EE/Ae 157 b. Polarimetric Synthetic Aperture Radar (1)

Transcription

1 EE/Ae 157 b Polarimetric Synthetic Aperture Radar (1)

2 OUTLINE PRINCIPLES OF POLARIMETRY Polarization representations Scatterer as polarization transformer Characterization of scatterers Implementation of polarimeters Polarization signatures APPLICATIONS OF POLARIMETRY Observed polarization signatures Unsupervised land cover characterization Cloude s decomposition theorem Soil moisture and surface roughness estimation Jakob van Zyl 2

3 INTRODUCTION SPATIAL INFORMATION RADAR IMAGERS IMAGING RADAR SPECTROMETERS IMAGING RADAR POLARIMETERS MULTI-FREQUENCY IMAGING RADAR POLARIMETERS POLARIZATION INFORMATION RADAR POLARIMETERS MULTI-FREQUENCY RADAR POLARIMETERS SPECTRAL INFORMATION RADAR SPECTROMETERS Jakob van Zyl 3

4 WAVE POLARIZATIONS: EXAMPLES HORIZONTAL (LINEAR) VERTICAL (LINEAR) RIGHT-HAND CIRCULAR LEFT-HAND CIRCULAR Jakob van Zyl 4

5 PRINCIPLES OF POLARIMETRY: FIELD DESCRIPTIONS POLARIZATION VECTOR POLARIZATION ELLIPSE STOKES VECTOR a h p a v 2 2,, a a h a v S S 0 S 1 S 2 S 3 v S 0 a h 2 a v 2 POLARIZATION ELLIPSE a v 2 2 S 1 a h a v S 2 2 a h a v * S 3 2 a h a v * S 0 cos2 cos 2 S 0 cos2 sin2 S 0 sin2 a h MAJOR AXIS MINOR AXIS Jakob van Zyl 5

6 WAVE POLARIZATIONS: GEOMETRICAL REPRESENTATIONS v LEFT-HAND CIRCULAR 45 0 S 3 LINEAR POLARIZATION (VERTICAL) ; 0 0 POLARIZATION ELLIPSE a v S 0 a h h 2 2 S 2 MAJOR AXIS MINOR AXIS S 1 LINEAR POLARIZATION (HORIZONTAL) 0 0 ; 0 0 RIGHT-HAND CIRCULAR 45 0 POLARIZATION ELLIPSE POINCARE SPHERE Jakob van Zyl 6

7 EXAMPLE POLARIZATIONS LINEAR HORIZONTAL LINEAR VERTICAL LEFT-HAND CIRCULAR v v v h h h p 1 0 p 0 1 p i 0 0 ; ; ; S S S 0 1 Jakob van Zyl 7

8 DEFINITION OF ELLIPSE ORIENTATION ANGLES WARNING Sometimes the polarization ellipse orientation angle is defined with respect to the vertical direction. In that case, linear horizontal polarization has an ellipse orientation angle of +90 degrees or -90 degrees, and linear vertical polarization is characterized by an ellipse orientation angle of 0 degrees. Both conventions are used in this viewgraph package. Jakob van Zyl 8

9 SCATTERER AS POLARIZATION TRANSFORMER Transverse electromagnetic waves are characterized mathematically as 2- dimensional complex vectors. When a scatterer is illuminated by an electromagnetic wave, electrical currents are generated inside the scatterer. These currents give rise to the scattered waves that are reradiated. SCATTERED WAVES INCIDENT WAVE SCATTERER Mathematically, the scatterer can be characterized by a 2x2 complex scattering matrix that describes how the scatterer transforms the incident vector into the scattered vector. The elements of the scattering matrix are functions of frequency and the scattering and illuminating geometries. Jakob van Zyl 9

10 SCATTERING MATRIX E h E v sc S hh S hv S vh S vv E h E v inc Far-field response from scatterer is fully characterized by four complex numbers Scattering matrix is also known as Sinclair matrix or Jones matrix Must measure a scattering matrix for every frequency and all incidence angles Jakob van Zyl 10

11 COORDINATE SYSTEMS All matrices and vectors shown in this package are measured using the backscatter alignment coordinate system. This system is preferred when calculating radar-cross sections, and is used when measuring them: Transmitting Antenna h ˆ t ˆ v t ˆ k t ˆ z ˆ h r ˆ k r ˆ v r Receiving Antenna i s Scatterer s ˆ y i ˆ x Jakob van Zyl 11

12 POLARIMETER IMPLEMENTATION To fully characterize the scatterer, one must measure the full scattering matrix: E h E v rec S hh S hv S vh S vv E h E v tr Setting one of the elements of the transmit vector equal to zero allows one to measure two components of the scattering matrix at a time: S hh S vh S hh S hv 1 0 S vh S vv inc ; S hv S vv S hh S hv 0 1 S vh S vv inc This technique is commonly used to implement airborne and spaceborne SAR polarimeters, such as AIRSAR and SIR-C. Jakob van Zyl 12

13 POLARIMETER IMPLEMENTATION BLOCK DIAGRAM TIMING Horizontal Transmission: Receiver Horizontal Vertical Transmitter Reception: Receiver Horizontal HH HH HH HV HV Vertical Vertical VH VV VH VV VH Jakob van Zyl 13

14 POLARIMETER IMPLEMENTATION In the implementation shown, the scattering matrix elements are measured in pairs. The two pairs are measured at different times, i.e. from (slightly) different viewing directions. Due to the speckle effect, this means that the two pairs of signals may not be fully correlated. In the AIRSAR case, this decorrelation is of little consequence, since the distance traveled by the aircraft between successive pulses (~ 50 cm) is small compared to the system resolution (~ 2-5 m). In spaceborne cases, like SIR-C, the decorrelation distance is much closer in length to the distance traveled by the antenna during the interpulse period. Therefore, significant decorrelation could occur. This could be corrected through resampling (interpolation) of one channel with respect to the other. Jakob van Zyl 14

15 MATHEMATICAL CHARACTERIZATION OF SCATTERERS: SCATTERING MATRIX The radiated and scattered electric fields are related through the complex 2x2 scattering matrix: E sc Sp rad The (complex) voltage measured at the antenna terminals is given by the scalar product of the receiving antenna polarization vector and the received wave electric field: V p rec Sp rad The measured power is the magnitude of the (complex) voltage squared: P VV * p rec Sp rad 2 NOTE: Radar cross-section is proportional to power Jakob van Zyl 15

16 MATHEMATICAL CHARACTERIZATION OF SCATTERERS: COVARIANCE MATRIX We can rewrite the expression for the voltage as follows: V p rec Sp rad p h rec p h rad S hh p h rec p v rad S hv p v rec p h rad S vh p v rec p v rad S vv A T p h rec p h rad p h rec p v rad p v rec p h rad p v rec p v rad S hh S hv S vh S vv The first vector contains only antenna parameters, while the second contains only scattering matrix elements. Using this expression in the power expression, one finds * ; P VV AT TA ATT A A C A C TT * * * * * The matrix C is known as the covariance matrix of the scatterer Jakob van Zyl 16

17 MATHEMATICAL CHARACTERIZATION OF SCATTERERS: STOKES SCATTERING OPERATOR The power expression can also be written in terms of the antenna Stokes vectors. First consider the following form of the power equation: P p rec E sc p rec E sc * rec sc rec sc rec sc rec sc p h E h p v E v p h E h p v E v p rec rec* h p h E sc sc* h E h p rec rec* v p v p rec rec* h p h E sc sc* h E h p rec rec* v p v p rec rec* h p v E sc sc* v E v E sc sc* h E v p rec rec* v p h E sc sc* v E h g rec X * E sc sc* v E v p rec rec* h p v E sc sc* h E v p rec rec * v p h E sc sc* v E h The vector X in the expression above is a function of the transmit antenna parameters as well as the scattering matrix elements. Jakob van Zyl 17

18 MATHEMATICAL CHARACTERIZATION OF SCATTERERS: STOKES SCATTERING OPERATOR Using the fact that E sc Sp rad, it can be shown that X can also be written as X Wg rad where W * S hh S hh * S vh S vh * S hh S vh * S vh S hh * S hv S hv * S vv S vv * S hv S vv * S vv S hv * S hh S hv * S vh S vv * S hh S vv * S vh S hv * S hv S hh * S vv S vh * S hv S vh * S vv S hh This means that the measured power can also be expressed as: P g rec Wg red Jakob van Zyl 18

19 MATHEMATICAL CHARACTERIZATION OF SCATTERERS: STOKES SCATTERING OPERATOR From the earlier definition of the Stokes vector, we note that the Stokes vector can be written as: * * p h p h* p v p v p h p h * p h p h* p v p v * p v p v S p h p v* p h* p v Rg g R 1 S * p h p v i( p h p v* p h* p v ) 0 0 i i p h* p v This means that we can express the measured power as: ~ P S rec R 1 W R 1 S rad S rec M S rad The matrix M is known as the Stokes scattering operator. It is also called Stokes matrix. Jakob van Zyl 19

20 POLARIZATION SYNTHESIS Once the scattering matrix, covariance matrix, or the Stokes matrix is known, one can synthesize the received power for any transmit and receive antenna polarizations using the polarization synthesis equations: Scattering matrix: Covariance Matrix: P p rec Sp rad 2 P ACA * Stokes scattering operator: P S rec MS rad Keep in mind that all matrices in the polarization synthesis equations must be expressed in the backscatter alignment coordinate system. Jakob van Zyl 20

21 POLARIZATION SIGNATURE The polarization signature (also known as the polarization response) is a convenient graphical way to display the received power as a function of polarization. Usually displayed assuming identical transmit and receive polarizations (copolarized) or orthogonal transmit and receive polarizations (cross-polarized). Jakob van Zyl 21

22 Polarization Response Another way to look at the signature is that the scatter changes the co-polarized return from a sphere (using the Poincare representation) to some other figure: Jakob van Zyl 22

23 POLARIZATION SIGNATURE OF TRIHEDRAL CORNER REFLECTOR ˆ v t l ˆ h t S c c k 0l C c M 1 2 c Jakob van Zyl 23

24 POLARIZATION SIGNATURE OF TRIHEDRAL CORNER REFLECTOR Jakob van Zyl 24

25 POLARIZATION SIGNATURE OF TRIHEDRAL CORNER REFLECTOR Using the Stokes matrix approach, we can write the received power of the trihedral corner reflector as P k 2 0l 4 1 cos 2 2 cos 2 2 sin 2 2 cos k 2 0l 4 1 cos k 2 0l cos 4 sin 2 2 sin 2 2 where the top sign is for the co-polarized case, and the bottom sign is for the cross-polarized case. It is immediately apparent that the received power is independent of the ellipse orientation angle, as shown by the signatures on the previous page. For the co-polarized case, the maximum is found when 0, i.e. for linear polarizations, and the minimum occurs when 450, i.e. for circular polarizations. The positions of the maxima and minima are reversed in the case of the cross-polarized signatures. This behavior is explained by the fact that the reflected waves have the opposite handedness than the transmitted ones. Therefore, if either circular polarization is transmitted, the reflected wave is polarized orthogonally to the transmitted wave, leading to maximum reception in the cross-polarized, and minimum reception in the co-polarized case. Jakob van Zyl 25

26 POLARIZATION SIGNATURE OF A DIHEDRAL CORNER REFLECTOR ˆ v t a 2b ˆ h t S c c k 0ab C c M 1 2 c Jakob van Zyl 26

27 Three-Dimensional Response Jakob van Zyl 27

28 POLARIZATION SIGNATURE OF A SHORT THIN CYLINDER vˆ t l hˆ t S c C c M 1 4 c c k 2 0 l 3 ; a radius of cylinder 3 ln4l / a 1 Jakob van Zyl 28

29 POLARIZATION SIGNATURE OF A SHORT THIN CYLINDER Jakob van Zyl 29

30 POLARIZATION SIGNATURE OF A SHORT THIN CYLINDER vˆ t l hˆ t S c C c M c c k 2 0 l 3 ; a radius of cylinder 6 ln4l / a1 Jakob van Zyl 30

31 POLARIZATION SIGNATURE OF A 45 o SHORT THIN CYLINDER Jakob van Zyl 31

32 POLARIZATION SIGNATURE OF RIGHT-HANDED HELIX ˆ v t ˆ h t S 1 1 i 2 i 1 1 i i 1 C 1 i 1 1 i 4 i 1 1 i 1 i i M Jakob van Zyl 32

33 RIGHT-HANDED HELIX Jakob van Zyl 33

34 POLARIZATION SIGNATURE OF LEFT-HANDED HELIX ˆ v t ˆ h t S 1 1 i 2 i 1 1 i i 1 C 1 i 1 1 i 4 i 1 1 i 1 i i M Jakob van Zyl 34

35 LEFT-HANDED HELIX Jakob van Zyl 35

36 SCATTERING MODELS AND OBSERVATIONS Jakob van Zyl 36

37 Signatures Jakob van Zyl 37

38 Rough Surface Scattering - Roughness Jakob van Zyl 38

39 Rough Surface Scattering Dielectric Constant Jakob van Zyl 39

40 Surface Characteristics Geometrical Properties Jakob van Zyl 40

41 Real Part of Epsilon POLARIMETRIC SYNTHETIC APERTURE RADAR 45 Surface Characteristics Dielectric Properties Hallikainen Wang and Schmugge Brisco et al Dobson et al Volumetric Soil Moisture Jakob van Zyl 41

42 POLARIZATION SIGNATURE OF SINGLE SHORT THIN CYLINDER vˆ t l hˆ t S c C c M 1 4 c c k 2 0 l 3 ; a radius of cylinder 3 ln4l / a 1 Jakob van Zyl 42

43 SHORT CYLINDERS ORIENTED COSINE SQUARED AROUND VERTICAL Jakob van Zyl 43

44 THIN CYLINDERS ORIENTED UNIFORMLY RANDONLY Jakob van Zyl 44

45 MODEL OF VEGETATION SCATTERING Vegetation Layer b, a, l,, p p p p p p, a, l,, p s s s s s Ground Surface, h, l, s g g l Jakob van Zyl 45

46 SCATTERING MECHANISMS i b i i i z Jakob van Zyl 46

47 OBSERVED POLARIZATION SIGNATURES: L-BAND POLARIZATION SIGNATURES OF THE OCEAN Jakob van Zyl 47

48 OBSERVED POLARIZATION SIGNATURES: SAN FRANCISCO Jakob van Zyl 48

49 OBSERVED POLARIZATION SIGNATURES: SAN FRANCISCO At each frequency the ocean area scatters in a manner consistent with models of slightly rough surface scattering, the urban area like a dihedral corner reflector, while the park and natural terrain regions scatter much more diffusely, that is, the signatures possess large pedestals. This indicates the dominant scattering mechanisms responsible for the backscatter for each of the targets. The ocean scatter is predominantly single bounce, slightly rough surface scattering. The urban regions are characterized by two-bounce geometry as the incident waves are twice forward reflected from the face of a building to the ground and back to the radar, or vice versa. The apparent diffuse nature of the backscatter from the park and natural terrain indicates that in vegetated areas there exists considerable variation from pixel to pixel of the observed scattering properties, leading to the high pedestal. This variation may be due to multiple scatter or to a distinct variation in dominant scattering mechanism between 10m resolution elements. Jakob van Zyl 49

50 OBSERVED POLARIZATION SIGNATURES: GEOLOGY Jakob van Zyl 50

51 OBSERVED POLARIZATION SIGNATURES: GEOLOGY The previous viewgraph shows polarization signatures extracted from an AIRSAR image of the Pisgah lava flow in California. Signatures correspond to the flow itself (a very rough surface), an alluvial fan (medium roughness), and from the playa next to the flow (a very smooth surface). Note that as the roughness of the surface increases, so does the observed pedestal height. This is quite consistent with the predictions of the slightly rough surface models, even though the surface r.m.s. heights exceed the strict range of validity of the model. The exception is the playa case, where the pedestal at P-band is higher than that at L-band. Possible explanations for this behavior include subsurface scattering due to increased penetration at P-band, or signal-to-noise limitations for the very smooth surface. Jakob van Zyl 51

52 OBSERVED POLARIZATION SIGNATURES: SEA ICE Jakob van Zyl 52

53 OBSERVED POLARIZATION SIGNATURES: SEA ICE Note the pedestal height as a function of frequency for the multi-year ice - the P-band pedestal is quite small, while the C-band pedestal is the greatest of the three. For the first year ice, exactly the opposite behavior is seen. The P-band signature shows the highest and the C-band signature the lowest pedestal. The multi-year ice behavior may be explained if we consider the ice to be formed of two layers, where the upper layer consists of randomly oriented oblong inclusions about the size of a C-band wavelength, several centimeters. The lower layer forms a solid, but slightly rough surface. In this situation the C-band signal would interact strongly with the diffuse scattering upper layer, giving rise to the high pedestal. The longer wavelength L- and P-band signals would pass through the upper and be scattered by the lower layer, which is smooth enough to exhibit fairly polarized backscatter. On the other hand, similar characteristics are also observed for simple rough surface scattering also, as previously explained. At present we have no model to explain the first year ice behavior. Jakob van Zyl 53