Keywords

1 Introduction

In airborne gravimetry, measurements of the Earth’s gravity are traditionally collected by means of a gravimeter based on a gyro-stabilized gimbal platform. The current trend is the use of strapdown airborne gravimeters based on a navigation-grade strapdown inertial navigation system or inertial measurement unit (IMU) (Stepanov and Peshekhonov 2022). The principle of measurements of such a system is based on measuring the specific force and angular velocity vector by the inertial sensors (accelerometer and gyroscope triads) of the gravimeter’s IMU. The gravimeter is also supplemented by a thermal stabilization system and global navigation satellite system (GNSS) receivers (one is onboard the aircraft and the others are the ground-based reference stations).

The well-known advantages of strapdown airborne gravimeters are light weight, small size, and low power consumption (Jensen and Forsberg 2018), which allows to install them in a small aircraft or drone. In addition, there are almost no technical limitations for using strapdown systems, in contrast to the traditional airborne gravimeters, in harsh dynamic conditions (draped flights over terrain of any complexity).

However, postprocessing of raw gravimeter data becomes more challenging in the case of strapdown gravimetry because the aircraft accelerations (during motion and manoeuvring) affect directly on the IMU inertial sensors’ measurements. Moreover, one needs to take into account the systematic errors of the IMU inertial sensors (bias, drifts, scale factor errors, etc.) in order to obtain reliable results (Becker 2016; Jensen and Forsberg 2018).

In 2020–2022, Lomonosov Moscow State University has developed the postprocessing methodology and algorithms for strapdown airborne gravimetry (Vyazmin and Golovan 2023) and at the moment completes developing the postprocessing software package. This work is on the base of our experience in developing (and supporting) postprocessing software for the GT-2A airborne gravimeter (manufactured by Gravimetric Technologies, LLC, Russia) (Parusnikov et al. 2008; Bolotin and Golovan 2013).

The developed strapdown airborne gravimetry methodology and algorithms were tested in a number of airborne gravimetry campaigns with using state-of-the-art strapdown gravimeters (manufactured by iMAR GmbH and a domestic company) and various carriers – fixed-wing aircrafts (An-30, An-3T, Cessna 208 B, and others) and helicopter-type unmanned aerial vehicles (UAVs) (Vyazmin and Golovan 2023).

In the paper, we briefly outline the key stages of the postprocessing methodology and present the numerical results from two airborne gravimetry campaigns carried out by a Russian geophysical company (Aerogeophysica JSC) using an aircraft (An-3T) flown in the draped mode and UAV (BAS-200) flown at a constant altitude.

2 Postprocessing Methodology and Algorithms

Postprocessing strapdown airborne gravimeter data is different from that developed for the traditional airborne gravimeters. First, in strapdown gravimetry, one has to process raw data (IMU inertial sensors’ measurements) recorded at a high sampling rate (300–400 Hz). Second, installing a strapdown gravimeter in a small aircraft or drone leads to higher angular velocities and accelerations during the flight than in the case of large-size aircrafts traditionally used in airborne gravimetry. This means that the systematic errors in the IMU inertial sensors’ readings must be determined as accurately as possible. For example, when flying in the draped mode, the aircraft’s roll and pitch angles can reach 30° and, hence, the horizontal accelerometer biases will noticeably affect the gravity disturbance estimates. For instance, the 10 mGal bias in one of the horizontal accelerometers will produce a 5 mGal error in the gravity estimate in the case of 30° rolls.

Third, the IMU initial alignment procedure (determination of the IMU attitude at the aircraft’s stop before the flight) is strongly affected by external disturbances (caused by wind at the aerodrome, turning on the aircraft’s engines, work of the crew, etc.). Hence, the IMU initial alignment algorithm should be operable under such conditions (vibrations).

Below are the key stages of the developed methodology for postprocessing raw data from a strapdown airborne gravimeter (Vyazmin and Golovan 2023):

  1. 1.

    quality control of raw data collected by the gravimeter’s IMU inertial sensors and GNSS receivers (check for possible data losses, analysis of system performance indicators, etc.);

  2. 2.

    raw GNSS data processing (determining velocities and position of the onboard GNSS receiver in the differential mode using pseudorange, Doppler pseudorange rate and carrier phase multi-frequency measurements) (Golovan and Vavilova 2007);

  3. 3.

    IMU initial and final alignment (determining the IMU attitude at the aircraft stops before and after the flight and the accelerometers biases and linear drifts) (Vyazmin et al. 2023);

  4. 4.

    IMU-GNSS integration (using the horizontal channels only) (Vavilova et al. 2020);

  5. 5.

    estimation of the gravity disturbance (or all three components of the gravity vector) on the aircraft’s flight path.

2.1 Key Aspects of the Algorithms

The IMU initial alignment algorithm (stage 3 of the methodology) is based on approximating the specific force in the inertial frame and admits angular motion of the IMU.

IMU-GNSS integration is performed using the Kalman filtering and optimal smoothing technique (Kailath et al. 2000). At this stage, the estimates of the attitude errors and systematic errors of the IMU inertial sensors are obtained. The system’s state vector also includes the GNSS antenna offsets with respect to the IMU and time-synchronization errors in the IMU and GNSS data.

The gravity estimation on the flight path (stage 5 of the methodology) is performed using the basic equation of airborne gravimetry (equation of the gravimeter’s proof mass motion projected onto the vertical axis of the navigation geodetic (east, north, up) frame) (Bolotin and Golovan 2013). Other choices of the navigation frame are possible (Becker 2016).

By introducing the IMU accelerometer errors (residual biases b f and random noise q f) and residual attitude errors k e, k n (misalignments of the vertical axis of the computed geodetic frame), we rewrite the airborne gravimetry basic equation in the form (Bolotin and Golovan 2013):

$$ \Delta {\dot{v}}_{up}=-\updelta g+{k}_e{f}_n-{k}_n{f}_e+{\textbf{n}}^T\left({\textbf{b}}_f+{\textbf{q}}_f\right), $$
(1)

where Δv up is the IMU vertical velocity error, n is the unit vector of the vertical axis of the geodetic frame expressed in the IMU frame, and f e, f n are the accelerometers readings projected onto the east and north directions. Additional error parameters (time-synchronization errors, GNSS antenna offsets, and the scale factor error of the vertical accelerometer) are omitted in Eq. (1) for simplicity sake. The estimate of the gravity disturbance δg and IMU systematic errors on the flight path are provided by the Kalman filter and optimal smoothing. The algorithm for estimating all three components of the gravity vector is presented in (Vyazmin and Golovan 2023).

3 Numerical Results

Below are the numerical results from two strapdown airborne gravimetry surveys carried out in 2021–2022 by Aerogeophysica JSC (Russia) for geophysical applications. A state-of-the-art strapdown gravimeter (Fig. 1) was used in both surveys. Postprocessing of the gravimeter raw data was performed by Lomonosov Moscow State University.

3.1 Aircraft-Based Survey

The survey was based on the Antonov An-3T aircraft (Fig. 2) flown in the drape mode over the mountainous area. The flight altitude above the ellipsoid varied from 300 m up to 1,200 m during the flights. The aircraft’s average speed at a flight line was 40 m/s. The flights were characterized by harsh dynamics conditions leading to roll and pitch angles of up to 30° at flight lines and extreme vertical accelerations of up to 2.5 g, which appears to be the first case in airborne gravimetry (Fig. 3). Three GNSS receivers from JAVAD (one is onboard and two placed on the ground) were used with the baseline lengths reaching 150 and 200 km.

Fig. 1
figure 1

A strapdown airborne gravimeter on board the aircraft

Full size image
Fig. 2
figure 2

The Antonov An-3T aircraft at the aerodrome

Full size image

Postprocessing of raw GNSS and IMU data was performed using the developed algorithms and software. The gravity estimation results and flight altitudes are presented in Fig. 4. The estimation accuracy from the cross-over statistics (without levelling) is 0.5 (mean) and 2.3 mGal (RMS error). The half-wavelength spatial resolution of the gravity estimates is 2.4 km (on the average), which was determined from the average flight speed and the Kalman filter cutoff frequency (equal to 1/120 Hz). The RMS of the flight height discrepancies at the intersection points is about 7 m.

Fig. 3
figure 3

The gravimeter’s IMU vertical accelerometer readings, m/s2

Full size image
Fig. 4
figure 4

The results of the aircraft-based survey: (a) the gravity disturbance estimates, mGal; (b) flight altitudes, m

Full size image

3.2 UAV-Based Survey

The survey flight was carried out on 19.08.2022 using a large-size helicopter-type UAV (BAS-200) shown in Fig. 5. Four repeat lines were flown at the altitudes of 340 m (the L1 and L2 lines) and 420 m (the L3 and L4 lines) above the WGS-84 ellipsoid. The line length was about 20 km. The flight speed was 25 m/s.

Fig. 5
figure 5

The BAS-200 UAV at the aerodrome

Full size image

The flights were characterized by strong vibrations (the STD of the vertical velocity at a line is 0.2 m/s), which were damped by shock absorbers inside a box containing the gravimeter (the black box in Fig. 5).

The gravity estimates at four repeat lines are shown in Fig. 6. The STD is 1.12 mGal for the difference between the L1 and L2 lines and 0.66 mGal for the difference between the L3 and L4 lines. The STD for the difference between the gravity estimates at the L1–L4 lines and the averaged estimate is 0.61 mGal. The Kalman filter cutoff frequency is 1/100 Hz, which is equivalent to the half-wavelength spatial resolution of 1.25 km.

Fig. 6
figure 6

The gravity disturbance estimates at repeat flight lines, mGal

Full size image

4 Conclusions

A new methodology and algorithms for strapdown airborne gravimeter data postprocessing are outlined. The key stages of the methodology are the IMU-GNSS integration (using the horizontal channels) and gravity estimation on the flight path (using the IMU vertical channel and IMU-GNSS results, namely, the estimates of the IMU attitude angles). The algorithms from the both stages are based on the Kalman filtering and optimal smoothing technique.

The methodology’s algorithms were tested in a number of strapdown airborne gravimetry campaigns (80,000 line km in total) carried out in 2020–2022. State-of-the-art strapdown gravimeters (by iMAR and a domestic company) and various carriers (fixed-wing aircrafts and UAVs) were used in the campaigns. For the UAV-based survey (one of the first in Russia), the gravity estimation accuracy at the sub-mGal level was reached, which is promising for surveying with this type of a carrier. The 2-mGal accuracy was achieved for the draped flight survey (using the Antonov An-3T aircraft), which is a good result for the flights with extreme vertical accelerations up to 2.5 g. The latter appears to be the first case in airborne gravimetry.