GPS Solutions

, Volume 16, Issue 2, pp 231–241

Altimetry with GNSS-R interferometry: first proof of concept experiment

Authors

    • Institut d’Estudis Espacials de Catalunya-CSIC
  • Oleguer Nogués-Correig
    • Institut d’Estudis Espacials de Catalunya-CSIC
  • Serni Ribó
    • Institut d’Estudis Espacials de Catalunya-CSIC
  • Estel Cardellach
    • Institut d’Estudis Espacials de Catalunya-CSIC
  • Santi Oliveras
    • Institut d’Estudis Espacials de Catalunya-CSIC
  • Enric Valencia
    • Institut d’Estudis Espacials de Catalunya-UPC
  • Hyuk Park
    • Institut d’Estudis Espacials de Catalunya-UPC
  • José Miguel Tarongí
    • Institut d’Estudis Espacials de Catalunya-UPC
  • Adriano Camps
    • Institut d’Estudis Espacials de Catalunya-UPC
  • Hans van der Marel
    • Institute of Earth Observation and Space Systems, Faculty of Aerospace EngineeringDelft University of Technology
  • Roel van Bree
    • Institute of Earth Observation and Space Systems, Faculty of Aerospace EngineeringDelft University of Technology
  • Bas Altena
    • Institute of Earth Observation and Space Systems, Faculty of Aerospace EngineeringDelft University of Technology
  • Manuel Martín-Neira
    • European Space Research and Technology Centre
Original Article

DOI: 10.1007/s10291-011-0225-9

Cite this article as:
Rius, A., Nogués-Correig, O., Ribó, S. et al. GPS Solut (2012) 16: 231. doi:10.1007/s10291-011-0225-9

Abstract

The Global Navigation Satellite System Reflectometry (GNSS-R) concept was conceived as a means to densify radar altimeter measurements of the sea surface. Until now, the GNSS-R concept relied on open access to GNSS transmitted codes. Recently, it has been proposed that the ranging capability of the technique for ocean altimetric applications can be improved by using all the signals transmitted in the bandwidth allocated to GNSS, which includes open access as well as encrypted signals. The main objective of this study is to provide experimental proof of this enhancement through a 2-day experiment on the Zeeland Bridge (The Netherlands). In the experiment, we used a custom built GNSS-R system, composed of high gain GPS antennas, calibration subsystem, and an FPGA-based signal processor which implemented the new concepts, an X-band radar altimeter and a local geodetic network. The results obtained indicate that the new approach produces a significant improvement in GNSS-R altimetric performance.

Keywords

Ocean altimetryGNSS reflectometryBistatic radars

Introduction

Space borne bistatic radars using opportunity signals reflected onto the earth’s surface were proposed in Martín-Neira (1993) to densify, in time and space, the mean sea level measurements provided by monostatic radar altimeters. This concept was described in terms of the measurements of the coherence of the direct and reflected signals, obtained using radiointerferometric techniques, and was termed the Passive Reflectometry and Interferometry System (PARIS). A detailed description of such radiointerferometric techniques and references to related applications, such as sea interferometry and moon reflectometry, can be found in Thompson et al. (2004).

When the sources of opportunity are the GNSS, the PARIS concept is called GNSS-Reflectometry (GNSS-R). In such a case, because the information to create some of the components of the transmitted signals is available, we could cross-correlate the reflected signals with their modeled replicas. Successful detection of GNSS-R signals from space has been reported in Lowe et al. (2002) and Gleason et al. (2005), after the cross-correlation of the reflected signals with their modeled replicas.

With this standard approach, GNSS-R ignores the encrypted components of the signals, collects less power in narrower bandwidth, and thus lowers the achievable performance of the technique. To overcome this limitation, the European Space Agency (ESA) has announced the PARIS In-Orbit Demonstrator (PARIS-IOD) space mission, as described in Martín-Neira et al. (2011), based on the cross-correlations of GNSS signals collected with two high gain antennas. In this paper, we will use the term GNSS-Ri to denote this interferometric approach.

An additional advantage of GNSS-Ri is that it is not sensitive to distortions in the transmitted signals. In the GNSS-R implementations, the modeled replicas differ slightly from the actual transmitted signals. This will distort the shape of the measured cross-correlations and will produce SNR losses and uncertainties in the delay measurements, inducing errors in the altimetric information sought. See a recent discussion in Wong et al. (2010), and in the references cited there, describing such deformations and showing that they are common.

We present the first GNSS-Ri proof of concept experiment performed to demonstrate the feasibility of the technique by sensing the sea tide from a bridge in the Netherlands. In the following sections, we describe the setup used to gather GNSS-Ri data and the waveform models developed to derive the altimetric information from the observations. Finally, we compare our results with the mean sea level estimates obtained from independent X-band radar altimeter measurements performed simultaneously.

Experiment setup

On July 7 and 8, 2010, in two sessions from 10:00 to 18:00 UTC (hereafter, Sessions A and B), we deployed on the Zealand Bridge (The Netherlands) a system composed of (a) a custom GNSS-Ri observational system, (b) an X-band radar altimeter (XBRA), and (c) a local geodetic network, to tie (a) and (b). The setup is shown in Fig. 1.
https://static-content.springer.com/image/art%3A10.1007%2Fs10291-011-0225-9/MediaObjects/10291_2011_225_Fig1_HTML.gif
Fig. 1

Logical arrangement of the Zealand Bridge setup. The system produced two independent measurements of the mean sea level H. It is composed of a GNSS-Ri Instrument, and an X-band radar altimeter (XBRA). A local reference frame was established using a TOPCON GPT-7003i Total Station (R1), and a GPS reference point (R2). The hemispherical antenna of a GPS receiver (N) was used to provide time reference to the PIR rack

GNSS-Ri observational system

The GNSS-Ri system deployed at the Zeeland Bridge was composed of:
  • Two fixed GPS L1-band directional antennas, U (up looking) and D (down looking), mounted at the end of a 4 m boom over the sea, with boresights (Azimuth, Elevation) equal to (310°, 70°) and (310°, −70°). This pointing was the result of a careful a priori study, which maximized for the experiment site and date the number of hours per day for which only one GPS space vehicle (SV) would pass through the antenna’s main beam, leaving the rest of satellites away from the main beam with a relative strength with respect to the radiation pattern peak of −15 dB at most. This amounted to about 2 h per day. The output of these U and D antennas is a pair of signals (Yd, Yr), where the subscripts d and r stand for direct and reflected, respectively.

  • A two-position switch, controlled by the PARIS Interferometric Receiver (PIR) rack, delivers the signal pair (Yx, Yy), equal to (Yd, Yr) or (Yr, Yd) depending on the switch position. This makes it possible to swap the antenna outputs as inputs of the receiver.

  • The PIR-signal processor (PIR-SP), placed inside the PIR rack, computes in real time the GNSS-Ri complex cross-correlations Zxy defined in Eq. 1, where (Yx, Yy) is the analytic representation of the real signals (Yx, Yy). These waveforms are sent and stored in real time to a laptop computer, not depicted in Fig. 1, which is external to the PIR rack.

  • A GPS receiver inside the PIR rack, not depicted in Fig. 1, provides the GPS time frame reference using the signals captured by the GPS antenna N.

The relevant parameters of the GNSS-Ri instrument are summarized in Table 1. Antennas only differ in terms of polarization, U was RHCP and D was LHCP. The gain was 6 dB lower than the directivity, due to the fact that the antenna had an ohmic efficiency of 0.25.
Table 1

GNSS-Ri system parameters

Parameter

Value

Parameter

Value

U/D antenna directivity

15 dBi

U/D antenna gain

9 dBi

U polarization

RHCP

D polarization

LHCP

Sidelobes

<−35 dB

Switching rate

1 Hz

Sampling rate

80 MHz

RF signal bandwidth

24 MHz

The main parts of the PARIS Interferometric Receiver are presented in Fig. 2. It is composed of the RF calibration switch, the down conversion chains, and the signal processor, which includes a 320-lag complex correlator with a sampling rate of Fs = 80 MHz, continuously delivering the complex correlations:
$$ {\mathbf{Z}}_{{{\mathbf{xy}}}} (t,\tau )\left| {_{{\tau = m \cdot T_{s} }} } = \int\limits_{{u = - T_{c} /2}}^{{u = + T_{c} /2}} {{\mathbf{Y}}_{{\mathbf{x}}} } (t + u){\mathbf{Y}}_{{\mathbf{y}}}^{*} (t + u - \tau ){\text{d}}u \right|_{{\tau = m \cdot T_{s} }} $$
(1)
with m ∈ [80, 239] and Ts ≡ 1/Fs = 12.5 ns. The correlation channel works with a fixed coherent integration time of Tc = 1 ms. The waveforms are sent in real time to the PIR laptop through the Ethernet link and saved. The cross-correlation function has a different meaning depending on the switch position, as commented previously. The PIR-SP is autonomous and does not receive any command from outside. The PIR-signal processor works on a sequence basis that repeats every GPS minute and controls the switch position. The GPS receiver inside the PIR rack provides the GPS time frame to the PIR-SP via a signal of one pulse per second (1 PPS) and time data logs that are synchronized with this 1PPS, which provide the GPS time tags in a week and second-of-week format. The GPS receiver clock is totally independent and incoherent with respect to the PIR rack 30 MHz reference clock. The downconverters are locked to that reference in order to generate their internal LO tones coherently. Also, the PIR-SP takes that clock reference and internally boosts it up to 80 MHz to generate its own working clock. We chose a 30 MHz clock reference because the 52nd and 53rd harmonics of this frequency fall at ±15 MHz around the GPS L1 carrier, avoiding the instrument self-interference inside the band that we experienced with a 20 MHz reference in our preliminary prototype designs of the receiver. Any tone added to the input signals in both correlator inputs contributes to the cross-correlation, while in the normal GNSS receiver, the signal is correlated with a model, removing this effect of the interference through the correlation process.
https://static-content.springer.com/image/art%3A10.1007%2Fs10291-011-0225-9/MediaObjects/10291_2011_225_Fig2_HTML.gif
Fig. 2

PIR receiver parts: the switch, the rack and the laptop

X-band radar altimeter

The X-band radar altimeter (XBRA) is a WaveGuide short range FMCW radar produced commercially by RADAC BV of The Netherlands. The radar operates in the 9.8–10.3 GHz frequency range and has a distance range of 1–75 m. The width of the radar beam is about 5°. The X-band radar measures the distance to the water surface, or heave, at a rate of 2.56 samples per second (391 ms interval). The height of the water level, significant wave height (SWH) and wave period, as well as several other parameters including energy density spectra, were computed from the raw heave (height) measurements. We found that SWH was in the order of 10 cm during both sessions.

The standard deviation of the raw heave data, computed from the first differences, was 4.12 cm for July 7 and 4.58 cm for July 8. These standard deviations include, besides measurement errors, the high frequency part of the waves. Normalized to 1 s, the precision of the heave data is better than 2.9 cm. In order to eliminate the effects of waves, the raw heave data have been smoothed. We have compared 10-min averages with 1 min averages and 10-s averages. The standard deviation of the differences with the 10 min smoothed data was about 0.5 cm for the 1 min smoothed data and less than 1 cm for the 10-s smoothed data. As a reference for the GNSS-Ri, we have selected the 10 min average sea level height H10, which is recorded at 1 min intervals. It is estimated that the sea-level height H10 is precise to less than 1 cm.

The X-band radar data have been compared with tide gauge data from two nearby tide gauges at distances of 7.4 and 15.3 km. The data from these tide gauges have been interpolated to the location of the X-band radar. There is a systematic difference between the interpolated sea level from the tide gauges and the X-band radar of about 1.3 cm with a slow oscillation of an amplitude of about 2 cm. This difference is well within the accuracy of the interpolation. According to the manufacturer’s specifications, the bias in the X-band radar should be within a few millimeters. The comparison with the tide gauge data confirms that there are no relevant biases in the X-band radar data.

Local survey measurements

A local survey was carried out on the bridge every 1–2 h to monitor the height differences between the antennas of the GNSS-Ri and the X-band radar, which were mounted 27 m from the GNSS-Ri and to connect these instruments to various stable reference points on the bridge. For the survey, we used a Topcon GPT–7003i Total Station. The precision of the local survey measurements, including the height difference between PIR and WG, were better than 1 mm in height. However, the Total Station measurements revealed that the area of the bridge where the PIR was mounted, including the boom itself, moved by 3 cm in height during the day. The motion of the X-band radar, which was closer to the main pillar, was smaller. The X-band radar data were corrected for the zero-point offset between the X-band and GNSS-Ri (~16 cm) and the smooth diurnal bridge deformation (~3 cm), in order to provide a “true” data set for the GNSS-Ri data.

Another part of the local survey measurements consisted of measuring GPS baselines to a nearby height benchmark in the national ordnance datum Normaal Amsterdams Peil (NAP). The GPS measurements were necessary in order to compare data from nearby tide gauges, which are given in the NAP datum, to the X-band radar data. The internal consistency between the completely independent GPS baselines was at the millimeter level. The measurements also showed that reference points of the stable part of the bridge were indeed stable.

Interferometric waveforms model and data analysis

There is a relevant difference between the nature of the waveforms obtained with GNSS-R or GNSS-Ri. In the first case, the reflected signal is correlated with a replica which is associated to a particular transmitter. In the case of GNSS-Ri, this association does not exist, and we need to rely on the discrimination provided by the antenna directivities and the relative delay and Doppler shifts applicable in the correlator. In our experiment, the relative delay was less than twice the height of the bridge (~18 m), and the relative Doppler shift was negligible due to the static nature of the experiment. Consequently, the discrimination could only be provided by the antenna directivity. We describe the analysis performed with our primary observables, the waveforms, to obtain derived observables like the relative delays.

Signal model

The GNSS-Ri observables are the mean power waveforms. The model used in our implementation is based on Zavorotny and Voronovich (2000), Garrison et al. (2002), and Martín-Neira et al. (2011). The complex Yx and Yy signals are divided into consecutive coherent integration periods of length Tc = 1 ms, to compute the complex cross-correlation Zxy as defined in Eq. 1. The mean \( \langle |{\mathbf{Z}}_{{{\mathbf{xy}}}} (t,\tau )|^{2} \rangle \) computed during an incoherent integration time Tnc are the waveforms wxy:
$$ w_{xy} (t,\tau ) = \frac{1}{{T_{nc} }}\int\limits_{{t - T_{nc} /2}}^{{t + T_{nc} /2}} \langle |{\mathbf{Z}}_{{{\mathbf{xy}}}} (t^{\prime } ,\tau )|^{2} \rangle {\text{d}}t^{\prime } $$
(2)
where t in Eq. 2 is the time tag associated to the incoherent integration interval, and τ is the cross-correlation delay variable, in the range −1 μs < τ < 3 μs (or −300 < τ < 900 m), sampled at a rate of 80 MHz.Assuming that the complex noise terms in Yx and Yy are uncorrelated, the waveforms in Eq. 2 are modeled as the sum of signal and noise terms:
$$ w_{xy} (t,\tau ) = w_{s} (t,\tau ) + w_{n} (t,\tau ) $$
(3)
with the signal term is
$$ w_{s} = \sum\limits_{{{\text{all}}\,:S}} {w_{s}^{S} } (t,\tau - \tau_{S} ) $$
(4)
where \( w_{s}^{S} \) is the signal waveform corresponding to satellite S, which according to Zavorotny and Voronovich (2000), Garrison et al. (2002), and Martín-Neira et al. (2011) could be written as:
$$ w_{s}^{S} (t,\tau ) = P_{T}^{S} P_{\text{r}}^{S} (\tau )*|W^{S} (\tau )|^{2} $$
(5)
For each transmitter S, \( P_{T}^{S} \) is the power of the received direct signal, the function \( P_{\text{r}}^{S} (\tau ) \) is the average reflected signal power from the sea surface scatterers arriving with a delay τ, and WS(τ) is the Woodward Ambiguity Function (WAF) associated to the complex baseband modulation of the signal and * is the convolution operator. The use of Eq. 4 is necessary when the differences between relative delays τS for different transmitters are smaller than the width of the WAF function, that is, when the waveforms of different satellites overlap.
If the footprint of the function \( P_{\text{r}}^{S} \) is much smaller than the footprint of \( |W^{S} |^{2} \) (or \( P_{\text{r}}^{S} \propto \delta \), the Dirac function), as in our experiment, neglecting constant factors and free-space losses, Eq. 5 takes the simple form:
$$ w_{s}^{S} (t,\tau ) = C^{S} |W^{S} |^{2} $$
(6)
with \( C^{S} = \left( {P_{T}^{S} G_{T}^{S} } \right)^{2} G_{{R,{\text{d}}}} G_{{R,{\text{r}}}} \sigma \), where GR,d and GR,r are the power gain of the transmitter, U and D antennas along the corresponding link path, and σ is a scattering coefficient.

Data analysis

During both sessions, the PIR-SP produced waveforms continuously, repeating each minute a sequence of different calibration switch positions. The data used in this research were collected each minute from seconds 10 through 29; the calibration switch was set to obtain wxy = wdr for the ten even seconds (10, 12, …, 28) and wxy = wrd for the ten odd seconds (11, 13, …, 29). Both data sets were integrated independently to have two waveforms each minute with an effective incoherent integration time of 10 s. To simplify the description, we ignore the 1-s difference between the time stamps of the waveforms, assuming that both waveforms wdr(t,τ) and wrd(t,τ) were obtained at the same time t, and carry the same altimetric information.

The Session B set of waveforms wdr(t,τ) is shown in Fig. 3, for delays τ in the interval [−20:100] m. We normalized each waveform by dividing it by its maximum value. A similar aspect is obtained for Session A. Figure 4 shows details of six waveforms extracted from this data set, for delays τ in the interval [−300:350] m, with the same normalization. The M-code component is seen with the highest contrast during the 14:00–15:00 UTC period in both Sessions A and B. We will refer to these 1-h intervals as slices A and B.
https://static-content.springer.com/image/art%3A10.1007%2Fs10291-011-0225-9/MediaObjects/10291_2011_225_Fig3_HTML.gif
Fig. 3

The normalized waveforms obtained during Session B, represented as a function of cross-correlation delay τ and the UTC). The M-code autocorrelation has a narrower peak than the P(Y) code and has important sidelobes that are approximately 30 m away from the peak. This is visible during intervals including 11:00 UTC, 14:30 UTC and 17:00 UTC. Some of these waveforms are represented in Fig. 4

https://static-content.springer.com/image/art%3A10.1007%2Fs10291-011-0225-9/MediaObjects/10291_2011_225_Fig4_HTML.gif
Fig. 4

Six normalized waveforms as a function of the correlator delay τ with the interval [−300:350]. These waveforms have been extracted from the data set obtained in Session B. The corresponding UTC epoch is indicated in each panel. Each waveform is the superposition of the codes present in the correlated signals. Near 11 UTC, 14 UTC and 15 h UTC there were M-code signals in addition to the C/A and P(Y) components. Because the C/A code contribution extends ±300 around the peak, the last values at the right end of each waveform could be taken as the noise to signal power. The changes in the relative noise level are partly because the transmitters had changing angular distances to the antennas’ boresight. The red line corresponds to a relative power equal to 0.64 of the peak power, used to characterize the waveform width β. Note the apparent reduction in β and the increase in the SNR when the M-codes are present, implying a substantial increase in delay resolution

We extracted from each waveform w, either wdr or wrd, its maximum value wmax, the delay associated with this value τpeak, the pointing waveform delay τpointing = 0.5(τ+), and the waveform width β(t) = τ+−τ, where τ+ are the solutions to the equation w(t,τ) = 0.64w(tpeak). The factor 0.64 has been selected to use data with large slopes to increase the sensitivity (Figs. 3, 4). These values are presented in the three lower panels of Fig. 5. With this choice, the width of a 24 MHz filtered WAF containing C/A, P(Y) and M signals, with the power levels given in Barker et al. (2000), is approximately 13.5 m. The upper panel of Fig. 5 shows the XBRA heights H10(t), smoothed with a 10-min sliding window, and referred to the GNSS-Ri reference point. The uncertainty of these 10-min measurements, including instrumental errors, is about 1 cm. According to Fig. 5, Panel 3, within the noise we have \( \tau_{\text{dr}}^{\text{peak}} (t) = \tau_{\text{dr}}^{\text{pointing}} (t) \), indicating that the shape of the waveforms is almost unaffected by the scattered power function Pr. The same equality is obtained with the data obtained when the switch was in the other position. These common values will be designated \( \tau_{\text{dr}}^{\text{PIR}} (t) \) and \( \tau_{\text{rd}}^{\text{PIR}} (t) \), which could be related with an instrument bias b and the desired calibrated delay observable τPIR using the equations:
$$ \tau^{\text{PIR}} = 0.5\left( {\tau_{\text{rd}}^{\text{PIR}} - \tau_{\text{dr}}^{\text{PIR}} } \right) $$
(7a)
$$ b = 0.5\left( {\tau_{\text{rd}}^{\text{PIR}} + \tau_{\text{dr}}^{\text{PIR}} } \right) $$
(7b)
The main sources for the observed variations of τPIR(t) are the mean sea level and the relatively faster changes in the positions of the contributing transmitters, as confirmed by the changes in wmax and β (Fig. 5, Panels 2 and 4). The variability of τPIR(t) at shorter scales (min) is noise-like. The instrumental bias b corresponds to the differential instrumental delay between the signal paths from the switch to the signal processor, including the downconversion chains (Fig. 2). The variability of b has been estimated for the three intervals in which the width β of the waveform was smaller and stable. The results are given in Table 2:
https://static-content.springer.com/image/art%3A10.1007%2Fs10291-011-0225-9/MediaObjects/10291_2011_225_Fig5_HTML.gif
Fig. 5

Observable quantities obtained in both sessions. Blue dots correspond to Session A, and red dots refer to Session B. Panel 1 contains the XBRA average mean sea level H10, referenced to the GNSS-Ri instrument. The remainder gives observable quantities extracted from the waveforms wdr. Panel 2 represents the observed waveform maximum power wmax. Panel 3 represents the delays τpointing. In the same panel, we have over plotted the delays τpeak in green and brown for Sessions A and B respectively, showing that, within the noise, τpeak = τpointing. Panel 4 gives the waveform width β(t)

Table 2

Instrumental differential bias b mean and standard deviation computed for three intervals during Session B

Interval (hours UTC)

Bias b (m)

SD of b (m)

10.3–11.4

3.64

0.06

14.0–15.0

3.59

0.04

16.9–17.7

3.61

0.08

GNSS-Ri altimetry model

As noted in the previous section, the GNSS-Ri observables are not assigned unambiguously to a particular satellite. In the GNSS-R applications, where a single transmitter is involved, the following expression is used, which relates the observable τS, the altimetric parameter H, and the elevation elS of the observed satellite:
$$ \tau_{S} \, = \, - 2 \cdot H \cdot \sin (el^{S} ) + n_{\tau } + n_{{\bmod {\text{el}}}} $$
(8)
This includes measurement nτ and model noise nmodel terms. Additional terms accounting for differential tropospheric, ionospheric delays are negligible because of the proximity of the antennas to the sea surface. Due the smallness of the SWH, the effects of sea roughness have also been neglected. Because the range of elS is reduced, the different parameters will show highly correlated effects with instrumental biases (Rius et al. 2010) and the use of this equation to obtain H requires additional information. The purpose now is to derive a generalization of Eq. 8 applicable to our GNSS-Ri observables.
We assume that, as a first approximation, each waveform around its peak could be described as a parabola. Assuming the validity of Eq. 6, Eq. 4 will read:
$$ w_{s} = \, \sum\limits_{{{\text{all}}\,\:S}} {C^{S} } \cdot (\tau - \tau_{S} )^{2} $$
(9)
where
$$ \tau_{S} = - 2 \cdot H^{\text{PIR}} \cdot \sin (el^{S} ) $$
A relation between the delay of the maximum of the waveform τPIR and the height using PIR data HPIR is easy to deduce, using a derivative of Eq. 9 at τ = τPIR of zero, and Eq. 8:
$$ \tau^{\text{PIR}} \, = \, - 2 \cdot H^{\text{PIR}} \cdot \sum\limits_{{{\text{all}}\,\:S}} {c^{S} } \cdot \sin (el^{S} ) + n_{\tau } + n_{{\bmod {\text{el}}}} $$
(10)
where \( c^{S} = {{C^{S} } \mathord{\left/ {\vphantom {{C^{S} } {\sum\nolimits_{{{\text{all}}\,\:S}} {C^{S} } }}} \right. \kern-\nulldelimiterspace} {\sum\nolimits_{{{\text{all}}\,\:S}} {C^{S} } }} \) are the normalized weights, and, as before, we have included terms to account for the measurement noise nτ and the model noise nmodel. This model noise term will depend mainly on the direction of the satellites, with a period of 1 sidereal day in a first approximation. Ignoring the noise terms, Eq. 10 could be used to obtain
$$ H^{\text{PIR}} = {{ - \tau^{\text{PIR}} } \mathord{\left/ {\vphantom {{ - \tau^{\text{PIR}} } {\left( {2 \cdot \sum\limits_{{{\text{all}}\:S}} {c^{S} } \cdot \sin \left( {el^{S} } \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2 \cdot \sum\limits_{{{\text{all}}\,\:S}} {c^{S} } \cdot \sin \left( {el^{S} } \right)} \right)}} $$
In the computation of the weights cS, see Eq. 6, we have assumed that the transmitter parameters \( P_{T}^{S} \) and \( G_{T}^{S} \) have common values for all satellites S and \( G_{{R,{\text{d}}}}^{S} \) and \( G_{{R,{\text{r}}}}^{S} \) have been computed assuming that the beam of both U and D antennas have a Gaussian pattern half-power beamwidth equal to 30°.
In Table 3, we indicate, for each GPS satellite above 15°, its PRN number, its launch year, approximate values of its a priori reflected to direct relative delay τapp, elevation elS and the normalized weights cS at the start and end times of slices A and B.
Table 3

Satellites observed during slices A and B. We show for each satellite, block type, and launch year with approximate a priori relative differential delays in meters, elevations el in degrees, and weights for the start and stop times of slices A and B

Space vehicle

14:00 UTC

15:00 UTC

PRN

Block

Year

τapp

el

cS

τapp

el

cS

9

IIA

1993

31.4

67

0.00

21.5

39

0.00

12

IIR

2006

31.6

68

0.81

33.7

83

0.18

14

IIR

2000

17.4

30

0.00

24.0

45

0.01

25

IIF

2010

30.2

63

0.19

34.0

87

0.75

27

IIA

1992

28.7

57

0.00

17.9

32

0.00

30

IIA

1996

17.2

30

0.00

28.3

56

0.06

After the experiment was performed, we noticed that on May 28 2010, a few days before our experiment, Block IIF SV with PRN 25 was launched and positioned in the same orbit and slot as Block IIR SV with PRN12. Both satellites were transmitting M-code signals. According to Table 3, during slices A and B, the relative delays between PRN 12 and PRN 15 changed between +1.4 and −0.3 m. Because the differences between the relative delays were smaller than the width of the autocorrelation of the code and their angular separation was smaller than the antenna beamwidth, we have applied the model given in Eq. 10 to this double source.

Comparison of GNSS-Ri and XBRA height estimates

In order to compare the results obtained, we will use the single and double-difference operators defined as
$$ \Updelta^{{}} H(t) = H^{\text{PIR}} (t) - H10(t) $$
(11a)
$$ \Updelta^{2} H(t) = \Updelta^{{}} H(t + 1\;{\text{Sidereal}}\;{\text{Day}}) - \Updelta^{{}} H(t) $$
(11b)
where H10(t) is the 10-min average sea level height measured by the XBRA.
The single difference will provide a measure of the goodness of our model, and the double difference should cancel model uncertainties, in order for the precision of the method to be determined. Figure 6 shows ΔH(t) and Δ2H(t) computed with the data obtained in slices A and B. The series of single differences indicates that there is a bias and a linear variation, and the series of double differences does not show significant differences.
https://static-content.springer.com/image/art%3A10.1007%2Fs10291-011-0225-9/MediaObjects/10291_2011_225_Fig6_HTML.gif
Fig. 6

Variation of the single-difference ΔH(t) for both slices. Blue dots indicate Slice A, and red refer to Slice B. The values of the double-difference Δ2H(t) are represented with green dots

We have found from Δ2H(t) in the interval 14:10–14:30 UTC that its mean and standard deviation were \( \left\langle {\Updelta^{2} H} \right\rangle = - 1.5\,{\text{cm}} \) and \( \sigma_{{\Updelta^{2} H}} = 3.6\,{\text{cm}} \), or \( \sqrt {10} \cdot3.6 = 11.4 \) cm if we reduce the integration time from 10 to 1 s. From Eq. 11a, we could derive \( \sigma_{{H^{\text{PIR}} }} = \sqrt {{{\sigma_{{\Updelta^{2} H}}^{2} } \mathord{\left/ {\vphantom {{\sigma_{{\Updelta^{2} H}}^{2} } {2 - \sigma_{H10}^{2} }}} \right. \kern-\nulldelimiterspace} {2 - \sigma_{H10}^{2} }}} = \sqrt {{{(11.4)^{2} } \mathord{\left/ {\vphantom {{(11.4)^{2} } {2 - 2.9^{2} }}} \right. \kern-\nulldelimiterspace} {2 - 2.9^{2} }}} = 7.5\,{\text{cm}} \), where we have used for σH10 a value of 2.9 cm, which is the standard deviation of the XBRA data normalized to 1 s as already shown in the X-band radar altimeter section.

Conclusions

The first GNSS-R interferometric altimetry experiment yielded a 7.5 cm uncertainty in 1 s measurements, despite the nonoptimal conditions of the experiment, driven by the static and low-altitude position of the receiver. These conditions introduced contamination of the altimetric observables by simultaneous reception of multiple satellite signals, with different delays, and difficulties in properly modeling and correcting this effect. Nevertheless, these results are a significant improvement in signal envelope-based GNSS-R altimetric performance with respect to the standard C/A code correlation approach, which is in the order of 1 m (Spilker 1996), and still improvable under other more realistic observation geometries that permit delay-Doppler filtering of the different satellite contributions.

The results of this experiment consolidate the interferometric processing as the baseline for a demonstration space mission. In an orbital scenario, other factors not present in the bridge experiment will tend to degrade the performance, mainly the larger free-space loss of the reflected path, the increased roughness of the ocean surface, and the ionospheric and tropospheric delays. Larger antennas are needed for an experiment from space, of the order of 1 m2, as well as careful compensation of the delay and Doppler shifts between the direct and reflected signals. On the other hand, the greater delay-Doppler separation between different GNSS satellites from orbital altitude will help discriminate different signals from different space vehicles, something that was not possible from the bridge.

Acknowledgments

The X Band WaveGuide radar altimeter was kindly provided to us by RADAC BV, The Netherlands. We also would like to thank Tom van der Vlucht (RADAC), for his help with the bridge measurements, and the Province of Zeeland for permission to use the bridge. This research was supported by the European Space Agency (RFQ/3-127-47/09/NL/JD) and the Spanish Ministry of Science and Innovation (AYA2008-05906-C02-01/02). E.C. belongs to the Spanish Ramón y Cajal Program.

Copyright information

© Springer-Verlag 2011