IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 8, AUGUST 2009
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An Efficient Algorithm to the Simulation of Delay–Doppler Maps of Reflected Global Navigation Satellite System Signals Juan Fernando Marchán-Hernández, Student Member, IEEE, Adriano Camps, Senior Member, IEEE, Nereida Rodríguez-Álvarez, Student Member, IEEE, Enric Valencia, Student Member, IEEE, Xavier Bosch-Lluis, Student Member, IEEE, and Isaac Ramos-Pérez, Associate Member, IEEE
Abstract—A new and efficient algorithm to compute delay– Doppler maps is presented. It improves by more than an order of magnitude the required computation time and memory resources. This approach is based on the derivation of explicit expressions of the space coordinates as a function of the delay offset and Doppler shift. Using this technique, the limitation posed by the number of sampling points of the observed surface is drastically attenuated, and a wide range of scenarios from low- to mediumheight airborne-to-spaceborne scenarios can now be simulated with standard desktop computers. Index Terms—Delay–Doppler map (DDM), reflected Global Navigation Satellite System (GNSS-R), reflectometer.
I. I NTRODUCTION
T
HROUGHOUT the past 15 years, research on the use of reflected Global Navigation Satellite System (GNSS-R) signals has yielded promising results: from altimetry applications [1], [2] to soil moisture determination [3], ice characterization [4], or sea state retrieval [5], [6]. So far, only the GPS satellite constellation is fully operational, and therefore, most of the research is particularized for this system. GPS was designed to provide global positioning at any time, and thus, it required a minimum of four satellites to simultaneously be visible. The space segment is composed of at least four satellites in each of the six orbital planes phased 60◦ [7]. The orbits are near circular and have an inclination of 55◦ , a semimajor axis of approximately 26 562 km, and an orbital period of about 12 h. The so-called Standard Positioning Service (SPS) is a positioning and timing service provided at the GPS L1 band (1575.42 MHz). Each GPS satellite transmits in this band a Manuscript received June 12, 2008; revised October 20, 2008 and December 9, 2008. First published April 10, 2009; current version published July 23, 2009. This work, which was conducted as part of the award “Passive Advanced Unit (PAU): A Hybrid L-band Radiometer, GNSS-Reflectometer and IR-Radiometer for Passive Remote Sensing of the Ocean” made under the European Heads of Research Councils and European Science Foundation European Young Investigator (EURYI) Awards scheme in 2004, was supported in part by the Participating Organizations of EURYI, by the European Commission Sixth Framework Program, by the Department of Universities of the Catalan Autonomous Government, by the European Social Fund, and by the Spanish Plan Nacional del Espacio project PAU in SeoSAT. The authors are with the Remote Sensing Laboratory, Department of Signal Theory and Communications, Universitat Politècnica de Catalunya (UPC), 08034 Barcelona, Spain (e-mail:
[email protected];
[email protected];
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2009.2014465
signal that contains a pseudorandom noise Coarse/Acquisition code (PRN-C/A) that is unique and known for each satellite plus the navigation message. It also contains a precision code (P) that is reserved for military uses, but this is not part of the SPS. A new civil signal in L2 (LC2) is already being transmitted from the new GPS satellites. GPS satellites broadcast right-hand circular polarization signals that, when impinging over a surface, change to a mostly left-hand elliptical polarization. Over a perfectly flat surface, the scattered signal comes from the specular reflection point, which is determined by the shortest distance between the transmitting GPS satellite and the receiver. However, the rougher the surface, the larger the region (known as the glistening zone) from where the scattered signals are collected. If this surface is a plane, the loci of constant delay (isorange) are a set of ellipses, and the loci of constant Doppler shift (iso-Doppler) are a set of hyperbolae. Thus, each surface point has a particular delay and Doppler (actually there are two points with the same delay and Doppler). In radar, the ambiguity function [8] provides a measure of the similitude between a signal and a delayed and Doppler-shifted version of it. The GNSS-R equivalent to this ambiguity function is known as a delay–Doppler map (DDM) and consists of the power distribution of the reflected signal over the 2-D space of delay offsets and Doppler shifts. This power distribution can be expressed as [9], [10] ||2 D2 ( ρ)Λ2 (Δτ ) |S(Δf )|2 q 4 ( ρ) |Y (τ, fd )|2 = Ti2 2 2 4πR0 ( ρ)R ( ρ) qz ( ρ) G
q⊥ × Pυ − d2 ρ (1) qz
where Ti is the coherent integration time; τ is the delay; fd Δ Δ is the Doppler frequency; Δτ = τ − τ ( ρ); Δf = fd − fd ( ρ); ρ is the position vector of a surface point; τ ( ρ) and fd ( ρ) are the delay offset and Doppler shift associated to a signal contribution reflected at the surface point ρ , respectively; is the Fresnel reflection coefficient at a given polarization; D is the antenna radiation pattern; R0 is the distance from the GPS transmitter to the scattering point; R is the distance from this point to the receiver; q is the scattering vector, which is ˆ i = ˆ q⊥ + qz zˆ, q =| ˆ q |; and n i and n s are defined as q = n ˆs − n unit vectors in the incident and scattering directions. Pυ is the probability of having a given sea surface slope, G is the integration domain (the space coordinates of the sea surface),
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and the triangle and |S| functions are defined as Λ = 1 − |τ |/τc if |τ | ≤ τc , and 0 elsewhere (τc is the length of a chip of the C/A code and has a length of 1 ms/1023), and |S(f )| = | sin(πf )/(πf )|. To obtain simulated DDMs, (1) can straightforwardly be implemented by defining the scenario (the reflecting surface, the transmitter, and the receiver) in a reference system and evaluating the functions inside the integrand to compute the integral for every (τ, fd ) coordinate. However, such approach becomes extremely time and resource consuming as the observed surface being simulated increases in size, as it happens for spaceborne receivers, since the glistening zone extends over hundreds of kilometers (∼240 km of diameter for a 10-m height wind speed U10 = 4 m/s and ∼400 km for U10 = 10 m/s at nadir incidence from a low-Earth-orbit (LEO) receiver [11]). A first step to accelerate the simulation consists of expressing the DDM as a 2-D convolution [11], [12] that can efficiently be computed by means of 2-D fast Fourier transforms, i.e., |Y (τ, fd )|2 = χ2 (τ, fd ) ∗ ∗Σ(τ, fd ). (2) In the radar technique, (2) is analog to the so-called radar mapping equation [13], whereas χ is known as the ambiguity function, i.e., χ2 (τ, fd ) Λ2 (τ ) · |S(fd )|2
(3)
where Λ and |S| have already been defined in (1). This ambiguity function is independent of the pixel position and can be computed in a straightforward manner and only once for all the pixels. χ can be understood as the impulse response to the scattered signal from a single delay–Doppler cell. In the GNSSR field, it usually appears as χ2 , since it is usual to work with the average square magnitude of the DDM. On the other hand, Σ is given by D2 ( ρ)σ 0 ( ρ) Σ(τ, fd ) = Ti2 2 4πR0 ( ρ)R2 ( ρ) G
× δ (τ − τ ( ρ)) δ (fd − fd ( ρ)) d2 ρ (4) and accounts for the surface geometry, the antenna patterns, and other terms from the bistatic radar equation, assigning a “weighting factor” to each delay–Doppler cell of the scene. The advantage of (4) over (1) is that instead of evaluating Λ2 and |S|2 at every (τ, fd ) coordinate, they are just computed for all the (τ, fd ) values at a time (to obtain χ2 ) and then convolved with Σ. Nevertheless, the integral in (4) is still a simulation bottleneck that may prevent computing DDMs at all. In this paper, a new (to the authors’ knowledge) and efficient approach to overcome this limitation is described. This paper is organized as follows. Section II describes the analytical approach that results in significantly lower processing times to compute the DDMs. Section III introduces the particular case of the flat Earth approximation, whereas Section IV deals with the spherical Earth scenario, which corresponds to the spaceborne situation. Finally, Section V summarizes the main conclusions of this paper. II. A NALYTICAL A PPROACH TO THE DDM S IMULATION Instead of performing the integration of (4) over the xy surface, it is possible to apply a change of variables using the
geometrical expressions that link τ −fd to x−y, i.e., τxy = τ (x, y) fd,xy = fd (x, y).
(5)
The surface differential d2 ρ in (4) becomes d2 ρ = |J(τxy , fd,xy )| · dfd,xy · dτxy
(6)
where dfd,xy and dτxy are the differentials of the new integration variables, and |J| stands for the absolute value of the Jacobian of the change of variables defined in (5). Substituting (6) in (4) yields Σ(τ, fd ) =
Ti2 G
D2 ( ρ(τxy , fd,xy )) σ 0 ( ρ(τxy , fd,xy )) 4πR02 ( ρ(τxy , fd,xy )) R2 ( ρ(τxy , fd,xy ))
· δ(τ − τxy )δ(fd − fd,xy ) |J(τxy , fd,xy )| · dfd,xy · dτxy
(7)
where G is the integration domain that results from applying the change of variables defined in (5) to the domain G. Applying the Dirac delta properties, (7) becomes Σ(τ, fd ) = Ti2
D2 ( ρ(τ, fd )) σ 0 ( ρ(τ, fd )) |J(τ, fd )| . (8) 2 2 4πR0 ( ρ(τ, fd )) R ( ρ(τ, fd ))
Finally, substituting (8) in (2) |Y (τ, fd )|2 = χ2 (τ, fd ) 2 ρ(τ, fd )) σ 0 ( ρ(τ, fd )) 2 D ( |J(τ, fd )| . ∗ ∗ Ti 4πR02 ( ρ(τ, fd )) R2 ( ρ(τ, fd ))
(9)
A similar result can also be derived by directly applying the defined change of variables to (1). Thus, the DDM can fully be determined with a 2-D integral (convolution) over the (τ, fd ) domain, which is much smaller than the (x, y) domain (scattering surface) used in (7) to obtain Σ. Therefore, it is only necessary to evaluate (9) at the desired delay–Doppler coordinates regardless of the size of the physical surface to simulate. This allows performing simulations corresponding to spaceborne scenarios that otherwise would require very high performance computers. In this approach, the Jacobian accounts for the area associated to a given delay–Doppler cell on the xy space (Fig. 1). As stated, the geometrical relationships that link a given surface coordinate and its associated delay and Doppler values at the defined scenario are the key to compute the Jacobians. Therefore, it is necessary to derive analytical expressions for them. Another remark is that the change of variables is not univocal, since a single delay–Doppler coordinate corresponds to two points in the physical space (Fig. 1). Thus, the modules of the Jacobians associated to each of the two solutions must be properly combined, since each of them is associated to a different section of the observed surface. Then, (8) results in ρ1 (τ, fd )) · σ 0 ( ρ1 (τ, fd )) Ti2 D2 ( Σ(τ, fd ) = |J1 (τ, fd )| 2 4π R0 ( ρ1 (τ, fd )) · R2 ( ρ1 (τ, fd )) D2 ( ρ2 (τ, fd )) · σ 0 ( ρ2 (τ, fd )) + 2 |J (τ, f )| (10) 2 d R0 ( ρ2 (τ, fd )) · R2 ( ρ2 (τ, fd ))
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MARCHÁN-HERNÁNDEZ et al.: EFFICIENT ALGORITHM TO THE SIMULATION OF DELAY–DOPPLER MAPS
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Fig. 1. Delay–Doppler mapping. Each delay–Doppler bin is associated to two delay–Doppler cells that may have different areas over the mapped surface.
where ρ1 and ρ2 are each of the two space points associated to one delay–Doppler bin. The Jacobians have the following expression: ∂x ∂xi i ∂τxy ∂fdxy (11) |Ji (τxy , fdxy )| = det ∂yi i ∂τxy ∂f∂ydxy where det is the determinant, and i = 1, 2 stands for the first or second solution of the xy coordinates. Its practical computation calls for substituting the derivatives by their associated finitedifference approximations, i.e., ∂x x(τxy + Δτ /2, fd,xy ) − x(τxy − Δτ /2, fd,xy ) ≈ ∂τxy Δτ ∂x x(τxy , fd,xy + Δfd /2) − x(τxy , fd,xy − Δfd /2) ≈ ∂fd,xy Δfd (12) where Δτ and Δfd are the delay and Doppler resolutions, respectively, of the DDM being computed. The derivatives of y with respect to τ and fd are obtained in an analog way. As explained in Section III, the computation of these Jacobians is critical around the borders of the DDM domain, and the central difference approximation is preferred. III. F LAT E ARTH S CENARIO For low- to middle-altitude airborne scenarios, the Earth’s surface can be considered flat around the specular point without incurring in a significant error. This approximation simplifies the expressions of the change of variables between the surface and the delay–Doppler domains, resulting in faster processing times. To derive the analytical expressions, it is first necessary to define the scenario geometry. A Cartesian coordinate system is centered at the specular point, with the transmitter–specular reflection point–receiver plane equal to the yz plane. The receiver height over the surface h and the transmitter elevation γ univocally define the scenario (Fig. 2). The positions of the receiver, the transmitter, and an arbitrary surface point are h0 Rt = 0, , h0 tan γ r = 0, − h , h R tan γ r = (x, y, 0) (13) where h0 and h are the transmitter and receiver heights over the flat surface, respectively. The absolute delay associated to a
Fig. 2. Flat and spherical Earth scenario geometry. The scenario is determined with the elevation γ and the receiver (Rx ) height h over the flat surface. Re and Rgps are the Earth and GPS orbit radii, respectively; h0 is the transmitter (Tx ) height over the flat surface; and n i and n s are unit vectors in the incident and scattering directions, respectively.
surface point is t | + |R r − r|. τxy,abs = |r − R Then, substituting (13) into (14) yields 2 h0 τxy,abs = x2 + y − + h20 tan γ + x2 + y +
h tan γ
(14)
2 + h2 .
(15)
The first square root can further be simplified considering that h0 x, y (very far away transmitter assumption) and keeping the terms up to first order only, i.e., 2 h h0 − y cos γ. + h2 + τxy,abs = x2 + y + tan γ sin γ (16) It is usual to express the code delay with respect to the specular point delay (surface point with x = y = 0); then τxy = τxy,abs (x, y, h, h0 , γ) − τxy,abs (0, 0, h, h0 , γ) 2 h h − y cos γ. (17) +h2 − = x2 + y + tan γ sin γ The Doppler shift of each point is derived in an analog way (very far away transmitter approximation), i.e., t · r · fd,xy = V ni − V ns = − Vty · cos γ − Vtz · sin γ
Vrx · x + Vry · y + tanh γ − Vrz · h + 2
x2 + y + tanh γ + h2
(18)
t and V t are the transmitter and receiver velociwhere V ties, respectively. To compute the Jacobians, it is necessary
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to express the xy coordinates as a function of τxy , fd,xy . After some lengthy but straightforward manipulations, the expressions f1 , f2 , g1 , and g2 for the change of variables between the delay–Doppler and the space domains are obtained (Appendix), i.e., t , V r ) x1 = f1 (τxy , fd,xy , h, γ, V t , V r ) y = g1 (τxy , fd,xy , h, γ, V 1 r ) x2 = f2 (τxy , fd,xy , h, γ, Vt , V (19) t , V r ) y2 = g2 (τxy , fd,xy , h, γ, V which allows computing the Jacobians associated to each solution using (11). The DDM simulation using the Jacobian approach starts by setting a Cartesian mesh over an area of the delay–Doppler space with Δτ and Δfd resolution steps. Then, the Jacobian value at each (τxy −fd,xy ) coordinate for each solution (x1 , y1 ) and (x2 , y2 ) (19) is obtained, along with the associated D, σ 0 , and propagation losses values. Eventually, applying (9), the DDM is obtained. The use of a Cartesian grid to map the delay–Doppler space implies that “forbidden” delay–Doppler coordinates (delay–Doppler pairs that do not match to any existing delay–Doppler bin over the surface) may also be evaluated. When that happens, the obtained complex xy coordinates must be discarded. The border between “forbidden” and “allowed” delay–Doppler coordinates follows the line of DDM maxima. The criterion of discarding all the complex points of the Jacobian (“allowed” delay–Doppler coordinates should produce pure real Jacobian values) fails around the mentioned border. Then, it is necessary to define a mask to set to zero the incorrect values: First the pure real points are selected, and then, the adjacent pixels to the selected ones are also considered (Fig. 3). These pixels provide the highest contribution to the DDM and cannot be discarded. To test this approach, a MATLAB script is written to generate DDMs by both the new Jacobian approach and by the classical approach reviewed in Section I (Fig. 4). Simulation parameters are γ = 60◦ , h = 680 km, Δτ = 0.1 chip, Δfd = 20 Hz, and the 10-m height wind speed U10 = 5 m/s. This last parameter is used to derive the directional mean square slope (DMSS) values that parameterize the sea surface slope’s probability density function. The DMSS values have been computed using [15], scaled by 0.45 as in [16]. A desktop computer with a 3.4-GHz Pentium-IV processor and 2-GB random access memory is used. The difference is computed after normalizing both DDMs by their peak value. As shown in Fig. 5, in the worst case, this difference is smaller than 1% the peak value. The computing times are ∼25 000 s for the classical approach and ∼5 s for the Jacobian one when meshing the xy space with a Cartesian 1500 × 1500 point mesh. The mesh density of the scattering surface grid has progressively been increased until convergence of the computed DDM using the classical approach. This DDM is used as a reference to assess the accuracy of the proposed method. The flat Earth approximation simplifies the Jacobian expressions but implies a mapping error that increases with the receiver height. For a satellite-borne receiver, the actual isolines significantly differ from the ones obtained using the flat Earth approximations (Fig. 6). This difference is clearly seen in the obtained DDM for the actual spherical Earth and the flat
Fig. 3. Mask to select the valid Jacobian points when working on the delay–Doppler domain. Pure real points appear in gray (associated to real xy coordinates), whereas the transition points (nonzero imaginary part, but still valid) appear in black.
approximation (Fig. 7). Therefore, a spherical Earth model would greatly improve the DDM modeling for spaceborne scenarios. IV. S PHERICAL E ARTH S CENARIO The reference system defined in Section III remains the same. However, the z values of the surface are no longer considered zero but are assigned a height according to
z = Re2 − x2 − y 2 − Re (20) where Re is the Earth’s radius. Thus, both the plane and sphere surfaces are tangential at the specular point. Within this new reference frame, the new expressions of delay and Doppler are easily derived considering a nonzero z, i.e., 2 h 2 + (h − z)2 τxy = x + y + tan γ h − y cos γ − z sin γ (21) − sin γ t · r · ni − V n s = −Vty · cos γ − Vtz · sin γ fd,xy = V
Vrx · x + Vry · y + tanh γ + Vrz · (z − h) + . (22) 2
h 2 2 x + y + tan γ + (h − z) Without loss of generalization, for the sake of simplicity, in the derivation of (22), the incident vector has been assumed to be constant and equal to n i = (0, − cos γ, − sin γ) for all surface points, as in (18). This assumption may seem less accurate, since now z = 0, but as it will be seen later on, the difference with the DDM obtained using (1) is acceptably small. The next step is the derivation of a similar expression to (19) from (20)–(22). However, no closed-form solutions with the following form have been found: t , V r ) xi = fi (τxy , fd,xy , h, γ, V t , V r ) yi = gi (τxy , fd,xy , h, γ, V t , V r ) zi = ki (τxy , fd,xy , h, γ, V
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(23)
MARCHÁN-HERNÁNDEZ et al.: EFFICIENT ALGORITHM TO THE SIMULATION OF DELAY–DOPPLER MAPS
Fig. 4.
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Obtained DDM with the (a) xy integration and (b) Jacobian approaches for the flat Earth.
Fig. 5. DDM difference between the xy and the Jacobian approaches after peak normalization (flat Earth).
Fig. 6. Isorange difference between the spherical Earth and the flat Earth approximations.
where i = 1, 2 denotes the first or second solution, and fi , gi , and ki denote the change of variables between the delay–Doppler and the space domains. A possible bypass to this limitation calls for considering z as another parameter (such as h or γ), so that it is possible to find explicit (although very long) expressions for t , V r , zi ) xi = fi (τxy , fd,xy , h, γ, V t , V r , zi ). yi = gi (τxy , fd,xy , h, γ, V
(24)
The problem is that the xyz values of the surface points are not a priori known, starting from a delay–Doppler mesh. The use of the following iterative procedure overcomes the problem. 1) A first-guess value of z1 = z2 = 0 (flat Earth) is chosen. 2) An approximation to the actual xy values is obtained using (24). 3) Equation (20) is then applied to retrieve the corresponding new values of z1 and z2 . 4) Steps 2 and 3 are repeated until convergence to obtain an accurate estimate of xy as a function of τxy , fd,xy . 5) These estimates are then used to compute the finitedifference approximation to the Jacobians [(11) and (12)], and to compute the DDM as for the flat Earth case.
Fig. 7. Difference of the DDM obtained with the Jacobian approach for flat Earth and the actual DDM (xy integration for the spherical Earth scenario). DDMs are normalized to 1 at the peak value.
An important issue is to determine the number of iterations needed so that the retrieved xyz approximations have a minimum error. To do so, every iteration each of the two retrieved values of z is compared to the actual z of the simulated scenario, and their mean square error (mse) is obtained. For h = 680 km
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Fig. 8. (a) MSE of the retrieved z at each iteration. (b) Difference between the actual z and the retrieved z1 over the simulated xy surface.
Fig. 9. (a) Obtained DDM with the Jacobian approach for spherical Earth and difference with the actual DDM (xy integration for the spherical Earth scenario).
and γ = 60◦ (LEO satellite), after seven iterations, the error is below 1 m (Fig. 8). The DDMs obtained by applying the classical and the Jacobian approaches for a spherical Earth surface are compared in Fig. 9. The difference is smaller than 7% (peak error) for the worst case and usually well below 4%. These values are significantly smaller than the measurement variances of a real system [14] and can be used to perform systematic simulations in different configurations and sea states. The result is significantly better than that obtained comparing the flat Earth Jacobian method with the precise but extremely resource-consuming method of integrating over the xy domain (2) using a spherical Earth surface (Fig. 7), where the error even exceeded 80%. Using the same computer described in Section III, the computing time of this iterative Jacobian approach is 600 s (seven iterations), which is significantly lower than the 25 000 s needed for the 1500 × 1500 points DDM. V. C ONCLUSION The simulation of DDMs in spaceborne scenarios is required to study the capabilities of the GNSS-R techniques to retrieve geophysical parameters. Those simulations increase the resource consumption (time and memory) as the surface gets larger, until becoming not affordable at all. The use of explicit expressions for the delay and Doppler values allows fast
computation of the DDMs using the Jacobians of this change of variables between space and delay–Doppler domains. This significantly reduces the number of points to be evaluated: from all the surface points (hundreds of thousands or millions) to just the number of bins of the desired DDM. Despite the approximations made on the isoline expressions, the obtained DDM differs by less than 1% with respect to the reference one (obtained without any approximation) for the flat Earth scenario. For the spaceborne case, the flat Earth approximation no longer holds, and an iterative approach to compute the Jacobians is introduced. In this case, the retrieved DDM differs by less than 7% (peak error) with respect to the reference one, which is well below the noise error in a real system [14]. This significant speed improvement allows one to undertake simulations of satellite reflections that were too slow or not feasible at all. The overall Jacobian approach described in Section II is understood as a general reference frame. Closed-form expressions for other scenarios apart of those described in this paper can be derived and used in a similar way, for instance, a low-height static receiver. A PPENDIX The expressions for the change of variables between the delay–Doppler and the flat Earth xy spaces are derived by
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MARCHÁN-HERNÁNDEZ et al.: EFFICIENT ALGORITHM TO THE SIMULATION OF DELAY–DOPPLER MAPS
considering the system of equations composed by (16) and (17) and solving it for x and y. There are two solutions for each τ −fd pair, i.e., x1,2
y1,2
√ sin γ · Vrx (τ + h · sin γ) α · cos γ + = ∓ Vrx · β β √ α · Vry × (fd · λ + Vty cos γ + Vtz · sin γ) ± Vrx · β Vrx · Vrz · h · sin3 γ − Vry · cos γ · (τ + h · sin γ) + β (25) 1 √ 2 =− ± α + 2 · Vty · fd · λ · h · cos γ + 2 · Vty β · sin γ · fd · λ · τ · cos2 γ + Vty · sin γ · Vrz · h · cos2 γ + 2 · Vty · sin γ · Vtz · h · cos2 γ − Vty · Vry · h · cos γ + 2 · Vty · Vtz · τ · cos2 γ − 2 · Vty · Vtz · τ · cos4 γ − Vty · Vry · h · cos3 γ − Vty · cos γ · τ · sin γ · Vry + Vty2 · sin γ · cos3 γ · τ + Vty2 · cos3 γ · h + 2 · Vtz · cos γ · fd · λ · τ − 2 · Vtz · cos3 γ · fd · λ · τ − Vtz · sin γ · h · Vry · cos2 γ − Vtz · τ · Vry − Vtz · sin γ · h · Vry − Vtz2 · sin γ · cos3 γ · τ + Vtz2 · sin γ · cos γ · τ + Vtz · τ · Vry · cos2 γ + Vtz · cos γ · Vrz · h − Vtz · cos3 γ · Vrz · h + 2 · Vtz · sin γ · cos γ · fd · λ · h + Vtz2 · cos γ · h − Vtz2 · cos3 γ · h − fd · λ · h · Vry · cos2 γ − fd · λ · h · Vry + fd2 · λ2 · cos γ · h − fd · λ · τ · sin γ · Vry + fd2 · λ2 · cos γ · sin γ 2 · τ + fd · λ · cos γ · sin γ · Vrz · h + Vry · h · cos γ − Vry · sin γ · Vrz · h − cos γ 2 (26) · sin γ · τ · Vrx
where α and β are defined as 2 α = Vrx · (−1 + cos2 γ)
· 2 · fd · λ · h2 · Vrz · sin γ − 2 · fd · λ · h2 · Vry 2 · cos γ + fd2 · λ2 · h2 − Vrx · τ 2 + fd2 · λ2 · τ 2 + 2 · fd · λ · τ · Vrz · h + 2 · fd2 · λ2 · h · τ · sin γ + 2 · Vty · cos γ · h2 · Vtz · sin γ + 2 · Vty · cos γ · h2 · fd · λ + 2 · Vty · cos γ · h2 · Vrz · sin γ + 2 · Vty · Vrz · h · cos γ · τ − 2 · Vty · h2 · Vry · cos2 γ − 2 · Vtz · sin γ · h2 · Vry · cos γ + 2 · Vtz · sin γ · h2 · fd · λ + 2 · Vtz · h2 · Vrz − 2 · Vtz · h2 · Vrz · cos2 γ + 2 · Vtz2 · sin γ · h · τ + 2 · cos γ · τ 2 · Vty · fd · λ + 2 · cos2 γ · sin γ · τ · Vty2 · h + 2
· Vty · sin γ · Vtz · τ 2 · cos γ + 4 · Vty · h · cos γ · sin γ · τ · (Vtz · sin γ + fd · λ) + Vty2 · cos2 γ · (h2 + τ 2 ) + 4 · Vtz · fd · λ · h · τ − Vtz2 · τ 2 · cos2 γ − Vtz2 · h2 · cos2 γ + 2 · Vtz · sin γ · fd · λ · τ 2 + 2 · sin γ · Vtz · τ · Vrz · h + Vtz2 · τ 2 + Vtz2 · h2 − 4 · Vtz · f d · λ · h · τ · cos2 γ − 2 · Vtz2 · sin γ
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2 2 · sin γ − 2 · Vry · h · τ · cos2 γ − 2 · h · τ · Vrx
· h · τ · sin γ − 2 · Vry · h2 · cos γ · Vrz · sin γ 2 + Vry · h2 · cos2 γ − 2 · Vry · Vrz · h · cos γ · τ 2 2 · τ 2 Vrz · h2 · sin2 γ (27) − Vry β = sin γ
2 2 · Vrx · sin2 γ + Vry + fd · λ · cos γ × (−2 · Vry + fd · λ · cos γ) + 2 · Vtz · sin γ · cos γ × (−Vry + fd · λ cos γ) + Vtz2 · cos2 γ sin2 γ + Vty · cos2 γ × (2 · cos γ · Vtz · sin γ + Vty · cos2 γ (28) + 2 · cos γ · fd · λ − 2 · Vry ) .
R EFERENCES [1] M. Martín-Neira, “A passive reflectometry and interferometry system (PARIS): Application to ocean altimetry,” ESA J., vol. 17, pp. 331–355, 1993. [2] S. Lowe, C. Zuffada, Y. Chao, P. Kroger, L. Young, and J. LaBrecque, “Five-cm precision aircraft ocean altimetry using GPS reflections,” Geophys. Res. Lett., vol. 29, no. 10, pp. 4359–4362, May 2002. [3] V. U. Zavorotny and A. G. Voronovich, “Bistatic GPS signal reflections at various polarizations from rough land surface with moisture content,” in Proc. Int. Geosci. Remote Sens. Symp., Honolulu, HI, Jul. 24–28, 2000, pp. 2852–2854. [4] A. Komjathy, J. Maslanik, V. U. Zavorotny, P. Axelrad, and S. J. Katzberg, “Sea ice remote sensing using surface reflected GPS signals,” in Proc. Int. Geosci. Remote Sens. Symp., Honolulu, HI, Jul. 24–28, 2000, pp. 2855–2857. [5] A. Rius, J. M. Aparicio, E. Cardellach, M. Martín-Neira, and B. Chapron, “Sea surface state measured using GPS reflected signals,” Geophys. Res. Lett., vol. 29, no. 23, p. 2122, Dec. 2002. [6] F. Soulat, M. Caparrini, O. Germain, P. Lopez-Dekker, M. Taani, and G. Ruffini, “Sea state monitoring using coastal GNSS-R,” Geophys. Res. Lett., vol. 31, no. 21, p. L21 303, Nov. 2004. [7] J. B. Y. Tsui, Fundamentals of Global Positioning System Receivers. A Software Approach. New York: Wiley Interscience, 2000. [8] M. Skolnik, Radar Handbook, 2nd ed. New York: McGraw-Hill, 1990. [9] V. U. Zavorotny and A. G. Voronovich, “Scattering of GPS signals from the ocean with wind remote sensing application,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 2, pp. 951–964, Mar. 2000. [10] H. You, J. L. Garrison, G. Heckler, and V. U. Zavorotny, “Stochastic voltage model and experimental measurement of ocean-scattered GPS signal statistics,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 10, pp. 2160–2169, Oct. 2004. [11] T. Elfouhaily, D. R. Thompson, and L. Linstrom, “Delay–Doppler analysis of bistatically reflected signals from the ocean surface: Theory and application,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 3, pp. 560– 573, Mar. 2002. [12] J. F. Marchan-Hernandez, N. Rodríguez-Álvarez, A. Camps, X. Bosch-Lluis, I. Ramos-Perez, and E. Valencia, “Correction of the sea state impact in the L-band brightness temperature by means of delay–Doppler maps of global navigation satellite signals reflected over the sea surface,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 10, pp. 2914–2923, Oct. 2008. [13] A. W. Rihaczek, Principles of High-Resolution Radar. New York: McGraw-Hill, 1969. [14] S. Gleason, S. Hodgart, Y. Sun, C. Gommenginger, S. Mackin, M. Adjrad, and M. Unwin, “Detection and processing of bistatically reflected GPS signals from low earth orbit for the purpose of ocean remote sensing,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 6, pp. 1229–1241, Jun. 2005. [15] C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sea glitter,” J. Opt. Soc. Amer., vol. 44, no. 11, pp. 838–850, Nov. 1954. [16] S. J. Katzberg, O. Torres, and G. Ganoe, “Calibration of reflected GPS for tropical storm wind speed retrievals,” Geophys. Res. Lett., vol. 33, no. 18, p. L18 602, Sep. 2006.
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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 8, AUGUST 2009
Juan Fernando Marchán-Hernández (S’04) was born in Barcelona, Spain. He received the M.S. degree in telecommunications engineering in 2004 from the Universitat Politècnica de Catalunya, Barcelona, Spain, where he is currently working toward the Ph.D. degree with the Passive Remote Sensing Group, Department of Signal Theory and Communications. In 2003, he was with the Laboratory of Space Technology, Helsinki University of Technology (TKK), Espoo, Finland. His current research activities are related to radiometry and signal reflections of GNSS-R.
Adriano Camps (S’91–A’97–M’00–SM’03) was born in Barcelona, Spain, in 1969. He received the M.S. and Ph.D. degrees in telecommunications engineering from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1992 and 1996, respectively. In 1991–1992, he was with the Ecole Nationale Superieure des Télécommunications de Bretagne, Brest, France, with an Erasmus Fellowship. In 1993, he was with the Electromagnetics and Photonics Engineering Group, Department of Signal Theory and Communications, UPC, as an Assistant Professor, and since 1997 as an Associate Professor. In 1999, he was on sabbatical leave at the Microwave Remote Sensing Laboratory, University of Massachusetts, Amherst. He is an Associate Editor of Radio Science. His research interests are focused on microwave remote sensing, with special emphasis on microwave radiometry by aperture synthesis techniques. Dr. Camps was the Chair of μCal 2001 and the Technical Program Committee Chair of the International Geoscience and Remote Sensing Symposium (IGARSS) 2007. From 2003 to 2006, he was the Editor of the IEEE GEOSCIENCE AND REMOTE SENSING NEWSLETTER and the PresidentFounder of the IEEE Geoscience and Remote Sensing Society Chapter in Spain. He has performed numerous studies within the frame of the European Space Agency SMOS Earth Explorer Mission that have received several awards. In 1993, he received the second national award of university studies; in 1997, the INDRA award of the Spanish Association of Telecommunication Engineering to the best Ph.D. in Remote Sensing; in 1999, the extraordinary Ph.D. award at UPC; in 2002, the Research Distinction of the Generalitat de Catalunya for contributions to microwave passive remote sensing; and in 2004, the European Young Investigator Award. In addition, as a member of the Microwave Radiometry Group at UPC, in 2000, 2001, and 2004, he received the following awards: the First Duran Farell and the Ciudad de Barcelona awards for Technology Transfer, and the “Salvà i Campillo” Award of the Professional Association of Telecommunication Engineers of Catalonia for the most innovative research project.
Nereida Rodríguez-Álvarez (S’07) was born in Barcelona, Spain. She received the M.S. degree in telecommunications engineering in 2007 from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, where she is currently working toward the Ph.D. degree in GNSS-reflectometry. She is with the Passive Remote Sensing Group, Department of Signal Theory and Communications, UPC.
Enric Valencia (S’07) was born in Sabadell, Spain. He received the B.S. and M.S. degrees in electronic engineering in 2007 from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, where he is currently working toward the Ph.D. degree in GNSSreflectometry. He is with the Passive Remote Sensing Group, Department of Signal Theory and Communications, UPC.
Xavier Bosch-Lluis (S’04) was born in Lleida, Spain. He received the M.S. degree in telecommunication engineering from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 2005. He is currently with the Passive Remote Sensing Group, Department of Signal Theory and Communications, UPC. His research activities are related to L-band radiometry, digital beamforming, calibration techniques, and reflectometry of GNSS-R.
Isaac Ramos-Pérez (S’04–A’06) was born in Barcelona, Spain. He received the M.S. degree in telecommunications engineering in 2005 from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, where he is currently working toward the Ph.D. degree in microwave radiometry, developing a synthetic aperture passive advanced unit, which is a new instrument to develop and test potential improvements for the future SMOS operational system. He was with Passive Remote Sensing Group, Department of Signal Theory and Communications, UPC.
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