# Bayesian approach for network adjustment for gravity survey campaign: methodology and model test

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## Abstract

The drift rate of relative gravimeters differs from time to time and from meter to meter. Furthermore, it is inefficient to estimate the drift rate by returning them frequently to the base station or stations with known gravity values during gravity survey campaigns for a large region. Unlike the conventional gravity adjustment procedure, which employs a linear drift model, we assumed that the variation of drift rate is a smooth function of lapsed time. Using this assumption, we proposed a new gravity data adjustment method by means of objective Bayesian statistical inference. Some hyper-parameters were used as trade-offs to balance the fitted residuals of gravity differences between station pairs and the smoothness of the temporal variation of the drift rate. We employed Akaike’s Bayesian information criterion (ABIC) to estimate these hyper-parameters. A comparison between results from applying the classical and the Bayesian adjustment methods to some simulated datasets showed that the new method is more robust and adaptive for solving problems caused by irregular nonlinear meter drift. The new adjustment method is capable of determining the time-varying drift rate function of any specific gravimeter and optimizing the weight constraints for every gravimeter used in a gravity survey. We also carried out an error analysis for the inverted gravity value at each station based on the marginal distribution. Finally, we used this approach to process actual gravity survey campaign data from an observation network in North China.

## Keywords

Adjustment of gravity data Objective Bayesian model Akaike’s Bayesian information criterion Gravity variation Gravity survey campaign Relative gravity instrument drift## 1 Introduction

Gravity survey campaign data generally refer to the gravity observed repeatedly at fixed stations with the same routes and similar time schedules. For each campaign, the less time needed to finish the entire loop of stations, the more acceptable the observation results are. During the campaign, it is assumed that the gravity does not change at the same location. A continental-scale gravity network has been established on the Chinese mainland. Gravity survey campaigns using this network require many observational teams working together in the scheduled period. The regional gravity variations detected by this terrestrial gravity network have been successfully used to study geoscience problems, including groundwater change (Kennedy et al. 2014), surface vertical deformation (Ballu et al. 2003), tectonic events (Lambert et al. 2006; Van Camp et al. 2016), and earthquakes (Chen et al. 1979, 2016; Kuo et al. 1993, 1999; Zhu et al. 2010).

In a time-lapse terrestrial gravity survey, the whole network is traveled with a fixed time interval (1/2 or 1 year) to measure gravity variation, which is at a scale of a few tens of microgals, at each station during the period between these two observations. The distance between two adjacent gravity stations is about several tens of kilometers, and road vehicles are used to travel between stations. The time interval between gravity measurements at two adjacent stations is about few hours. Realistically, only about ten measurements can be taken per day. For improving survey productivity, portable relative gravimeters are widely used.

Since their invention in the early 1960s, relative gravimeters have come to be used in terrestrial gravity surveys all over the world. These gravimeters mainly employ zero-length springs as the core gravitational sensor. Because the drift of the zero-length spring is typically low and can be predicted with a linear fitting in a short time period, the observations by the high-precision microgravity meters are processed by the technique of gravity network adjustment (Crossley et al. 2013). Currently, the absolute gravimeters (FG-5 and A10 made by Micro-G, Inc.) are the best alternative approach for the terrestrial gravity measurement. However, they are still much more expensive and more time-consuming to use in the field compared with relative gravimeters, although their accuracy is much higher. In contrast, relative gravimeters, which are equipped with spring sensors, are economical and portable, and they do not have specific environmental requirements. These reasons make them still irreplaceable for the foreseeable future.

Drift is one of the inherent features of all kinds of relative gravimeters. It is the phenomenon that the gravity measurement varies with the passage of time. The drift rate should be frequently estimated by repeating measurements at the base station or stations with known precise gravity values throughout the entire survey period. The estimation error of instrument drift rate will significantly affect the accuracy of network adjustment. In general, the gravity survey campaign refers to sets of in situ gravity measurements at a stable station at fixed time intervals year after year. The campaign time of a gravity survey often spans weeks or months. The drift rate cannot be modeled with a simple linear function if the campaign time is more than 24 h. The repeatability of ± 0.005 mGal is often requested for the high-precision microgravity and time-lapse gravity research. But a regional survey may only require a repeatability of ± 0.050 mGal. Since many factors can affect the drift rate of the gravimeter, such as the spring age, temperature inside the instrument, and transportation, using repeated observations is a practical approach for estimating the drift rate. However, the need to frequently return to the base station obviously reduces the efficiency of the survey. Moreover, loop errors at the base station, tares, offset, and outliers are also potential causes of errors in gravity data.

All relative gravimeters that use spring sensors, including Scintrex CG5/6, Locaost-Romberg model D/G, and ZLS Burris, have a complicated drift feature, especially for nonlinear long-term drift. Besides adding more absolute gravity observations and increasing the number of loops in the survey network, which are time-consuming and expensive, an economical and efficient way to obtain more accurate estimation of gravity is improving the network adjustment methods. Currently, most of gravity network adjustment methods cannot provide the adaptive drift rate inversion and the weight on strict optimization for multiple gravimeters. It is one of the key limitations on the improvement of gravity estimation accuracy.

Gravity survey campaigns have been carried out many years on the Chinese mainland. To accomplish a continental-scale terrestrial gravity survey, thousands of stations need to be observed every year by means of tens of relative gravimeters. The key aim of gravity surveys is to acquire high-accuracy gravity changes with a magnitude of tens of microgals. The network adjustment is essential and necessary for processing the relative gravity survey data, which calculates least-squares solutions (optimal estimates of gravity values) from the redundant observations based on the assumptions of linear drift. Some improvements of the classical adjustment method have been proposed for some particular applications (Hwang et al. 2002; Kennedy et al. 2016). However, how to fit nonlinear drift and balance the accuracy and productivity of gravity surveys is rarely discussed.

In this paper, we proposed a new approach for adjusting gravity survey data using objective Bayesian analysis and minimizing Akaike’s Bayesian information criterion (ABIC) (Sakamoto et al. 1988; Malinverno 2000; Mitsuhata 2004). The origin of the ABIC was in weather data analysis (Akaike 1977, 1980), and it has been widely used for the problems of estimating parameters of seismicity models (Ogata et al. 1993), 3D tomography (Inoue et al. 1990), and geodetic data inversion (Fukahata et al. 2004, 2008; Murata 1993; Nawa et al. 1997). For example, the BAYTAP-G is one representative example of the application of ABIC to analyzing the earth tide in a continuous relative gravity data at a fixed station (Tamura et al. 1991).

In this study, we rewrote the network adjustment equations by introducing new trade-off parameters that balance the residual of gravity survey campaign data and the drift rate of the relative gravimeter. This new method was tested with some synthetic datasets that were prepared with different drift models based on an actual gravity observation network. A comprehensive analysis of the fitting residuals and the accuracy of adjustment was carried out. The gravity survey route and time schedule at each station all came from actual gravity field work. We also performed a sensitive analysis for the uncertainty parameters of earth tide factor (TF) and atmospheric admittance (AA). At last, we evaluated this method by applying it to two observation datasets and compared the adjustment results to the absolute gravity observation.

## 2 Observations from the gravity survey campaigns and nonlinear drift of relative gravimeters

The accuracy of gravity variation data is a primary demand for many potential studies. It depends to a great degree on the gravity data processing method. In practice, the variation of the drift rate of relative gravimeters has been regarded as the primary error source.

Generally, to reduce the effects of drift rate variation and offsets, the relative gravity difference (GD) between adjacent station pairs is used instead of the actual gravity measurements obtained at each station as the input information for the adjustment. Another advantage of using GD is that, for station pairs with a relatively short elapsed time between observations, common-mode signals can be removed automatically (Kennedy et al. 2016).

### 2.1 Observation data

For ground-based relative gravity measurement, the observation equation can be written as (Torge 2001):

### 2.2 Drift

Figure 1b shows a continuous record of gravity reductions over 55 days by the same CG5 gravimeters, from which we can see the high-ordered drift. If a second-order drift model is employed to fit these 55 days of time-series data, the gravity residual is about ± 150 μGal and significantly related to the temporal variance of the meter drift rate. Moreover, if a segmental linearization is applied to fit the gravity records for each day, the residual gravity variation decreases to about ± 10 μGal with a few outliers, as shown Fig. 1c. The residual this time basically contains the internal noise of the instrument with obviously random characteristics. However, the range of estimated drift rate increased to 0.3 mGal/day. In this case, we can see that the gravimeter’s long-term and short-term drift rates obviously differ from the observation noise.

## 3 Methodology

In this section, we introduce a new adjustment method for estimating gravity values in the case of long-term gravity survey campaigns with multiple gravimeters. The method mainly focuses on the processing of temporal variation of the drift rate and stochastic noise. The input consists of relative GD data with noises of unknown variance and absolute gravity data with errors of known variance. We assume that the effects caused by the earth tide, ocean tide, air pressure, and polar motion are reduced before the network adjustment. Transient processes including precipitation, surface deformation, and temperature fluctuations at the stations during the campaign are not taken into account for the time being.

### 3.1 Basic definitions and notation

We adopt the following notation throughout this article. The numbers of observed stations, relative gravimeters, absolute gravity observations, and GD observations are \( N, P, K_{g} \), and \( K, \) respectively, and \( K > N \) based on the redundant observations. The number of drift rate estimates is \( M \).

#### 3.1.1 Data vector definitions

For the linear drift model, \( \varvec{v}_{i} = \left[ {v_{i} } \right] \) and when \( M \) time intervals are used for estimating the nonlinear drift model for gravimeter \( i \), \( \varvec{v}_{i} = \left[ {v_{i1} , v_{i2} , \ldots , v_{iM} } \right]^{\text{T}} . \)

If *P* gravimeters are used in the survey, the total number of GDs can be calculated by \( \mathop \sum \nolimits_{i = 1}^{P} {\text{size}}\left( {\varvec{y}_{i} } \right) = K_{1} + K_{2} + \cdots + K_{p} = K \),where \( K_{i} ,i = 1, 2, \ldots ,P, \) are the numbers of GD observations by each gravimeter.

#### 3.1.2 Matrix definitions

The matrices of relative gravity observation configurations, observed durations of each GD, smoothness for drift rate variations, and absolute gravity observation configuration are denoted by \( {\mathbf{A}},\varvec{ }{\mathbf{D}},\varvec{ }{\mathbf{B}}, \) and \( {\mathbf{G}} \), respectively.

The variances of observational noises, drift rate variations, and absolute gravity observations are denoted by \( \sigma^{2} , \sigma_{b}^{2} , \) and \( \sigma_{g}^{2} \), respectively. The weighted diagonal matrix of relative gravimeters, drift rate variation, and absolute gravity are denoted by \( {\mathbf{W}}, {\mathbf{W}}_{\varvec{b}} ,\varvec{ } \) and \( \varvec{ }{\mathbf{W}}_{\varvec{g}} \), respectively. In this case, \( {\mathbf{W}}_{g} \) with respect to the absolute gravity or basement station needs to be known before adjustment.

If there are \( P \) relative gravimeters in total, the variances of each observation value and the drift rate for the \( i \)-th gravimeter are, respectively, denoted by \( \sigma_{i}^{2} \) and \( \sigma_{bi}^{2} \), where \( i = 1, 2, \ldots ,P \). The corresponding matrices for the relative gravity observation configuration, observed durations of each GD, smoothness for drift rate variations, weights of relative gravimeters, weights of drift rate variations, and weights for absolute gravity are denoted by \( {\mathbf{A}}_{\varvec{i}} ,\varvec{ }{\mathbf{D}}_{i} ,\varvec{ }{\mathbf{B}}_{i} , {\mathbf{W}}_{i} ,\varvec{ } \) and \( {\mathbf{W}}_{{\varvec{b}i}} \), respectively. Please see Appendix A for a summary of this notation.

### 3.2 Gravity network adjustment with linear drift model

In gravity surveys, the GD data need to be adjusted because of the existence of observation noise and gravimeter drift. The absolute gravity observations can be used to establish the network datum and improve the precision of adjustment. Network adjustment, which combines the absolute gravity and redundant relative gravity observations, generally determines an optimal gravity value for each station by a least-squares criterion (Torge 1989).

#### 3.2.1 Basic equations

#### 3.2.2 Likelihood function

#### 3.2.3 Solution for gravity values

If all the covariance matrices are given, the least-squares network adjustment is equivalent to solving the following general linear equation system.

In summary, by investigating the traditional gravity observation network adjustment method in the framework of statistical multivariate Gaussian models, an approach for estimating the optimal weighted coefficient related to relative gravimeter by means of minimizing Eq. (17) with the MLE method is proposed naturally.

### 3.3 Bayesian gravity adjustment with nonlinear drift model

In the above model, the variation of gravimeter drift is assumed to be linear, that is, over the entire campaign period, the drift rate of each gravimeter is expressed as a single value \( \varvec{v}_{i} = v_{i} , i = 1,2, \ldots , P \). In fact, most relative gravimeters have drift rates that change with time during the gravity survey campaign. In this paper, we introduced a new nonlinear model to incorporate the nonlinear drift variation for a relative gravimeter.

#### 3.3.1 Smoothness prior for instrument drift

According to our knowledge of the drift rate of the spring-type gravimeter, we assume that the variation of the drift rate is smooth. In practice, the drift rate for one gravimeter can be regarded as a constant over a short time interval. If we divide the whole survey duration into *T* intervals, and denote the drift rate in the \( j \)-th period \( \left( {j = 1,2, \ldots ,T} \right) \) by \( v_{ij} \), the drift rate vector can be written in the form of \( \varvec{v}_{i} = \left[ {v_{i1} ,v_{i2} , \ldots ,v_{iT} } \right]^{\text{T}} \).

Hereby, we estimate the optimal value of \( \sigma_{b,i}^{2} \) for each gravimeter \( i \) by employing the empirical Bayesian method together with other trade-off parameters.

#### 3.3.2 Posterior distribution and ABIC

In this model, hyper-parameters \( \theta = \left\{ {\sigma_{1}^{2} ,\sigma_{2}^{2} , \ldots ,\sigma_{p}^{2} ,\sigma_{b1}^{2} ,\sigma_{b2}^{2} , \ldots ,\sigma_{bP}^{2} } \right\} \), including the \( 2P \) unknown variances, need to be estimated by minimizing the ABIC. If the number of hyper-parameters is small, the direct searching method with a multiplier step can be used for finding the minimum ABIC (Tamura et al. 1991). But in this case, the number of hyper-parameters depends on the number of gravimeters and days in the survey campaign. Thus, a high-dimensional space searching problem has to be faced. The nonlinear optimization methods need to be used for speeding up the minimization of the ABIC. Therefore, we employ the Nelder–Mead simplex nonlinear optimization method (Nelder et al. 1965; see, Wright 1996, for a review) in this study.

#### 3.3.3 Solution of gravity vector

## 4 Model tests for the simulated data

In this section, we use a practical network of gravimeters northwest of Beijing, China, and simulate the observation data according to a practical measurement scheme. The simulated gravity data can help us to understand the efficiency of Bayesian adjustment. Gaussian white noise and the instrument drift rate were generated to evaluate the capability of our adjustment theory. In the following tests, the observations between two adjacent stations are the input data for the adjustment. Five tests on simulation datasets were designed, including (1) investigating how to select the optimal weight constraints related to multiple gravimeters in the combined adjustment by means of MLE method, (2) verifying the validity of the Bayesian adjustment for the stochastic nonlinear drift rate model, (3) extending the Bayesian adjustment method to the case of multiple gravimeters, and (4) analyzing the sensitivity of input prior parameters in the adjustment.

In all the simulations, an actual observation route and time scheme were used. Such a survey campaign generally takes about 16 days. The intervals between the observations at two successive stations range from 0.3 to 4 h, with an average of about 0.7 h. In the simulation, the gravity values at all stations are known, the drift models are predefined, and noises with different features were added to the known gravity value at each station. In the synthetic model data, the simulated time spans between gravity observations cause meter drift and stochastic noise is added to the observation values at each station. The variation of random noise in the simulated time-lapse gravity data was controlled within a level similar to that of an actual CG5 m. We also simulated and added the gravity variation derived by the theory earth tide and air pressure related to specific times and locations based on recordings from an actual field gravity survey campaign. In total, 202 GD observation values, which form up to eight loops, were simulated for each gravimeter, and these were used to estimate the gravity values at the 91 stations.

### 4.1 Case 1: Comparison between optimally and equally weighted adjustment methods for linear drift model with instrument-dependent observational noise

Results from optimally weighted by the conventional linear drift adjustment model in Case 1

Gravimeter 1 | Gravimeter 2 | |||
---|---|---|---|---|

Drift rate (mGal/day) | SD of GD noise (μGal) | Drift rate (mGal/day) | SD of GD noise (μGal) | |

True values | 0.60 | 11.63 | 0.30 | 4.21 |

Adjustment results | 0.59 ± 0.02 | 11.57 | 0.30 ± 0.01 | 2.89 |

Difference between the adjustment and real gravity values at all stations for Case 1 (Unit: μGal)

Max | Min | SD | Mean | |
---|---|---|---|---|

Equally weighted | 14.08 | − 10.70 | 5.30 | 0.32 |

Optimally weighted | 6.85 | − 8.00 | 3.09 | − 0.92 |

### 4.2 Case 2: Nonlinear drift rate model

In this case, a duration of 16 days was used for the entire gravity survey campaign. Bayesian and classical adjustments were applied and compared. The smoothness of nonlinear drift rates can be estimated by the Bayesian approach (minimizing the ABIC) as shown in Fig. 6.

### 4.3 Case 3: Multi-gravimeter adjustment using Bayesian adjustment method

In this case, we simulated a dataset in which two gravimeters are used to test the Bayesian adjustment algorithm. Practically, double gravimeters are used in a gravity survey campaign to reduce the effects by the offset change, tares, outliers, and other random or system errors caused by instruments and environments. This simulation model with double gravimeters is designed for testing the potential advantages and efficiency of the Bayesian adjustment.

Comparison between results from two adjustments and true gravity values in the simulation for all the stations in Case 3 (Unit: μGal)

Adjustment | Max | Min | SD | Mean |
---|---|---|---|---|

CLS | 5.32 | − 21.71 | 4.72 | − 4.12 |

BAY | 3.30 | − 11.75 | 3.03 | − 2.92 |

### 4.4 Case 4: Sensitivity analysis for adjustment of input parameters

In cases 1–3, the uncertainties caused by the reductions of the tide and air loading are considered. To reduce such effects, two suitable parameters, generally called earth TF and AA, need to be used. In this section, we employ a sensitivity analysis approach to evaluate the potential effects caused by the uncertainty in a priori parameters.

Significant daily gravity changes are caused by tidal forces acting on the solid earth and oceans, and they range up to 0.3 mGal peak to peak (Hinze et al. 2013). The ocean tide loading is generally less than 10% of the solid earth tide. The value of TF generally varies over time and space owing to the fact that the actual earth is not a rigid body. The regional tidal factors can be estimated by the long-term continuous gravity observations, such as for 1 year or longer. The TF value can be approximated as 1.16 because its value is 1.1562 in the two-degree preliminary reference earth model (Dehant et al. 1999). An accurate TF value is often used to generate a time series of tidal gravity variation at a specific spatiotemporal location. But the TF value is likely to change, due to various factors, including the solid earth model, ocean tide loading model, and latitude dependence. The general variation is possibly about 2% (Hinze et al. 2013).

However, only the non-tidal gravity signal is of interest in a gravity survey campaign, and data are insufficient to estimate the TF. Therefore, the TF value needs be given as an a priori parameter before the gravity adjustment.

In addition, the atmosphere also affects gravity variation, which adds up to 10% of the TF over a wide frequency range, from minutes to seasonal periods (Crossley et al. 2013). A scalar admittance as a good approximation relates observed gravity to local pressure variation. A nominal AA value widely used is -0.3 μGal/hPa. The relationship includes the attraction of the atmosphere above the gravimeter and loading from air masses that elastically deform the crust (Warburton and Goodkind 1977). The AA value may change between − 0.27 and − 0.43 μGal/hPa using different local-zone models (Merriam 1992). A frequency-dependent admittance is also proposed and seen as a better approximation. The resulting difference between the two alternative approaches is not significant at the 0.5-μGal level (Hinderer et al. 2007). In general, the simple AA value is enough to remove the atmospheric effects, and the AA correction is straightforward.

*e*is the change of RMSE, p is the parameter value in the reference run, and ∆p is the parameter change.

First, 5% and 10% errors of TF were applied to tidal correction before the network adjustment. The periodic earth tide signal was mixed with the gravity time-series data. The smoothed feature was more similar to the nonlinear drift gravity variation, so it can be an error source that affects the accuracy of the adjustment. Second, 5 and 10% errors of AA were applied to the atmospheric effects correction before the network adjustment.

Sensitivity test of uncertainty in prior parameters in Case 4

+ 5% | + 10% | − 5% | − 10% | |
---|---|---|---|---|

TF RMSE (μGal) | 4.36 | 4.93 | 4.30 | 4.82 |

SS | 1.12 | 1.94 | 0.81 | 1.67 |

AA RMSE (μGal) | 4.13 | 4.13 | 4.13 | 4.14 |

SS | 0.0071 | 0.0078 | 0.017 | 0.016 |

But the SS values are also related to the noise level in the observed gravity data. If the noise feature can be estimated in the actual gravity survey campaign, we can select the appropriate method to estimate the a priori parameters by means of the sensitivity test. The corrections related to the controlled uncertainty of a priori parameters should be applied to reduce their effect on gravity before the gravity network adjustment processing.

## 5 Processing actual gravity survey data

In this section, we apply the Bayesian adjustment algorithm to two gravity observation datasets from North China.

### 5.1 Example 1

We applied both the Bayesian and the classical adjustment approaches to this dataset. The spatial differences between the linear and nonlinear drift rate models for adjustment are shown in the geometrical route map in Fig. 11a. Results showed the existence of gravity differences of a few tens of microgals. Significant differences between the two approaches appear in regions close to the edge of the network in Fig. 11a.

The fitting residuals for the actual GDs are plotted as Fig. 11b and 11c, and a histogram of the residuals is shown in Fig. 11d. It is obvious that the residuals from the Bayesian method are smaller and more like stochastic noise than those from the classical method. If using a simplistic linear drift model to estimate the gravity values, the residuals in Fig. 11c do not satisfy the assumptions of stochastic noise; rather, they show a systematic trend with time.

Thus, the existence of nonlinear drift in the actual gravity survey data has been demonstrated. Figure 11e shows the estimated temporal changes of the drift rates using the Bayesian adjustment approach with the related uncertainties. The variations of the drift rates are nonlinear and different from each other. The variation pattern of actual drift rates is similar to the simulated cases. Therefore, the Bayesian adjustment is effective and robust for the practical case.

### 5.2 Example 2

Absolute gravity stations

AG station | Absolute gravity value (μGal) | Measurement date | Duration (h) | Drops |
---|---|---|---|---|

ZJK | 1002.0 ± 2.1 | July 27, 2015 | 17 | 3400 |

DX | 7524.8 ± 1.7 | July 25, 2015 | 15 | 3000 |

BJT | 544.5 ± 1.5 | September 12, 2015 | 30 | 6000 |

Adjustment result when using different absolute gravity stations as known (unit: μGal)

AG stations used in calculation | AG station | BAY | CLS | ||
---|---|---|---|---|---|

Est. value | Diff | Est. value | Diff | ||

BJT | ZJK | 1001.7 ± 13.53 | − 0.3 | 1000.5 ± 14.33 | − 1.5 |

DX | 7574.4 ± 20.69 | 49.6 | 7585.7 ± 21.43 | 60.9 | |

BJT | 544.5 ± 1.54 | 0.0 | 544.5 ± 1.54 | 0.0 | |

ZJK | ZJK | 1002.0 ± 2.07 | 0 | 1002.0 ± 2.07 | 0 |

DX | 7574.7 ± 17.56 | 49.9 | 7587.1 ± 17.95 | 62.3 | |

BJT | 544.8 ± 13.60 | 0.3 | 546.0 ± 14.39 | 1.5 | |

DX | ZJK | 952.0 ± 17.49 | − 50.0 | 939.6 ± 17.91 | − 62.4 |

DX | 7524.8 ± 1.67 | 0 | 7524.8 ± 1.67 | 0 | |

BJT | 494.9 ± 20.67 | − 49.6 | 483.6 ± 21.44 | − 60.9 | |

BJT & DX | ZJK | 983.7 ± 11.39 | − 18.3 | 977.6 ± 11.91 | − 24.4 |

DX | 7525.1 ± 1.66 | 0.3 | 7525.1 ± 1.67 | 0.3 | |

BJT | 544.3 ± 1.54 | − 0.2 | 544.2 ± 1.54 | − 0.3 |

## 6 Conclusions

Relative gravimeters have been widely used in terrestrial gravity measurement. Designing and using a suitable model to estimate the nonlinear drift of relative gravimeters is a key point of gravity data adjustment. The drift and noise are the two main causes of errors and can be estimated by using redundant gravity observations, including adding more absolute gravity observations and increasing the number of loops in the survey network, with suitable data processing methods. To deal with the drift problem, we presented a novel approach for gravity campaign data processing, based on the Bayesian analysis theory. The basic assumption is that the variation of instrument drift rate is smooth over the entire campaign period for any relative gravity meters that are in good conditions. Such a prior constraint of the smoothness for the drift rate has been introduced to replace and to improve the linear drift model. On the basis of the “principle of parsimony,” appropriate unknown hyper-parameters were used to estimate the model using ABIC to balance the fitting residuals and the smoothness of the drift rate. Our Bayesian adjustment approach gives a good trade-off to avoid over-fitting problems.

In our results from model testing and error analysis, this new Bayesian approach for the network gravity adjustment was shown to be effective and straightforward for estimating the temporally complicated variation of meter drift rate. Particularly, if only linear drift exists in the relative gravity measurement, then there is little difference between this approach and the classical adjustment method. The testing results also showed that the Bayesian adjustment method is robust and adaptive for solving a similar problem for irregular drift rates. The new approach balances the survey productivity and the accuracy of gravity measurement and is especially powerful when returning to a base station takes more than 24 h in long-distance gravity surveys, when the gravimeter has a high-ordered drift rate, and when the environment or transportation is complex during the campaign.

In our experience, the computational load was affordable and could be handled by the majority of modern personal computers. A potential difficulty with this approach is that the dimensionality could increase dramatically if a large number of different gravimeters are used in the same survey campaign. The Nelder–Mead simplex nonlinear optimization method used in this work is more effective than other direct searching methods to optimize the joint likelihood function. Parallel computational techniques will be used to improve the computational efficiency in future work.

In summary, the objective Bayesian approach is suitable and powerful for reducing the effects of meter drift and stochastic noise in terrestrial gravity surveys performed in a large-scale and complicated network.

## Notes

### Acknowledgements

This study is partially supported by the National Key R&D Program of China (2017YFC1500503), National Natural Science Foundation of China (41774090), Science Foundation of Institute of Geophysics, CEA from the Ministry of Science and Technology of China (Nos. DQJB18B02; DQJB16A05), and China Earthquake Science Experiment (No. 2016CESE0202). The authors are grateful to the Associate Editor, Roman Pasteka, and two other anonymous reviewers for their encouragement and constructive comments. We also thank Yosihiko Ogata and Kunio Tanabe for their helpful discussions and Bryan Schmidt for his careful copy-editing work.

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