A traditional but effective method is tie line levelling. Comparing the flight line data with tie line data at the same survey point, the operators correct the crossover point based on the differences between the tie lines and flight lines. The accuracy of tie line levelling mainly relies on whether the differences match the levelling errors. Many geophysicists have proposed algorithms to improve matching precision (Foster et al., 1970; Yarger et al., 1978; Bandy et al., 1990; Mauring et al., 2002; Srimanee et al., 2020). However, the flight line data and the tie line data are flown in different aircraft configurations and external environments. Moreover, airborne electromagnetic data are relatively sensitive to altitude compared with airborne magnetic data. The levelling error is not the only reason for the accumulation of the differences in the crossover point. It is hard to separate the levelling errors from the differences. Furthermore, virtual tie lines (Huang and Fraser, 1999; Fan et al., 2016; Zhang et al., 2018) are skilfully constructed to level geophysical data instead of tie lines.

From the definition of levelling error, the inconsistent data level in flight lines is attributed to levelling errors that are not continuous between adjacent flight lines. However, as survey area geology changes quite slowly, it is reasonable to assume that the natural survey points are correlated in a certain region. The levelling errors can then be derived from line to line based on the differences between adjacent flight lines (Green, 2003; Huang, 2008; Zhu et al., 2020). Moreover, geophysicists skilfully constructed one-dimensional (1D) flight line windows and two-dimensional (2D) planar windows, considering the statistical parameters difference between the flight line data and region data. The levelling errors are calculated from point to point by matching the difference between the 1D and 2D window values (Mauring and Kihle, 2006; Beiki et al., 2010; Ishihara, 2015). Moreover, the geophysical data can be micro-levelled using the statistical approach in a designated moving window (Davydenko and Grayver, 2014; Groune et al., 2018).

The line-to-line and point-to-point methods only level small amounts of data in each loop that can be deemed as block processing methods. A common problem is cumulative inaccuracies that develop when the levelled data are used to level in the next loop. In contrast, global processing methods operate the entire region data instead of only part data in every iteration. The global processing methods available mainly focus on airborne magnetic data levelling based on the separated long-wavelength components (Urquhart, 1988; Nelson, 1994; Luo et al., 2012; White and Beamish, 2015; Zhang et al., 2021). The directional filters are designed to level the geophysical data (Minty, 1991; Ferraccioli et al., 1998; Siemon, 2009; Gao et al., 2021).

In summary, the conventional block processing methods would inevitably transfer errors. The global processing methods mainly focus on levelling airborne magnetic data. As with the levelling error properties discussed above, the levelling error is an additive drift, presented as the inconsistent data level among the flight lines. These inconsistencies are affected by a variety of factors that make it hard to construct a mechanism model for levelling errors. However, the striping errors would increase the total variation of the measuring area. Total variational theory inspires us to level the data by using an energy functional model. The proposed method is described below.

The theoretical basis of most levelling techniques is that the geophysical field is continuous. The observed data tend to show significant correlations with their neighbouring points. However, the levelling errors are not continuous between adjacent flight lines (Huang, 2008). When the assumption is valid, the geophysical data with levelling errors will have a large variation amplitude compared with nature geophysical data. It is thus advisable to estimate the levelling error components based on total variation model.

We simply deemed the survey data consists of two parts: 1S(i,j)=E(i,j)+D(i,j), where S(i,j) is the ith survey data in the jth flight line, E(i,j) is the levelling error component of the survey point, and D(i,j) is the levelled data. Here the survey data are considered a 2D function in the entire region Ω, while (i,j) defines the ith survey data in the jth flight line.

Rudin et al. (1992) introduced total variation norm and proposed Rudin–Osher–Fatemi (ROF) total variation model, which has been widely used in image-denoising applications. Based on total variational model, we can estimate the levelling error components by constructing an energy functional, 2F(D)=∫ΩS-D2+λTV(D), where λ is the regularisation coefficient that quantifies the degree of smoothness. Based on the multiscale hierarchical decomposition theory (Tadmor et al., 2003), we can determine the regularisation coefficient by the spatially adaptive multi-scale model (Zhang et al., 2022). TV(D) is the total variation of the estimated solution D expressed as follows: 3TV(D)=∫Ω∇D=∫ΩdDdx2+dDdy2dxdy. In the total variational model, ∫ΩS-D2 is a fidelity term which ensures the similarity between the original data S and the clear data D. In Eq. (2), L-2 norm is selected to build the fidelity term due to its excellent edge-preserving performance.

TV(D) serves as the regularisation term, aimed at penalising undesirable damage in data. The regularisation term is the total variation of the estimated solution. It means that sparse gradient domain constraints are imposed along the horizontal and vertical directions. Combined with prior information, the levelling error components can be computed by minimising the total variation model in Eq. (2). The total variational model has been applied to the striping noise removal (Zhang and Zhang, 2016; Liu et al., 2019).

Accurately extracting levelling errors requires us to combine the total variation model with as much prior information about the levelling error as possible. Levelling errors present a significant directional property and show up as a striping pattern along the flight direction. We can the design an anisotropic Gabor filter with the principal axis directed by the levelling error.

In geophysical exploration, the levelling error model should estimate the intensity at each survey point, which can be modelled as follows: 4E=α⋅G, where α is the weight coefficient that describes the intensities of levelling error, while G is the noise pattern. We model stripes as an anisotropic Gaussian function defined by 5G(i,j)=e-xi2σi2-yj2σj2xi=icos⁡θ+jsin⁡θyj=-isin⁡θ+jcos⁡θ. In Eq. (5), (i,j) defines the location of the ith survey data in the jth flight line. θ represents the normal's orientation to the Gabor function's parallel stripes, that is, the flight line direction. σi and σj are the Gaussian envelope's standard deviation in the x direction and flight line direction, respectively.

The pattern of levelling error is mainly described by the Gaussian function. We can obtain the parameters in Eq. (5) combined with prior shape information. The weight coefficient defines the levelling error intensity that is necessary to solve from the overall view.

When we guide the total variational model levelling by levelling error model, the structured variational model provides an accurate geophysical processing design. We obtain the following objective function: 6F(α,λ)=∫Ωα⋅G2+λTV(S-α⋅G). Equation (6) contains two coefficients, α and λ, to balance the fidelity term and regularisation term. It is permitted to reasonably merge the two coefficients and express Eq. (6) as follows: 7F(α)=∫Ωα⋅G2+TV(S-α⋅G). We then use alternating direction method of multipliers (ADMM) to solve non-convex optimisation problems. ADMM converts the original problem into subproblems with closed-form solutions. It is an effective approach in a sequence of iterative sub-optimisations (Bertsekas, 1982).

While the levelling error intensity for each survey point is solved, we complete the data levelling using Eqs. (1) and (4) under the structured variational model.

In exploration field, airborne geophysical measurement data contains a large amount of noise due to atmospheric flow, lightning, aircraft vibration, and unstable speed factors (Yin et al., 2015). In addition to levelling errors, different kinds of noises damage the measurement data simultaneously. Here, we simply assume the measurement data contains levelling errors and Gaussian white noise. In that case, the proposed levelling method has an obvious advantage compared with other levelling methods. The proposed method constructs an energy functional as Eq. (2). For other noise, we can consider the denoising problem under the framework similarly. The noise model in Eq. (4) describes the noise distribution and geometrical structure. When we try to remove several kinds of several kinds of disturbances, Eq. (4) is extended as follows: 8E=∑i=1nαi⋅Gi, where n is the number of noise type.

For Gaussian white noise, it can be obtained by convolving a Dirac function with a sample of white Gaussian noise. The proposed method simultaneously removes the levelling errors and Gaussian white noise in one step processing that helps to improve electromagnetic exploration accuracy.

Thus, there are three evident advantages in proposed levelling method.

The total variation model designs the total energy as a constraint condition and obtains the constrained gradient minimisation using the regularisation coefficient. When we use total variational model to deal with survey area data, it can reasonably remove the levelling errors that increase the gradient of survey area data.

We also tested the levelling method on the apparent conductivity data provided by Ontario Airborne Geophysical Surveys. The dataset used in the paper is formed by 70 flight lines named L310–L1000 as a part of Geophysical Data Set 1076 measured in the surveys (Ontario Geological Survey, 2014). Geotech Limited carried out a helicopter-borne combined aeromagnetic and electromagnetic survey for the Ministry of Northern Development and Mines in 2014 in the Nestor Falls area in northwestern Ontario. Based on the resistivity depth imaging (RDI) technique (Meju, 1998), Geotech Limited converted the EM profile decay data into an equivalent resistivity versus depth cross-section via deconvolution of the measured TEM data. Data compilation and processing were carried out using Geosoft^® OASIS montajTM and programs proprietary to Geotech Ltd (Ontario Geological Survey 2014).

Figure 4 presents the apparent conductivity data before and after levelling processing. As Fig. 4a shows, there are only slight striped errors along the flight line direction in the apparent conductivity data. While the electromagnetic data are transformed into conductivity parameters, the altitude sensitivity is strongly weakened (Fraser, 1972; Huang and Fraser, 1999).

We then applied the structured variational model to the apparent conductivity data and got the levelling results as Fig. 4b shown. In the analysis of the data, it is assumed that the levelling error is the only noise source. Figure 5 illustrates the transient data to compare the results in greater detail. Two parts of the data are plotted corresponding to the black rectangle in Fig. 4. The first part includes 10 flight line datasets as shown in Fig. 5a. We then selected a smaller data scope (0–2.4×10-3 S m-1) to locally enlarge and draw the data in Fig. 5b. Figure 5c and d are drawn using the five flight line dataset that are correspond to the dashed–dotted black rectangle in Fig. 4.