Analytical Computation of the Spatial Resolution in GNSS-R and Experimental Validation at L1 and L5

Article Analytical Computation of the Spatial Resolution in GNSS-R and Experimental Validation at L1 and L5

Adriano Camps 1,2,* and Joan Francesc Munoz-Martin 1,2

1 CommSensLab-UPC, Department of Signal Theory and Communications, UPC BarcelonaTech, c/Jordi Girona 1-3, 08034 Barcelona, Spain; [email protected] 2 Institut d’Estudis Espacials de Catalunya-IEEC/CTE-UPC, Gran Capità, 2-4, Ediﬁci Nexus, despatx 201, 08034 Barcelona, Spain * Correspondence: [email protected]

Received: 3 November 2020; Accepted: 24 November 2020; Published: 28 November 2020

Abstract: Global navigation satellite systems reflectometry (GNSS-R) is a relatively novel remote sensing technique, but it can be understood as a multi-static radar using satellite navigation signals as signals of opportunity. The scattered signals over sea ice, flooded areas, and even under dense vegetation show a detectable coherent component that can be separated from the incoherent component and processed accordingly. This work derives an analytical formulation of the response of a GNSS-R instrument to a step function in the reflectivity using well-known principles of electromagnetic theory. The evaluation of the spatial resolution then requires a numerical evaluation of the proposed equations, as the width of the transition depends on the reflectivity values of two regions. However, it is found that results are fairly constant over a wide range of reflectivities, and they only vary faster for very high or very low reflectivity gradients. The predicted step response is then satisfactorily compared to airborne experimental results at L1 (1575.42 MHz) and L5 (1176.45 MHz) bands, acquired over a water reservoir south of Melbourne, in terms of width and ringing, and several examples are provided when the transition occurs from land to a rough ocean surface, where the coherent scattering component is no longer dominant.

Keywords: GNSS-R; spatial resolution; diﬀraction; experiment; airborne; L1; L5

1. Introduction Global navigation satellite systems reflectometry (GNSS-R) is a relatively new remote sensing technique that uses navigation signals as signals of opportunity in a multi-static radar configuration [1]. The spatial resolution of GNSS-R instruments was first analyzed in [2], and it was estimated from the dimensions of the area determined by the intersection of the first iso-delay and iso-Doppler lines. In this case, for a Low Earth Orbit (LEO) satellite at 700 km, the predicted spatial resolution for an instrument using the GPS L1 C/A (Coarse/Acquisition) code would be in the range of 36 to 54 km, for elevation angles from 90◦ to 40◦. Later on, the definition of spatial resolution was revised and “an effective spatial resolution derived as a function of measurement geometry and delay–Doppler (DD) interval, and as a more appropriate representation of resolution than the geometric resolution previously used in the literature” was proposed in [3]. This effective spatial resolution was defined following an approach similar to the definition of effective bandwidth used in filter theory. An increase in the effective spatial resolution is observed for increasing incidence angle, receiver altitude, and delay–Doppler window. With this definition, the effective spatial resolution is estimated to range from 25 to 37 km for a 700 km height receiver (Figure 4, [3]).

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The above definitions implicitly assume that the scattering is incoherent, as the radar equation used is the one that corresponds to surface scattering. However, the development of advanced GNSS-R receivers,Remote Sens. capable 2020, 12 of, x FOR processing PEER REVIEW the coherent component and separating it from the incoherently2 of 15 scattered component, showed that much finer features can be distinguished [4,5]. In fact, when coherent GNSS-R receivers, capable of processing the coherent component and separating it from the scattering occurs, the GNSS reflectivity values have a very high spatial resolution, as most of the power incoherently scattered component, showed that much finer features can be distinguished [4,5]. In fact, is mainly coming from the first Fresnel zone [6], an ellipse of semi-major axis a (Equation (1a)) and when coherent scattering occurs, the GNSS reflectivity values have a very high spatial resolution, as semi-minor axis b (Equation (1b)): most of the power is mainly coming from ther first Fresnel zone [6], an ellipse of semi-major axis a RT RR (Equation 1a) and semi-minor axis b (Equationa = 1b):λ · (1a) RT + RR 𝑅a ·𝑅 𝑎=b =𝜆 (1a) 𝑅 +𝑅, (1b) cos(θi) where λ is the electromagnetic wavelength, R𝑏=T and RR, are the distances from the transmitter(1b) and receiver to the specular reflection point, and θi is the incidence angle. This means that even from an where λ is the electromagnetic wavelength, RT and RR are the distances from the transmitter and LEO satellite, the spatial resolution can be as small as ~400–500 m. receiver to the specular reflection point, and θi is the incidence angle. This means that even from an In [7] the spatial resolution under coherent scattering was theoretically analyzed in detail, including LEO satellite, the spatial resolution can be as small as ~400–500 m. the effectIn of [7] the the di ffspatialerent Fresnelresolution zones. under It coherent was shown scattering that up was to ~45theoretically◦ most of analyzed the power in isdetail, coming fromincluding an area determined the effect of bythe thedifferent first Fresnel Fresnel zone,zones. although It was shown a power that up equivalent to ~45° most to the of the one power received is in free-spacecoming conditions from an area is actually determined coming by the from first a regionFresnel ~0.58–0.62 zone, although times a thepower first equivalent Fresnel zone. to the However, one significantreceived contributions in free-space fromconditions regions is actually farther coming away than from the a region first Fresnel~0.58–0.62 zone times also the contribute first Fresnel to the receivedzone. power,However, making significant its interpretation contributions from more regions difficult. farther This away region than wasthe first ~800 Fresnel m away zone atalso a 60◦ elevationcontribute angle to for the an received LEO satellite power, atmaking 500 km its height. interpretation more difficult. This region was ~800 m awayIn this at work,a 60° elevation we have angle gone for one an step LEO further satellite and at 500 analyzed km height. the spatial resolution from a theoretical point of view,In this using work, known we have principles gone one of electromagnetism. step further and analyzed The use ofthe these spatial results resolution makes from it possible a to computetheoretical in apoint very of compact view, using form known the expression principles of of electromagnetism. the ringings and The oscillations use of these observed results makes when the it possible to compute in a very compact form the expression of the ringings and oscillations observed specular reflection point transitions from the land into the ocean or vice versa. Analytical results are when the specular reflection point transitions from the land into the ocean or vice versa. Analytical validated with airborne experimental data gathered at L1 and L5 with the Microwave Interferometric results are validated with airborne experimental data gathered at L1 and L5 with the Microwave ReflectometerInterferometric (MIR) Reflectometer instrument (MIR) when instru crossingment the when coastline crossing [8 ].the coastline [8]. 2. Methodology 2. Methodology TheThe scattering scattering geometry geometry is shownis shown in in Figure Figure1, where1, where the the receiver receiver (R) (R) and and transmitter transmitter (T) (T) lielie inin the YZ plane,the YZ and plane, are and at heights are at heights hR and hR h andT over hT over a flat a flat surface. surface. RT RandT and R RRare are the the distancesdistances to the the origin origin of coordinatesof coordinates where where the specular the specular reflection reflection point is,point and is,θ iandis the θi is incidence the incidence angle, angle, which which for the for specular the reflectionspecular point reflection (origin point of coordinates) (origin of coordinates) is equal to is the equal reflected to the anglereflectedθr .angle The diθr.ff Theerent different disks represent disks the Fresnelrepresent zones, the Fresnel with thezones, odd with ones the (1, odd 3, 5,ones etc.) (1, colored3, 5, etc.) dark colored gray, dark and gray, the and even the ones even (2, ones 4, 6,(2, etc.) colored4, 6, white.etc.) colored white.

FigureFigure 1. Global 1. Global navigation navigation satellite satellite systems systems reflectometryreflectometry (GNSS-R) (GNSS-R) scattering scattering geometry geometry showing showing the the transmitter,transmitter, receiver, receiver, specular specular reflection reflection point point (origin(origin of of coordinates), coordinates), and and the the different different Fresnel Fresnel zones. zones.

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InIn what follows,follows, a a few few principles principles of electromagnetismof electromagnetism are reviewed. are reviewed. They willThey be will used be to computeused to computea closed-form a closed-form solution solution of the GNSS-R of the GNSS-R response response to a step to function, a step function, such as insuch the as transition in the transition between betweenwater and water the ocean.and the ocean.

2.1. Review of Some Basic Principles of Electromagnetism 2.1. Review of Some Basic Principles of Electromagnetism 2.1.1. Principle of Equivalence and Image Theory 2.1.1. Principle of Equivalence and Image Theory The principle of equivalence [9] allows us to compute the solutions in the region of interest The principle of equivalence [9] allows us to compute the solutions in the region of interest (half- (half-space in front of the conducting plane) by replacing the plane conductor with the images of the space in front of the conducting plane) by replacing the plane conductor with the images of the dipoles. The image sources must satisfy the boundary condition that the tangential electric field at the dipoles. The image sources must satisfy the boundary condition that the tangential electric field at conducting surface should be zero. The total electric field above the conducting plane (Figure2) is then the conducting surface should be zero. The total electric field above the conducting plane (Figure 2) computed as the sum of the one created by the original currents and the image ones, while the total is then computed as the sum of the one created by the original currents and the image ones, while the electric field below the conducting plane is zero. Applying this principle to our problem, the transmitter total electric field below the conducting plane is zero. Applying this principle to our problem, the can be replaced by its “image” underneath the plane, as illustrated in Figure3. transmitter can be replaced by its “image” underneath the plane, as illustrated in Figure 3.

Figure 2. Left: linear and loop currents over a perfect conductor. Right: the conductor has been Figure 2. Left: linear and loop currents over a perfect conductor. Right: the conductor has been replaced by the image currents. Note that the sign reversal of the currents parallel to the conducting replaced by the image currents. Note that the sign reversal of the currents parallel to the conducting plane are responsible for the polarization change of the “reflected” wave. plane are responsible for the polarization change of the “reﬂected” wave.

FigureFigure 3. 3. GNSS-RGNSS-R scattering scattering geometry geometry in in Figure Figure 11 modifiedmodified according according to to the the principle principle of of equivalence: equivalence: thethe transmitter transmitter is is replaced replaced by by its its image. image. Computed Computed electric electric fields fields ar aree only only valid valid in in the the region region z z >> 0.0. 2.1.2. The Huygens Principle and Knife-Edge Diffraction 2.1.2. The Huygens Principle and Knife-Edge Diffraction The Huygens principle states that every point (red dots in Figure4) on a primary wave front The Huygens principle states that every point (red dots in Figure 4) on a primary wave front (blue curve in Figure4) may be regarded as a new source of spherical secondary waves, and that at (blue curve in Figure 4) may be regarded as a new source of spherical secondary waves, and that at each point in space these waves propagate in every direction at a speed and frequency equal to that of each point in space these waves propagate in every direction at a speed and frequency equal to that of the primary wave, and in such a way that the primary wave front at some later time is the envelope of these secondary waves (red curve in Figure 4).

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The Huygens principle can be applied to compute the electric fields when part of the trajectory is partially obstructed by a conducting plane as illustrated in Figure 5, leaving a clearance h with respectRemote Sens. to the2020 line, 12, 3910of sight. This academic case is known as the “knife-edge diffraction”, and it is 4used of 15 in radio communications to assess the propagation losses with respect to the free space caused by the Remoteobstruction Sens. 2020 by, 12 a, xmountain, FOR PEER REVIEW building, etc. In the latter this will represent the transition from4 ofone 15 the primary wave, and in such a way that the primary wave front at some later time is the envelope of medium to another [6]. theseThe secondary Huygens waves principle (red can curve be inapplied Figure to4). compute the electric fields when part of the trajectory is partially obstructed by a conducting plane as illustrated in Figure 5, leaving a clearance h with respect to the line of sight. This academic case is known as the “knife-edge diffraction”, and it is used in radio communications to assess the propagation losses with respect to the free space caused by the obstruction by a mountain, building, etc. In the latter this will represent the transition from one medium to another [6].

Figure 4. Graphical representation of the Huygens diffraction principle. Figure 4. Graphical representation of the Huygens diffraction principle. The Huygens principle can be applied to compute the electric fields when part of the trajectory is partially obstructed by a conducting plane as illustrated in Figure5, leaving a clearance h with respect to the line of sight. This academic case is known as the “knife-edge diffraction”, and it is used in radio communications to assess the propagation losses with respect to the free space caused by the obstruction by a mountain, building, etc. In the latter this will represent the transition from one medium to anotherFigure [6]. 4. Graphical representation of the Huygens diffraction principle.

Figure 5. Huygens principle applied to the “knife-edge diffraction” case. The light-gray area partially blocks the propagation of the signal, hiding parts of some Fresnel zones. h is the clearance of the obstruction.

The development of an analytical solution can be found in many text books, and only the final

result is provided below: FigureFigure 5. 5. HuygensHuygens principle principle applied applied to to the the “knife-edge “knife-edge di diffraction”ﬀraction” case. case. The The light-gray light-gray area area partially partially blocksblocks the the propagation propagation of of the the signal, signal, 𝐸hiding hiding=𝐸 parts parts of ofsome some·𝐹 Fresnel𝑣 Fresnel zones. zones. h is htheis clearance the clearance of the of (2) the obstruction. where:obstruction. The development of an analytical solution can be found in many text books, and only the ﬁnal The development of an analytical solution can be found in many text books, and only the final result is provided𝐹𝑣 = below: 𝑐𝑜𝑠 𝑡 𝑑𝑡 − 𝑗 𝑠𝑖𝑛 𝑡 𝑑𝑡 = −𝐶𝑣−𝑗 −𝑆𝑣, (3) result is provided below: E = E F(v) (2) 𝐶𝑣 and 𝑆𝑣 are the so-called Fresneldi f fintegrals,f ree space which· are given be: 𝐸 =𝐸 ·𝐹𝑣 (2) where: where: "Z Z # 1 + j ∞ π ∞ π 1 + j 1 1 F(v)=ˆ cos t2 dt j sin t2 dt = C(v) j S(v) , (3) 𝐹𝑣 = 2 v 𝑐𝑜𝑠 2𝑡 𝑑𝑡− − 𝑗 v 𝑠𝑖𝑛 2𝑡 𝑑𝑡 = 2 −𝐶2 −𝑣−−𝑗 2−𝑆−𝑣, (3) 𝐶𝑣 and 𝑆𝑣 are the so-called Fresnel integrals, which are given be:

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C(v) and S(v) are the so-called Fresnel integrals, which are given be: Remote Sens. 2020, 12, x FOR PEER REVIEW 5 of 15 Z ν π C(v) = cos t2 dt, (4a) 0 2 𝐶𝑣 = 𝑐𝑜𝑠 𝑡 𝑑𝑡, (4a) Z ν π 2 S𝑆(v𝑣)== 𝑠𝑖𝑛sin 𝑡tdt𝑑𝑡,, (4b) (4b) 0 2 andand v𝑣is is the the Fresnel–Kirchho Fresnel–Kirchhoffff parameter: parameter: s 22·(RT𝑅++𝑅RR) v𝑣=ℎ·= h · (5(5)) · λ𝜆·𝑅RT R·𝑅R · · FigureFigure6 6 shows shows thethe absoluteabsolute valuevalue of 𝐹F(𝑣v) inin Equation Equation (3) (3) in in linear linear units. units. When When h =h 0,= half0, half of the of theelectric electric field field is blocked, is blocked, which which translates translates into into a apower power loss loss of of 6 6 dB dB with with respect to thethe free-spacefree-space conditions.conditions. Note the ringing for h << 0 due to the didifferentfferent FresnelFresnel zoneszones beingbeing blockedblocked oror notnot andand contributingcontributing toto thethe totaltotal electricelectric fieldfield withwith didifferentfferent phases. phases.

Figure 6. Absolute value of F(v). Negative v indicates that the line-of-sight (LoS) trajectory is Figure 6. Absolute value of 𝐹𝑣 . Negative 𝑣 indicates that the line-of-sight (LoS) trajectory is unobstructed, while positive v indicates an LoS obstruction (see Figure5). unobstructed, while positive 𝑣indicates an LoS obstruction (see Figure 5). 2.1.3. Babinet’s Principle 2.1.3. Babinet’s Principle Babinet’s principle [10] states that the diffraction pattern from an opaque body is identical to that fromBabinet’s a hole of theprinciple same size [10] and states shape, that exceptthe diffraction for the overall pattern forward from an beam opaque intensity. body The is identical concept isto expressedthat from graphicallya hole of the in Figuresame 7size[11 ].and shape, except for the overall forward beam intensity. The conceptBabinet’s is expressed principle graphically can be used in Figure to compute 7 [11]. the electric field diffracted when the signal emitted by the transmitter’s image is diffracted by the light-gray opaque area in the z = 0 plane (the white area being “transparent”), and when it is diffracted by the white opaque area in the z = 0 plane (the light-gray area now being “transparent”). Remote Sens. 2020, 12, x FOR PEER REVIEW 5 of 15

𝐶𝑣 = 𝑐𝑜𝑠 𝑡 𝑑𝑡, (4a) 𝑆𝑣 = 𝑠𝑖𝑛 𝑡 𝑑𝑡, (4b) and 𝑣 is the Fresnel–Kirchhoff parameter:

2·𝑅 +𝑅 𝑣=ℎ· (5) 𝜆·𝑅 ·𝑅

Figure 6 shows the absolute value of 𝐹𝑣 in Equation (3) in linear units. When h = 0, half of the electric field is blocked, which translates into a power loss of 6 dB with respect to the free-space conditions. Note the ringing for h < 0 due to the different Fresnel zones being blocked or not and contributing to the total electric field with different phases.

Figure 6. Absolute value of 𝐹𝑣 . Negative 𝑣 indicates that the line-of-sight (LoS) trajectory is unobstructed, while positive 𝑣indicates an LoS obstruction (see Figure 5).

2.1.3. Babinet’s Principle Babinet’s principle [10] states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape, except for the overall forward beam intensity. The Remote Sens. 2020, 12, 3910 6 of 15 concept is expressed graphically in Figure 7 [11].

Figure 7. Graphical representation of the concept of Babinet’s principle and the three equivalent obstacles: electric ﬁeld incident on an arbitrary obstacle (left), associated opaque screen (center), and complementary screen (i.e., hole in the inﬁnite plane; right).

2.1.4. Principle of Superposition Once the electric fields diffracted by the semi-plane (the white area in the plane z = 0 in Figure5) are computed, the electric fields diffracted by the complementary semi-plane (the light-gray area in the plane z = 0 in Figure5), when it is not opaque, can be computed using Babinet’s principle, and then the total electric field arriving at the receiver can be computed as the sum of both.

The extension of the Fresnel reflection zones is small enough (a few hundred meters from a low • Earth orbiter) so that the variations of the local incidence angle can be neglected over a few Fresnel zones; The plane z = 0 is composed of two flat half-planes of different materials, and thus exhibiting • different reflection coefficients; and The transmission coefficient for the wave emitted by the image transmitter (Figure5) is equal to • the reflection coefficient evaluated at the local incidence angle at the origin.

Then, as illustrated in Figure8, the total electric ﬁeld can be immediately computed as the sum of the following two terms:

E = E [F(v) ρ(ε , θ ) + F( v) ρ(εr , θ )], (6) di f f f ree space· · r,1 i − · ,2 i where ρ(εr,n, θi) is the Fresnel reﬂection coeﬃcient, and εr,n is the dielectric constant of the half-plane n. The received power is proportional to the square of the absolute value of Equation (6).

2 P = P F(v) ρ(εr,1, θi) + F( v) ρ(εr,2, θi) , (7) di f f f ree space· · − · Remote Sens. 2020, 12, x FOR PEER REVIEW 6 of 15

Figure 7. Graphical representation of the concept of Babinet’s principle and the three equivalent obstacles: electric field incident on an arbitrary obstacle (left), associated opaque screen (center), and complementary screen (i.e., hole in the infinite plane; right).

Babinet’s principle can be used to compute the electric field diffracted when the signal emitted by the transmitter’s image is diffracted by the light-gray opaque area in the z = 0 plane (the white area being “transparent”), and when it is diffracted by the white opaque area in the z = 0 plane (the light-gray area now being “transparent”).

2.1.4. Principle of Superposition Once the electric fields diffracted by the semi-plane (the white area in the plane z = 0 in Figure 5) are computed, the electric fields diffracted by the complementary semi-plane (the light-gray area in the plane z = 0 in Figure 5), when it is not opaque, can be computed using Babinet’s principle, and then the total electric field arriving at the receiver can be computed as the sum of both.

2.2. Computation of the Electric Field Scattered by the Discontinuity between Two Media Assuming that an electromagnetic wave impinges over the z = 0 plane around (x,y,z) = (0,0,0), where the specular reflection point is, and: • The extension of the Fresnel reflection zones is small enough (a few hundred meters from a low Earth orbiter) so that the variations of the local incidence angle can be neglected over a few Fresnel zones; • The plane z = 0 is composed of two flat half-planes of different materials, and thus exhibiting different reflection coefficients; and • The transmission coefficient for the wave emitted by the image transmitter (Figure 5) is equal to the reflection coefficient evaluated at the local incidence angle at the origin. Then, as illustrated in Figure 8, the total electric field can be immediately computed as the sum of the following two terms:

𝐸 =𝐸 ·𝐹𝑣 ·𝜌𝜀,,𝜃+𝐹−𝑣 ·𝜌𝜀,,𝜃, (6) where 𝜌𝜀,,𝜃 is the Fresnel reflection coefficient, and 𝜀, is the dielectric constant of the half-plane n. The received power is proportional to the square of the absolute value of Equation (6). Remote Sens. 2020, 12, 3910 7 of 15 𝑃 =𝑃 ·𝐹𝑣 ·𝜌𝜀,,𝜃+𝐹−𝑣 ·𝜌𝜀,,𝜃 , (7)

(a) (b)

Figure 8. Graphical representation of the principle of superposition and its application to the computationFigure 8. Graphical of the total representation electric ﬁeld of incident. the principle Opaque of screensuperposition on the (anda) right its andapplication (b) left to sides the Remote Sens. 2020, 12, x FOR PEER REVIEW 7 of 15 blockcomputation radiation. of the total electric field incident. Opaque screen on the (a) right and (b) left sides block radiation. Following Equation (7), an example is shown in Figure 9. The red and blue lines correspond to Following Equation (7), an example is shown in Figure9. The red and blue lines correspond |𝐹 𝑣| and |𝐹 −𝑣| in decibels. The black line corresponds to the square of the absolute value of the to F(v) and F( v) in decibels. The black line corresponds to the square of the absolute value term within brackets− in Equation (7), computed for 𝜌𝜀 ,𝜃= , and 𝜌𝜀 ,𝜃=√0.1, which , 2 , √ of the term within brackets in Equation (7), computed for ρ(εr,1, θi) = 3 , and ρ(εr,2, θi) = 0.1, whichcorresponds corresponds to the to normalized the normalized peak peakpower power that would that would be collected be collected by a GNSS-R by a GNSS-R instrument. instrument.

| | | | FigureFigure 9. 9.Graphical Graphical representationrepresentation of of the the 𝐹 F(𝑣v) (red)(red) and and F𝐹(−𝑣v) (blue) (blue) functions, functions, and and total total power power loss loss − computed as 𝐹𝑣·𝜌𝜀 ,𝜃 +𝐹−𝑣 ·𝜌𝜀 ,𝜃 2 (black)for 𝜌𝜀 ,𝜃= and 𝜌𝜀 ,𝜃=√0.1. computed as F(v) ρ ε , ,θ + F( v) ρ(ε , θ,) (black) for ρ ε , ,θ = 2 and ρ(ε ,,θ )= √0.1. · r,1 i − · r,2 i r,1 i 3 r,2 i 3.3. Results Results

3.1.3.1. Spatial Spatial Resolution Resolution Computation Computation BasedBased on on the the above above results results the the spatial spatial resolution resolution of of a a GNSS-R GNSS-R system system can can now now be be deﬁned defined as as the the

2 2 2 2 widthwidth of of the the transition transition from from the the 10% 10% above above ρ 𝜌𝜀(εr,2,, θ,𝜃i) ,≜ ρ|max𝜌 ,| and, and 90% 90% of of ρ 𝜌𝜀(εr,1,, θ,𝜃i) ,≜ ρ |min𝜌 | | | | | 𝐹 𝑣% ·𝜌 +𝐹 −𝑣% ·𝜌 2 =0.9· 𝜌 2 , (8a) F(v90%) ρmax + F( v90%) ρmin = 0.9 ρmax , (8a) · − · · (8b) |𝐹𝑣% ·𝜌 +𝐹−𝑣% ·𝜌| =1.1· |𝜌| 2 2 F(v10%) ρmax + F( v10%) ρmin = 1.1 ρmin (8b) The solution of the above equations· has− to be ·performed numerically,· and it actually depends on the values of 𝜌 and 𝜌 . The difference between 𝑣% and 𝑣% leads to the width of 𝑣, ∆𝑣 =% 𝑣 −𝑣%, which is linked to the spatial resolution by means of Equation (9). For the sake of simplicity, assuming an airborne GNSS-R instrument, so that 𝑅 ≫𝑅 , and approximating 𝑅 ℎ⁄𝑐𝑜𝑠𝜃, the following is an approximated formula for the spatial resolution:

· · Δ𝑣 = Δ𝑥 ⟹Δ𝑥=Δ𝑣 . (9) · ·

Figure 10 shows the evolution of ∆𝑣 =% 𝑣 −𝑣% as a function of Δ𝜌 = 𝜌 / 𝜌 in decibels. It varies from 0.74 for 𝜌/𝜌 = −3 dB to 1.7 for 𝜌/𝜌 = −20 dB. This means that for a plane flying at hR = 1000 m height, at GPS L1/Galileo E1 (λ = 19 cm), the spatial resolution ranges from 7.2 m/cos(θi) to 16.6 m/cos(θi), depending on the contrast of the reflection coefficients in the transition.

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The solution of the above equations has to be performed numerically, and it actually depends on the values of ρ and ρmax. The diﬀerence between v and v leads to the width of v, ∆v = v v , min 10% 90% 10% − 90% which is linked to the spatial resolution by means of Equation (9). For the sake of simplicity, assuming an airborne GNSS-R instrument, so that RT RR, and approximating RR hR/cos(θ ), the following ≈ i is an approximated formula for the spatial resolution: s s 2 cos(θi) λ hR ∆v = ∆x · ∆x = ∆v · . (9) λ hR ⇒ 2 cos(θ ) · · i

Figure 10 shows the evolution of ∆v = v v as a function of ∆ρ = ρ /ρmax in decibels. 10% − 90% min It varies from 0.74 for ρ /ρmax = 3 dB to 1.7 for ρ /ρmax = 20 dB. This means that for a plane min − min − flying at hR = 1000 m height, at GPS L1/Galileo E1 (λ = 19 cm), the spatial resolution ranges from Remote Sens. 2020, 12, x FOR PEER REVIEW 8 of 15 7.2 m/cos(θi) to 16.6 m/cos(θi), depending on the contrast of the reflection coefficients in the transition.

∆𝒗 𝜌 𝜌 Figure 10. Width of the ∆v parameter for a transition from 10%10% aboveabove ρminto to 90%90% ofofρ max. .

3.2.3.2. ExperimentalExperimental ValidationValidation FigureFigure 11 11 shows shows thethe measuredmeasured reflectivityreflectivity ((ΓΓ)) inin aa transitiontransition betweenbetween landland andand calmcalm waterwater inin thethe DevilbendDevilbend Reservoir, Reservoir, south south of of Melbourne, Melbourne, Victoria, Victoria, Australia. Australia. The The data data was was acquired acquired during during one one of the of testthe test flights flights of the of the MIR MIR [8] instrument[8] instrument on on 30 30 April Apri 2018.l 2018. The The top top plot plot corresponds corresponds to to one one of of the the fourfour beamsbeams atat GPSGPS L1,L1, andand thethe bottombottom plotplot atat L5.L5. FlightFlight speedspeed waswas 7575 mm/s,/s, flightflightheight heightwas wash hRR == 1000 m,m, andand GNSS-RGNSS-R data data were were coherently coherently integrated integrated for for 1 ms1 ms (GPS (GPS L1 CL1/A C/A code), code), and and incoherently incoherently for T forinc =Tinc20, = 40,20, 100,40, 100, and and 200 200 ms. ms. At At 20 20 ms, ms, the the blurring blurring produced produced by by the the aircraft aircraft movement movement is is 1.5 1.5 m,m, whichwhich isis negligiblenegligible compared to to the the best best spatial spatial resolution resolution co computedmputed in Section in Section 3.1. 3.1 Inspecting. Inspecting Figure Figure 11, at 11 20, atand 20 and40 ms 40 incoherent ms incoherent integration integration time, time, one one can can clearly clearly observe observe some some reflectivity reflectivity ripples ripples that areare reminiscentreminiscent of of those those seen seen in in Figure Figure9. At9. At 100 100 ms, ms, the the blurring blurring is 7.5 is m,7.5 whichm, which is now is now comparable comparable to the to bestthe best spatial spatial resolution, resolution, and theand ripplesthe ripples can nocan longer no longer be appreciated. be appreciated. At 200 At ms200 the ms reflectivity the reflectivity plot isplot even is even smoother. smoother. In the In land the partland (leftpart part (left ofpart the of plots), the plots), note that note the that L5 the reflections L5 reflections exhibit exhibit a more a markedmore marked fading fading pattern pattern than the than L1 the ones. L1 Thisones.may This bemay because be because L5 penetrates L5 penetrates a bit morea bit more in the in land the (andland vegetation),(and vegetation), and it and is moister it is moister soil underneath soil undernea the surface.th the surface. Although Although it is not it veryis not clear, very the clear, width the ofwidth the reflectivity of the reflectivity step is approximately step is approximately 0.3 s, which 0.3 corresponds s, which roughlycorresponds to 22.5 roughly m, as compared to 22.5 tom, the as 20.7compared m predicted to the value20.7 m using predicted Equation value (9) usin andg FigureEquation 10 for(9) and a ~ Figure15 dB reflectivity.10 for a ~−15 dB reflectivity. −

Figure 11. Land-to-water transition during a Microwave Interferometric Reflectometer (MIR) test flight over Devilbend Reservoir (38.29°S, 145.1°E): (a) GPS L1, (b) GPS L5. Horizontal axis units: time and distance. Vertical dashed line: land–water transition.

Remote Sens. 2020, 12, x FOR PEER REVIEW 8 of 15

Figure 10. Width of the ∆𝒗 parameter for a transition from 10% above 𝜌 to 90% of 𝜌.

3.2. Experimental Validation Figure 11 shows the measured reflectivity (Γ) in a transition between land and calm water in the Devilbend Reservoir, south of Melbourne, Victoria, Australia. The data was acquired during one of the test flights of the MIR [8] instrument on 30 April 2018. The top plot corresponds to one of the four beams at GPS L1, and the bottom plot at L5. Flight speed was 75 m/s, flight height was hR = 1000 m, and GNSS-R data were coherently integrated for 1 ms (GPS L1 C/A code), and incoherently for Tinc = 20, 40, 100, and 200 ms. At 20 ms, the blurring produced by the aircraft movement is 1.5 m, which is negligible compared to the best spatial resolution computed in Section 3.1. Inspecting Figure 11, at 20 and 40 ms incoherent integration time, one can clearly observe some reflectivity ripples that are reminiscent of those seen in Figure 9. At 100 ms, the blurring is 7.5 m, which is now comparable to the best spatial resolution, and the ripples can no longer be appreciated. At 200 ms the reflectivity plot is even smoother. In the land part (left part of the plots), note that the L5 reflections exhibit a more marked fading pattern than the L1 ones. This may be because L5 penetrates a bit more in the land (and vegetation), and it is moister soil underneath the surface. Although it is not very clear, the width of the reflectivity step is approximately 0.3 s, which corresponds roughly to 22.5 m, as Remote Sens. 2020, 12, 3910 9 of 15 compared to the 20.7 m predicted value using Equation (9) and Figure 10 for a ~−15 dB reflectivity.

Figure 11. Land-to-water transition during a Microwave Interferometric Reﬂectometer (MIR) test ﬂight Figure 11. Land-to-water transition during a Microwave Interferometric Reflectometer (MIR) test over Devilbend Reservoir (38.29 S, 145.1 E): (a) GPS L1, (b) GPS L5. Horizontal axis units: time and flight over Devilbend Reservoir◦ (38.29°S,◦ 145.1°E): (a) GPS L1, (b) GPS L5. Horizontal axis units: time distance. Vertical dashed line: land–water transition. and distance. Vertical dashed line: land–water transition. Taking a closer look to the periodicity of the ripples, it can be observed that the period in Figure 11b is larger than that in Figure 11a, as it can be inferred from Equation (5), and Figures6 and9, parameterized in terms of v. In Figure 11a, the separation between consecutive peaks in time is ~100 ms, while in Figure 11b it is ~130 ms, which, for an aircraft speed of v = 75 m/s corresponds to 7.50 and 9.75 m, respectively. More detailed results are shown in Table1 for several peaks. Replacing ∆x(t) by v ∆t in Equation (10), where v = v cos(ϕ), ϕ being the angle formed between the aircraft ⊥· ⊥ · ground-track and the coastline, one obtains: s 2 cos(θi) ∆v = v ∆t · . (10) ⊥· λ hR ·

Table 1. Peak positions in Figure6 and peak spacing ( ∆v ), peak positions in time for L1 and L5 beams, time spacing and estimated peak spacing in v.

t ∆t ^ t ∆t ^ v |∆v | peaks, L1 peaks, L1 ∆v peaks, L5 peaks, L5 ∆v peaks peaks [s] [ms] peaks,L1 [s] [ms] peaks,L5 1.22 - Not visible - - 7.140 - − 2.34 1.12 13.290 - - 7.310 170 0.95 − 3.08 0.74 13.400 110 0.71 7.450 140 0.67 − 3.68 0.60 13.500 100 0.65 7.570 120 0.78 − 4.18 0.50 13.580 80 0.52 7.700 130 0.67 −

Assuming hR = 1000 m, θi = 45◦, and substituting ∆t for the time difference between consecutive peaks in Figure 11 (except for roundoff errors due to the discrete sampling), this ratio closely matches the squared root of the ratio of the wavelengths at L1 and L5: √1575.42 MHz/1176.45 MHz = 1.16. Other sources of error are variations of the flight height (approximately hR = 1000 m), variations of the incidence angle, and most likely the fact that the velocity vector is not perpendicular to the coastline (as implicitly assumed in Figure5). Remote Sens. 2020, 12, 3910 10 of 15

These results are corroborated by many other transitions. In what follows a few more examples are presented to illustrate this effect. A minimum incoherent integration time of 40 ms is used, since blurring is still negligible, and these results are almost identical to when 20 ms is used. Figure 12 shows a geo-located double transition from land–calm water–land, the temporal evolution of the reflection coefficient at L1, and two zooms around the transitions for different incoherent integration times. Figure 13 shows another example at L5. Note that for large incoherent

Remoteintegration Sens. 2020 times, 12, x (i.e.,FOR PEER 200 ms) REVIEW the ringings cannot be identiﬁed for both the L1 and L5 examples.10 of 15

FigureFigure 12. 12. Land-to-waterLand-to-water transition transition during during an an MIR MIR at at L1 L1 test test flight flight over over Devilbend Devilbend Reservoir Reservoir (38.29°S, (38.29◦S, 145.1°E).145.1◦E). (a (a) )Geo-located Geo-located reflectivity, reflectivity, and and approximated approximated repr representationesentation of of the the first first four four Fresnel Fresnel zones; zones; (b(b) )time time evolution evolution of of the the reflectivity, reflectivity, and and a a zoom-in zoom-in of of the the step step transition transition on the (cc)) left left and and ( (d) right.

Remote Sens. 2020, 12, 3910 11 of 15 Remote Sens. 2020, 12, x FOR PEER REVIEW 11 of 15

FigureFigure 13. Land-to-water 13. Land-to-water transition transition during during a MIR a at MIR L5 te atst L5 flight test flightover Devilbend over Devilbend Reservoir Reservoir (38.29°S, (38.29 ◦S, 145.1°E).145.1 (a◦)E). Geo-located (a) Geo-located reflectivity, reflectivity, (b) time (b) evolution time evolution of the ofreflectivity, the reflectivity, and a and zoom-in a zoom-in of the of step the step transitiontransition on the on (c the) left (c and) left (d and) right. (d) right.

Figure 14 presents another example on the same flight. In this case, there is an additional transition to a small portion of a peninsula being crossed by the GNSS-R track. It can be noticed that at short integration times, some ripples are produced when changing from land to water, at the beginning and the end of the track. In addition, in the central part, when passing over the peninsula the combination of the different responses produces very large amplitude ripples. In this case, large incoherent integration times (i.e., 200 ms) are not blurring the ripples in the center of the image. However, if even larger integration times were used (i.e., 1 s) no ringings could be identified.

Remote Sens. 2020, 12, 3910 12 of 15 Remote Sens. 2020, 12, x FOR PEER REVIEW 12 of 15

FigureFigure 14. 14. Land-to-waterLand-to-water transition transition during during an an MIR MIR at at L1 L1 test test flight flight over over Devilbend Devilbend Reservoir Reservoir (38.29°S, (38.29◦S, 145.1 E). (a) Geo-located reflectivity, and (b) time evolution of the reflectivity. 145.1°E).◦ (a) Geo-located reflectivity, and (b) time evolution of the reflectivity. Finally, Figure 15 shows another double transition when passing over Sorrento, south of Melbourne. The left part corresponds to an open ocean area (Sorrento ocean beach), while the right part of the image is inside Port Phillip Bay (Sorrento front beach). Looking at the ocean–land transition in Figure 15c on the left-hand side, the reflectivity step is larger, but longer over time, and reflectivity values are noisy because of the increased surface roughness and speckle noise associated with a mostly incoherent scattering. The ripples described in the previous sections and illustrated in the previous examples can no longer be identified. However, as compared to the ocean side in Figure 15c, in Figure 15d the reflectivity step is smaller (calm water), but shorter in time (i.e., sharper), and similar ripples to the lake case can now be identified.

Remote Sens. 2020, 12, 3910 13 of 15 Remote Sens. 2020, 12, x FOR PEER REVIEW 13 of 15

Figure 15. Figure 15. Land-to-waterLand-to-water transition transition during during an an MIR MIR at at L1 L1 test flight flight over Point Nepan, Melbourne (38.35 S, 144.74 E). (a) Geo-located reflectivity, (b) time evolution of the reflectivity, and a zoom-in of (38.35°S,◦ 144.74°E).◦ (a) Geo-located reflectivity, (b) time evolution of the reflectivity, and a zoom-in of the step transition on the (c) left and (d) right. the step transition on the (c) left and (d) right. 4. Discussion 4. Discussion To the authors’ best knowledge, this is the first analytical study of the response of a GNSS-R systemTo tothe a authors’ reflectivity best step. knowledge, The analytical this is expression the first analytical of the measured study of reflectivity the responseΓˆ can of bea GNSS-R derived Γ systemfrom Equation to a reflectivity (7): step. The analytical expression of the measured reflectivity can be derived from Equation (7): 2 Γˆ = F(v) ρ(εr,1, θi) + F( v) ρ(εr,2, θi) , (11) · − · Γ =𝐹𝑣 ·𝜌𝜀 ,𝜃+𝐹−𝑣 ·𝜌𝜀 ,𝜃 , (11) , , 2 2 with v given by Equation (5). It shows a smooth transition between ρ(εr,1, θi) and ρ(εr,2, θi) , withand ripples𝑣 given associated by Equation to di(5).fferent It shows Fresnel a smooth zones transition passing from between one area 𝜌𝜀 to, the,𝜃 other and one. 𝜌𝜀 The,,𝜃 width , and of ripplesthe transition, associated as measuredto different from Fresnel the 10%zones above passing the from lowest one reflectivity area to the values other one. to 90% The of width the highest of the transition,reflectivity as value, measured has been from found the 10% to be above dependent the lowest on the reflectivity amplitude values of the to reflectivity 90% of the step highest itself, reflectivityalthough it value, is quite has flat been for a found wide range to be ofdependent reflectivity on steps the (Figureamplitude 10): of∆ vthe 1.5reflectivity step itself, ≈ Δ𝑣 1.5 although it is quite flat for a wide range ofs reflectivity stepss (Figure 10): λ·hR ·λ hR ∆x Δ𝑥= ∆v = Δ𝑣 · 1.51.5 · . .(12) (12) 2 cos·(θ) ≈ ·2 cos(θ ) · i · i The spatial resolution and the position of the pe peaksaks of the ripples was validated using airborne data at the L1 and L5 bands. It is worth mentioning that the theoretical analysis assumes that coherent scattering is the dominant mechanism, which is the case when transitioning from land to/from calm water. However, when transitioning into open ocean, where incoherent scattering is more important, the presence of

Remote Sens. 2020, 12, 3910 14 of 15

It is worth mentioning that the theoretical analysis assumes that coherent scattering is the dominant mechanism, which is the case when transitioning from land to/from calm water. However, when transitioning into open ocean, where incoherent scattering is more important, the presence of the ripples is not that evident, and the reflectivity exhibits random fluctuations associated mostly to the speckle noise. These results should allow us to improve the determination of the extent of the flooded areas using GNSS-R observations as in [12–15], and they can also be used to dynamically optimize the coherent and incoherent integration times in future GNSS-R instruments (airborne or spaceborne) depending on the application or target area, as opposed to the approach implemented in past and current missions (1 ms coherent integration time, and 1 s incoherent integration time) such as UK TDS-1 [16] and NASA CyGNSS [17] missions.

5. Conclusions In this study, the GNSS-R response to a step function in the reflectivity was derived analytically using known principles of electromagnetism. However, the evaluation of the Fresnel integrals has to be performed numerically. Analytical/numerical predictions match well with the experimental results at L1 and L5, acquired during an airborne experiment using the MIR (Microwave Interferometric Reflectometer) instrument over a water reservoir south of Melbourne and in a transition from the ocean into Port Phillip Bay, in terms of width and ringing, if incoherent integration times multiplied by the speed of the aircraft are shorter than the width of the Fresnel zone. Larger incoherent integration times blur the step response and degrade the achievable spatial resolution. Considering the size of the first Fresnel zone (tens of meters from a plane, to a few hundred meters from a spacecraft) the analysis presented is applicable to most coastlines.

Author Contributions: Conceptualization, A.C.; methodology, A.C.; software, J.F.M.-M. and A.C.; validation, A.C.; formal analysis, A.C.; investigation, A.C.; resources, A.C. and J.F.M.-M.; data curation, J.F.M.-M.; writing—original draft preparation, A.C.; writing—review and editing, A.C. and J.F.M.-M.; visualization, A.C. and J.F.M.-M.; supervision, A.C.; project administration, A.C.; funding acquisition, A.C. All authors have read and agreed to the published version of the manuscript. Funding: This work was funded by the Spanish MCIU and EU ERDF project (RTI2018-099008-B-C21/AEI/10.13039/501100011033) “Sensing with pioneering opportunistic techniques” and grant to ”CommSensLab-UPC” Excellence Research Unit Maria de Maeztu (MINECO grant MDM-2016-600). Acknowledgments: The authors would like to express their gratitude to R. Onrubia and D. Pascual who flew and operated the MIR instrument during the field experiment, and to C. Rüdiger, J.P. Walker, and A. Monerris for their support during the execution of the campaign. Conflicts of Interest: The authors declare no conflict of interest.

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