performance of using the Wn = 1/χ2n weight function, and the
dashed lines, the performance of using constant weights.
0 5 10 15 20SNR [dB]
0.0
0.5
1.0
1.5
2.0
Do
pp
ler
stan
dar
d d
evia
tio
n [
Hz] Image size = 1000x1000 Pixels
PRF = 1650 Hz
Kappa = 0.7, 0.8, 0.9
Figure 12: Doppler standard deviation as function of additive noise. The different linesrepresents different values of κ (the lower lines represents κ = 0.9). The solid linesrepresents the use of the Wn = 1/χ2
n weight function, and the dashed lines, the use ofconstant weights.
57
10 20 30 40 50Doppler width [Hz]
0
5
10
15
20
Do
pp
ler
wid
th S
TD
[H
z]
Image size = 1000x1000 Pixels
PRF = 1650 Hz
Kappa = 0.8
SNR = 0 dB, 6 dB, 12 dB, 18 dB, 24 dB
Figure 13: Doppler width standard deviation as function of Doppler width. Upper linerepresents SNR=0 dB and lower line SNR=24 dB.
58
