# Lake Altimetry Using Long Coherent Integration Carrier Phase Measurements from Airborne GNSS Reflectometry

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Site: Lake of Carcans-Hourtin (France)

#### 2.2. On-Board Equipment

#### 2.3. Innovative Data Processing

#### 2.3.1. First Step: Dedicated Reflectometry Software

#### 2.3.2. Second Step: Prolongation of the Coherent Integration

^{2}= −1. The data sign is extracted from the direct In-phase and Prompt correlator channel and multiplied to the reflected correlators so as to remove the sign inversions. This operation is crucial for increasing the coherent integration time as shown in Figure 5. Once this has been carried out, we can extend the coherent integration while applying an apodization window to remove high-frequency noise. The Hamming window ${W}_{M}\left(u\right)$ applied on $M$ samples is defined as:

#### 2.3.3. Third Step: Computation of Corrections, Residuals, and Integer Values for Ambiguities

#### Lever Arm Correction

#### Tropospheric Delay Correction

_{hydr}(i.e., dry gases) and wet N

_{wet}(i.e., water vapor) components such as $N={N}_{hydr}+{N}_{wet}$. It can then be expressed as a function of air total pressure P [hPa], temperature T [K] and water vapor pressure e [hPa] as:

#### Pseudo-Observation Computation to Fix Ambiguities

## 3. Results

#### 3.1. Data Processing

#### 3.2. Ambiguity Resolution

#### 3.3. Reflection Surface Height Computation

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Map of Carcans-Hourtin Lake with locations of the buoy and reference station in relation to the two flight profiles. (

**b**) Piper Aztec plane with direct GNSS antenna Trimble AV39 (

**c**) on top of its fuselage. (

**d**) Dedicated reflectometry antenna as seen from the aircraft cabin and iXblue inertial navigation system.

**Figure 2.**(

**a**) Reference station for RTK positioning of the in situ GNSS buoy (

**b**). (

**c**) Measurements of buoy heights obtained by RTK positioning when flying over D1–D2 profile. The mean value of the buoy height is represented by the red dotted line and is equal to 59.98 m.

**Figure 3.**Signal processing architecture of modified GNSS SDR for a reflectometry application. The same carrier and code replicas are used for direct and reflected processing.

**Figure 4.**Drawing of the reflection for an airborne reflectometry campaign. The model considered here is the simplest, with a plane reflection surface and satellite at an infinite distance with parallel rays.

**Figure 5.**Examples of graphs showing the effect of increasing the coherent integration time for the direct signal (solid blue line) and the reflected signal (solid red line). The amplitude of the reflected signal appears larger than the direct signal because the reception chain of the reflected channel had more gain than that of the direct channel. (

**a**) Vector magnitude for a CIT of 20 ms without data wipe-off. (

**b**) Vector magnitude for a CIT of 20 ms after data wipe-off. (

**c**) Vector magnitude for a CIT of 500 ms without data wipe-off. (

**d**) Vector magnitude for a CIT of 500 ms after data wipe-off. We can observe that a data wipe-off operation is required to extend coherent integration over 20 ms.

**Figure 6.**Schematic illustration of tropospheric delay. As can be seen, the reflected signal passes through an additional layer of the troposphere, causing an additional delay in the propagation of the signal.

**Figure 7.**(

**a**) The 21 correlator outputs of raw reflected signal over time. (

**b**) All the complex correlators were filtered with a prolongation of coherent integration over 500 ms as described in Section 2.3.2. It can be seen that the maximum correlator is around the 10th correlator. (

**c**) Phasor diagram of the strongest raw correlator output for the reflected signal of satellite G02 over time. (

**d**) The signal was filtered with a prolongation of coherent integration over 500 ms as described in Section 2.3.2. The loss of amplitude at the end of the flyover is consistent with the transition period between water and land.

**Figure 8.**(

**a**) Lever arm correction computed along the D1–D2 profile using the inertial unit data and the relative position of both antennas. (

**b**) Tropospheric delay corrections along the D1–D2 profile as described in Section 2.3.3. The tropospheric delays are consistent with GNSS satellite elevation and aircraft altitude.

**Figure 9.**(

**a**) Unwrapped phase for each satellite in view during D1–D2 flyover. (

**b**) Aircraft heights for D1–D2 flyover obtained from GNSS RTK positioning. The unwrapped phases are consistent with the aircraft height which is what we expect since the elongation of the reflected path depends on ĥ from Equation (1).

**Figure 10.**(

**a**) Residuals measurement for flyover D1–D2, with aircraft altitude of approximately 300 feet. The residuals are computed from the right-hand side of Equation (19) using the ambiguity integer values estimation from Equation (20). (

**b**) This time, ambiguities were fixed by subtracting 1 from the estimated ambiguity value of satellite G06, the residuals are now consistent with the satellite elevation (G06 is the satellite with the lowest elevation).

**Figure 11.**(

**a**) Least-squares estimated reflection surface heights along flyover D1–D2. The black cross represents the mean buoy height and is timed when the aircraft is closest to the buoy. (

**b**) Least-squares estimated reflection surface heights along flyover D3–D4. The black cross represents the mean buoy height and is timed when the aircraft is closest to the buoy. The orange cross represents the value of each cross flyover at the intersection.

**Figure 12.**Slope comparison of the height values obtained from least-squares estimation for signal GPS L1 C/A, and a local altimetric conversion surface RAF20. The slopes for the estimated values are 7 mm/km and 8.4 mm/km for RAF20.

**Figure 13.**(

**a**) Estimated heights for D1–D2 flyover at 300 ft using GPS signals L1 and L5, and GALILEO signals E1 and E5. Heights computed from L1, E1 and E5 are highly consistent with each other and with the buoy measurements with less than 1 cm of error. Values estimated from L5 are shifted, but the estimation is performed using the only two satellites in sight. (

**b**) Estimated heights for the D3–D4 flyover. Values are still consistent between L1, E1 and E5 and the buoy measurement is approximately 1 cm less.

**Figure 14.**Simulations of the optical path difference between a reflection on a flat earth model and an ellipsoidal model. For a 2000 ft flight with a satellite at 20° of elevation, the optical path difference is 0.1503 m.

**Figure 15.**(

**a**) GNSS buoy measurements of the reflection surface heights from RTK positioning. Colored areas represent the aircraft’s flyover times at different altitudes. (

**b**) Estimated heights for D1–D2 flyover and L1 signal at 300 ft. Buoy measurement and D3–D4 flyover measurement are, respectively, represented by the black and orange cross. (

**c**) Estimated heights for D1–D2 flyover and L1 signal at 500 ft. The beginning of the profile match with a height variation observed on the buoy measurement. (

**d**) Estimated heights for D1–D2 flyover and L1 signal at 1000 ft. Values are highly consistent with the 300 ft estimated values. (

**e**) Estimated heights for D1–D2 flyover and L1 signal at 2000 ft. Even if the values are consistent with other profiles, an unidentified stall can be observed around 16:39:35.

**Table 1.**Difference between observed values for the two profiles above Carcans-Hourtin Lake with an aircraft at 300, 500, 1000 and 2000 ft. Values in the table are discrepancies between the comparative method and the reflectometry estimate. Buoy and cross-profile discrepancies are calculated using a 2 s window around the closest point to the aircraft.

Buoy—Reflectometry | Cross-Profile—Reflectometry | RAF20 Geoid Slope—Reflectometry Slope | |
---|---|---|---|

300 ft—D1D2 | 3.6 mm | 7.8 mm | 1.3 mm/km |

300 ft—D3D4 | −20.9 mm | −7.8 mm | 3 mm/km |

500 ft—D1D2 | 14.2 mm | 4 mm/km | |

1000 ft—D1D2 | 11.9 mm | 1.1 mm/km | |

2000 ft—D1D2 | −8.2 mm | −1.9 mm/km |

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## Share and Cite

**MDPI and ACS Style**

Varais, N.; Verdun, J.; Cali, J.; Lestarquit, L.
Lake Altimetry Using Long Coherent Integration Carrier Phase Measurements from Airborne GNSS Reflectometry. *Remote Sens.* **2023**, *15*, 4192.
https://doi.org/10.3390/rs15174192

**AMA Style**

Varais N, Verdun J, Cali J, Lestarquit L.
Lake Altimetry Using Long Coherent Integration Carrier Phase Measurements from Airborne GNSS Reflectometry. *Remote Sensing*. 2023; 15(17):4192.
https://doi.org/10.3390/rs15174192

**Chicago/Turabian Style**

Varais, Nolan, Jérôme Verdun, José Cali, and Laurent Lestarquit.
2023. "Lake Altimetry Using Long Coherent Integration Carrier Phase Measurements from Airborne GNSS Reflectometry" *Remote Sensing* 15, no. 17: 4192.
https://doi.org/10.3390/rs15174192