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Gravity surveys are an important research topic in geophysics and geodynamics. This paper investigates a method for high accuracy large scale gravity anomaly data reconstruction. Based on the airborne gravimetry technology, a flight test was carried out in China with the strap-down airborne gravimeter (SGA-WZ) developed by the Laboratory of Inertial Technology of the National University of Defense Technology. Taking into account the sparsity of airborne gravimetry by the discrete Fourier transform (DFT), this paper proposes a method for gravity anomaly data reconstruction using the theory of compressed sensing (CS). The gravity anomaly data reconstruction is an ill-posed inverse problem, which can be transformed into a sparse optimization problem. This paper uses the zero-norm as the objective function and presents a greedy algorithm called Orthogonal Matching Pursuit (OMP) to solve the corresponding minimization problem. The test results have revealed that the compressed sampling rate is approximately 14%, the standard deviation of the reconstruction error by OMP is 0.03 mGal and the signal-to-noise ratio (SNR) is 56.48 dB. In contrast, the standard deviation of the reconstruction error by the existing nearest-interpolation method (NIPM) is 0.15 mGal and the SNR is 42.29 dB. These results have shown that the OMP algorithm can reconstruct the gravity anomaly data with higher accuracy and fewer measurements.

The Earth's gravity field is a fundamental physical field, which reflects the distribution, motion and variety of the Earth's interior matter. The gravity has connections with all the physical events on Earth and in its near space, and thus provides the basic information to support research on many subjects. Gravity surveys can support fundamental geophysical investigations, which are beneficial to determine the density of the Earth's interior matter and help explain many physical phenomena of the Earth. Meanwhile, gravity surveys are also significant in the exploitation of mineral resources and modern military science,

Airborne gravimetry is a method of determining the Earth's gravity by using instruments on board an aircraft such as accelerometers, global navigation satellite systems (GNSS), altimeters, and attitude sensors [

Airborne gravimetry is essentially a discrete digital sampling method. The theoretical foundation of discrete digital sampling on continuous-time band-limited signals was developed by Nyquist and Shannon [

Based on airborne gravimetry technology, a flight test was carried out in China with the strap-down airborne gravimeter (SGA-WZ) developed by the Laboratory of Inertial Technology of the National University of Defense Technology [

Airborne gravimetry can be classified as airborne scalar gravimetry, airborne vector gravimetry and airborne gradient gravimetry. Our work in this paper studies airborne scalar gravimetry (we study vertical component of the vector only).

An object's gravity is the composition of forces of gravitation caused by the Earth and other celestial bodies and the inertial centrifugal force caused by the Earth's rotation. The non-uniform distribution of the density of the Earth's interior matter makes the gravity vary with the position. In gravity prospecting, the gravity variations caused by the non-uniform density distribution of the Earth's interior rocks and minerals are called gravity anomalies. In fact, the airborne gravity measurements include two parts: the gravity anomaly (denoted as the free air anomaly here) and the normal gravity (which is the reference gravity field of a conventional ellipsoid). Therefore, the gravity anomaly can be expressed as:

The principle of strap-down airborne scalar gravimetry is based on Newton's equation of motion in the gravitational field of the Earth, utilizing the principle of relative gravity measurement. First, we must use a terrestrial gravimeter to connect national gravity and a point on the parking apron to obtain its gravity and take it as the gravity reference point. Before the aircraft takes off, static gravimeter data must be recorded so that gravity observations in the air can be associated with the gravity reference point on the parking apron. Then, the strap-down airborne scalar gravimetry model can be written as:

Obviously, _{D}_{D}_{b}_{E}

Compressed sensing (CS) is an exciting, rapidly growing field that has attracted considerable attention in fields as diverse as electrical engineering, applied mathematics, statistics, sensor technology and computer science. CS offers a framework for simultaneous sensing and compression of finite dimensional signals. Quite surprisingly, it predicts that sparse high-dimensional signals can be recovered from highly incomplete measurements by using efficient algorithms. CS also holds promise for increasing resolution by exploiting the signal structure. Especially, reducing the sampling rate or increasing resolution in airborne gravimetry can improve survey efficiency, increase data transfer rate and improve data quality.

Let ^{n}_{1},_{2},⋯,_{m}^{T}^{m}_{1},_{2},⋯,_{m}^{T}^{m}^{×}^{n}_{1},_{2},⋯_{m}^{T}^{m}^{n}^{×}^{n}_{i}

When _{0} = |{_{i}_{0} ≤

Let ^{−1} ∈ ^{m}^{×}^{n}

OMP is a greedy pursuit algorithm. Given that ^{−1} = [_{1},_{2},⋯,_{n}_{i}^{m}

To identify the signal

Given the matrix

Step 1: Initialize the residual _{0} = _{0} = Ø and the iteration counter

Step 2: Find the index _{t}

If the maximum occurs for multiple indices, break the tie deterministically.

Step 3: Augment the index set and the matrix of chosen atoms: _{t}_{t}_{−1}∪{_{t}_{t}_{t}_{−1} _{λt}_{0} is an empty matrix.

Step 4: Solve a least squares problem to obtain a new signal estimate:

Step 5: Calculate the new residual:

It is important to note that the residual _{t}_{t}_{t}_{−1} is nonzero, the algorithm selects a new atom at iteration _{t}_{t}

Step 6: Increment

Step 7: The estimate _{m}_{t}_{m}

Step 8: The estimate for the unknown signal ^{−1}

The strap-down airborne gravimeter and the flight test results are presented in this section. By comparing NIPM for the gravity anomaly data reconstruction, the superiority of the OMP algorithm will be shown in this section.

The strap-down airborne scalar gravimeter called SGA-WZ mentioned in this paper is the first system with this type in China. It was developed by the Laboratory of Inertial Technology of the National University of Defense Technology [

The flight tests were carried out in Shandong Province using SGA-WZ from April 2010 to May 2010. The hardware was installed in the aircraft six days before the flight tests. Two GNSS ground stations were located near the airport where the aircraft took off and landed. GNSS receivers installed on the ground and aircraft can be used to determine the vehicle position, velocity, and acceleration. The strap-down airborne scalar gravimeter onboard a Cessna 208 aircraft was used to collect the data.

The pilots controlled the aircraft with the autopilot, and the test was implemented in days with good weather to minimize the effects of air turbulence. The average flight altitude was approximately 400 m above sea level with a fluctuation of 20 m. The average speed during the flight was 60 m/s. The sampling rate of raw SINS readings was 100 Hz and 2 Hz for the GNSS sampling rate. After being installed in the aircraft, SGA-WZ worked all day for over a month. In the whole flight test campaign there were eight flights. The first and second flights were repeated lines that flew along the same trajectory to test the repeatability of the system, and the valid length of each repeated profile was about 100 km. The other six flights left were grid flights consisting of three flights of survey lines and three flights of control lines. The spacing between survey lines was about 2 km and the spacing between control lines was about 9 km.

As previously mentioned, the gravity anomaly was determined from the difference between the specific force and the vehicle acceleration using

The first and second flights were repeated lines surveys denoted by F401 and F402, respectively. Each of them consisted of six lines.

Based on the theory of CS, this study analyzes the sparsity of airborne gravimetry by DFT.

To evaluate the OMP algorithm performance and the gravity anomaly data reconstruction precision, we now define the performance indices with standard deviation and SNR. SNR can be calculated by:

In this study, the compressed measurement number was set as

For comparison purposes, we also made the gravity anomaly data reconstruction with NIPM, which is one of the commonly used methodologies for data reconstruction.

This paper has presented a new investigation on a method for high accuracy large scale gravity anomaly data reconstruction. Based on the airborne gravimetry technology, a flight test was carried out in China using a custom designed strap-down airborne gravimeter. Taking into account the sparsity of airborne gravimetry by DFT, this paper proposes a method for gravity anomaly data reconstruction using CS theory. The test results have revealed that the compressed sampling rate is approximately 14%, the standard deviation of the reconstruction error by OMP is 0.03 mGal and SNR is 56.48 dB. In contrast, the standard deviation of the reconstruction error by NIPM is 0.15 mGal and SNR is 42.29 dB. These results have shown that OMP can reconstruct the gravity anomaly data with higher accuracy and fewer measurements. In future investigations, the following considerations should be taken into account:

The boundary effect exists in the result of gravity anomaly data reconstruction with OMP. Although we can ignore the boundary data, that will result in a waste of data and increase the survey costs.

The discrete Fourier transform was effective for the 1-D gravity anomaly data sparse transform in this study. In consideration of the 2-D gravity anomaly data reconstruction, future work can pay attention to the Curvelet transform.

This work was supported by the National High Technology Research and Development Program of China under Grant No. SS2013AA060402.

In this paper, Yapeng Yang analyzed the basic principle of Compressed Sensing and Airborne Gravimetry, implemented the flight test, compiled program codes and wrote the paper. Meiping Wu analyzed the basic principle of Airborne Gravimetry and implemented the flight test. Jinling Wang analyzed test data and modified English errors. Kaidong Zhang developed the SGA-WZ and implemented the flight test. Juliang Cao developed the SGA-WZ and implemented the flight test. Shaokun Cai implemented the flight test and analyzed test data.

The authors declare no conflict of interest.

Appearance of the SGA-WZ.

Cessna 208 fixed-wing small aircraft.

Grid flight lines.

Gravity anomaly before low-pass filtering.

Gravity anomaly after low-pass filtering.

Gravity anomaly curves of F401.

Gravity anomaly curves of F402.

Amplitude spectrum curves of F401.

Amplitude spectrum curves of F402.

Reconstruction result of OMP in F401.

Reconstruction error of OMP in F401.

Reconstruction result of NIPM in F401.

Reconstruction error of NIPM in F401.

The comparison of the algorithm performance for F401.

| |||||
---|---|---|---|---|---|

OMP | NIPM | OMP | NIPM | ||

Line 1 | 3050 | 0.03 | 0.15 | 56.91 | 42.06 |

Line 2 | 3150 | 0.03 | 0.16 | 55.76 | 41.03 |

Line 3 | 2979 | 0.02 | 0.14 | 58.15 | 42.81 |

Line 4 | 3120 | 0.03 | 0.14 | 56.65 | 43.00 |

Line 5 | 2964 | 0.03 | 0.16 | 57.08 | 42.23 |

Line 6 | 3170 | 0.04 | 0.15 | 54.31 | 42.60 |

Average | 3072 |