We present a new geometric gravity inversion algorithm specifically designed for modeling sub-ice shelf bathymetry and its uncertainty. Synthetic testing, derived from real Antarctic data, shows the relative importance of known bathymetry point measurements, gravity data quantity and quality, and regional gravity field variability. For regional fields with low variability, we suggest a focus on collecting gravity data, while for highly-variable regional fields, we suggest a focus on collecting additional seismic bathymetry measurements.
Over the past few decades (1992–2020), Antarctica has been losing a significant amount of ice, averaging 92 billion tons per year, adding over 7 mm to global sea-level rise
In these cold-water shelves, basal melt tends to happen at the ice front and along the deep grounding zones, the region where ice transitions from grounded to floating. This grounding zone melt occurs because the melting point of ice drops under pressure, so even cold, salty waters can melt ice at depth
Techniques for collecting sub-ice-shelf bathymetry data are limited. Radar (ground or airborne) cannot image through the ocean cavity due to signal attenuation caused by water. Conventional over-ice seismic surveys provide accurate measurements, but can only cover small spatial scales (10's of kilometers), unless significant funding and time are available
There are two main types of gravity inversions. Density inversions estimate how density changes underground, often used in mineral exploration. Geometry inversions (also called structural inversions) estimate the shape of a surface which represents a change in density, like the seafloor beneath an ice shelf. They are commonly used to model the boundary between Earth's crust and mantle
We use an open-source Python-based geometry inversion tool called
Geophysical inversion involves determining a set of physical parameters that, when forward modeled, produce predicted data that closely match observed measurements. Forward modeling is the calculation of what data would be observed given a model of physical parameters. In this study, we perform a least-squares inversion to solve the matrix equation
Prior to inversion, gravity data must be reduced to isolate the signal relevant to the density contrast of interest. The reduction and preprocessing workflow (Fig.
Field corrections: Measured gravity is corrected for instrument drift, flight line leveling inconsistencies, aircraft maneuvers, and the effects of measuring gravity on a moving platform (Eötvös correction) Normal gravity correction: We then subtract the gravity of a reference Earth model (denoted Terrain correction: To isolate internal density structure, the effects of known topographic masses are removed from the disturbance. These terrain mass effects reflect all topographic deviations between the
Lastly,
Defining the terrain mass effect
For conventional gravity inversions targeting internal interfaces (e.g., the Moho or sediment-basement contact),
This removes the gravitational effects of ice and water, while retaining the effects of the rock surface (onshore topography and offshore bathymetry). A rigorous discussion of this formulation and its implementation is provided in Appendix
Schematic workflow diagram for
Both gravity reduction and inversion require forward modeling of the gravity field associated with a given topographic surface. The surface is discretized into an array of adjacent vertical rectangular prisms, and the total gravitational field at any given observation point is computed as the sum of the individual contributions of each prism. For prisms of constant density, gravity is calculated using the analytical solutions of
Equivalent methods of discretizing a density contrast
The gravity misfit used in the inversion is defined as
The regional component of the gravity misfit can be estimated using several methods, most of which assume that the regional field consists of longer-wavelength anomalies, while the residual field dominates the short-wavelength components. This assumption underpins techniques such as upwards continuation or the use of high-altitude gravity data to estimate the long-wavelength signals
The last method, which we utilize in this paper, leverages locations with known bathymetry depths, referred to as
This technique has key advantages. It imposes a form of smallness regularization, preventing significant changes to the depth at constraint points. Aside from the choice of interpolation method, it avoids the subjective parameter selection required by the other techniques, such as low-pass filter widths or upward continuation heights. However, it assumes the wavelength of the regional field is larger than the spacing between constraint points; shorter-wavelength regional anomalies will not be captured and may be absorbed into the residual (i.e., the inverted bathymetry). This approach, often referred to as a “DC-shift”
where
The first step of the inversion (Fig.
With the Jacobian and the gravity residuals, Eq. (
Inversion problems involving potential fields are inherently ill-posed: many solutions can satisfy the data. Regularization helps stabilize the solution by incorporating prior knowledge or desirable properties
Smoothness regularization is enforced through damping in the LSQR solver. The damping parameter restricts the magnitude of surface corrections in each iteration, reducing noise and promoting realistic bathymetry. While this gives smoother results, it requires extra iterations. Section
After each iteration, the surface corrections (possibly weighted) are applied to update the bathymetry. The new model is re-discretized into prisms, forward-modeled, and a new residual is computed. This process repeats until one of the termination conditions is met.
The inversion terminates when the first of the following conditions is met:
The maximum number of iterations ( The The changed in
Hyperparameters are parameters whose values need to be chosen (not calculated) and that also have a significant effect. Two key hyperparameters affect inversion quality: the regularization damping parameter and the uniform density contrast value (if it is not previously known). Below, we describe how to determine their optimal values.
The damping parameter balances solution stability and data misfit. Too high a value, the solution under-fits the data; too low a value, the data are overfit and noisy results are produced. To find an optimal value, we use cross-validation of the input gravity data
To construct the training and testing sets, the data are interpolated onto a grid at half the target resolution using equivalent sources
Workflows for selecting the optimal values for the two inversion parameters:
If no seafloor density contrast is known, it can be estimated via the inversion (Fig. Cross-validation: Split constraints into K-folds of training and testing subsets Alternative regional estimation: Temporarily use a different regional field method during optimization, then revert to Constraint Point Minimization once the best density contrast is found.
Both hyperparameter estimation methods rely on computing a score for each candidate value.
We use a Bayesian framework, specifically Monte Carlo simulation, to assess the spatial uncertainty of the inverted bathymetry
Schematic workflow diagram for the Monte Carlo uncertainty analysis.
Defining input parameter distributions is critical. For point data (gravity, constraints), we assume normal distributions based on user-provided or estimated uncertainties. For gravity, uncertainties often come from survey crossover errors; for constraints, uncertainty may be based on data source or be proportional to depth. For hyperparameters (density contrast and damping), the results of the optimizations can be used to define each parameter's distribution. For a well-defined best score, a normal distribution centered on that value with a small standard deviation can be used. For an even distribution of similar scores, a uniform (Boxcar) distribution can be used with the width defined by the range of the similar scores. To assess uncertainty in the regional field, we also assign distributions to parameters used in its estimation (e.g., equivalent source depth or damping in Constraint Point Minimization). Probability distributions for these parameters can be derived from the respective optimization procedures.
Since full inversion workflows are computationally intensive, we use Latin Hypercube Sampling to efficiently cover the input parameter space with a small number of runs
In this study, we use a few unique metrics to describe the distribution of data and assess model performance. There are many ways to describe how data points are spaced out, whether it is airborne gravity data or over-ice seismic survey points, but not all metrics work equally well for all survey types. For example, spacing (usually the average distance between points) only makes sense if your data are evenly spread across a grid. If data follow tracks, like flight lines or ship surveys, the mean distance between points is skewed toward unrealistically small values. Spatial density (how many points fall within a certain area) has a similar issue; it is strongly affected by the sampling frequency along those tracks. Line spacing, the average distance between survey lines, only works well for grid-based surveys and does not handle gaps, tie lines, or irregular coverage very well. To enable robust comparisons of data distribution with varying collection types and spatial distributions, we use a metric we call
To evaluate the performance of various techniques in our synthetic tests, we use root mean square error (RMSE). RMSE is useful because it ignores the sign of the errors and is less sensitive to outliers than the mean. For example, the RMSE between a “true” bathymetry grid and an inverted grid tells us how far off our model is, on average. Finally, we introduce a
Lastly, throughout this study, we discuss the relative amplitudes of regional gravity fields. Mathematically, we quantify this as the standard deviation of the regional field, but in text we use the term
To set up our synthetic tests realistically, we needed to choose values for several key parts of the bathymetry inversion process, such as how accurate the data are, how the data are spread out, and how variable the regional gravity field is. To choose which values to test, we analyzed all 164 Antarctic ice shelves to figure out the typical ranges for these variables. For each shelf, we collected nearby bed measurements from IBCSO v2
For each ice shelf we used AntGG-2021 gravity data
To assess the performance of the inversion software, showcase its use, and explore the sensitivity of the inversion to certain parameters, we ran a series of tests using a semi-synthetic model. This model is based on bathymetry data from Antarctica's Ross Sea (Fig.
Ross Sea topographic data used for synthetic inversions.
Across different tests, we used the full-resolution gravity grid, added synthetic noise, or sampled the data along simulated flight lines to mimic an airborne survey. In some cases, we also added a regional gravity component by forward modeling the effect of basement topography using ANTOSTRAT seismic data
We ran five core inversion tests: Test 1: the ideal case, using full-resolution, noise-free gravity data with no regional component. Test 2: adds noise to the gravity data. Test 3: uses simulated airborne gravity data instead of the full grid. Test 4: includes a regional gravity component. Test 5: combines all the above factors for a more realistic test case.
We also ran four inversion ensembles, each with 100 runs: Ensemble 1: explores the relationship between the number of constraint points and the variability of the regional field. We test 10 different amounts of the number of constraint points, from 4 points to 650 points, and 10 different levels of the regional gravity field variability, from very low variability ( Ensembles 2–4: look at how gravity noise and flight line spacing affect results. For these ensembles, we test 10 values of gravity noise, from none to 5 mGal of random noise, and 10 numbers of gravity flight lines, from 6 to 60 lines across the region. These values are tested with no regional gravity field, and medium variability field, and a highly variable field (ensembles 2, 3, and 4, respectively).
To make sure our tests reflect real-world conditions, we set all these parameters (e.g., constraint proximity, gravity noise, and flight line spacing) based on the ranges we found across Antarctica's ice shelves and in published gravity surveys. Each test and ensemble has a corresponding Jupyter notebook, available in the supplementary material.
The inversion algorithm we use in this study is implemented in Python as the open-source package
All of the experiments were run in Jupyter notebooks
To assess gravity inversion suitability across Antarctica, we compiled all available bathymetry measurements and created grids of three key variables for each ice shelf: constraint proximity, topography-corrected gravity disturbance, and regional gravity misfit. To avoid bias from the large number of small shelves, we restricted our analysis to the 50 largest ice shelves (those exceeding 1000 km2) out of the 164 named shelves in Antarctica. Among these, the median constraint proximity ranged from 2.5 km (Thwaites) to 32 km (Conger-Glenzer), with an overall mean of 10 km and median of 8 km (Fig.
We also evaluated the regional gravity field variability (amplitude of variation) using two measures: the standard deviation of the topography-corrected gravity disturbance and the standard deviation of the regional gravity misfit. We found the former to be a more reliable indicator, as it is less influenced by the number and distribution of constraint points. Although it can be affected by the resolution of the topographic model, our focus is on long-wavelength signals, which are well captured by the BedMap2 topography grids. Across the 50 shelves, regional field variability (based on topography-corrected gravity disturbance) ranged from 7 mGal (Holmes) to 66 mGal (Ekström), with a mean of 23 mGal and median of 18 mGal (Fig.
To better understand typical survey parameters, we reviewed published studies on airborne gravity data collection over Antarctic shelves. Reported flight line spacings typically range from 5 to 20 km, with 10 km being the most common. Estimating gravity uncertainty, however, is more complicated. Most studies use crossover analysis, but methods vary widely: some report the mean, others the standard deviation or RMS of crossover differences; some apply leveling (of varying orders), while others do not, making comparisons difficult. In general, reported uncertainties range between 1 and 5 mGal. To provide a consistent metric, we used the AntGG-2021 uncertainty grid to calculate the RMS uncertainty across each ice shelf. The lowest shelf-wide RMS uncertainty we found was 3.8 mGal (Pine Island), highlighting the inconsistency in reporting practices.
Antarctic ice shelf analysis;
Results of the inversions and uncertainty analyses for each of the five scenarios (each column is a scenario). Thefirst row
Comparison of the five inversion scenario results. Each scenario has an associated color, shown in the legend.
We construct a low-resolution bathymetry model that serves as both the initial model and the basis for generating synthetic gravity data in Tests 1–5. This model is derived by interpolating individual bathymetry point measurements (constraint points). To simulate realistic situations, we include 42 evenly spaced interior constraints representing an over-ice seismic survey, along with a surrounding border of constraints mimicking denser measurements near the ice shelf periphery, such as airborne radar over grounded ice or multibeam bathymetry offshore the calving front (Fig.
This test represents an idealized, best-case scenario, establishing a baseline for inversion performance. The resulting errors are expected to be minimal and primarily reflect limitations in estimating the two key inversion hyperparameters: the regularization damping parameter and the density contrast. We generate a synthetic gravity disturbance dataset by forward-modeling the full-resolution gravity response of the real bathymetry. No noise or regional gravity component is added. To calculate the gravity misfit used in the inversion, we apply the starting bathymetry model using a prism-layer method with an incorrect density contrast of 1350 kg m−3, while the synthetic data is generated using the true value of 1476 kg m−3.
We identify the optimal damping value by performing cross-validation over eight candidate values (0.001 to 0.1), finding an optimal value of approximately 0.01 (Fig.
To assess uncertainty, we performed a Monte Carlo analysis by sampling damping and density contrast values from probability distributions (see Appendix
In this scenario, we expand upon Test 1 by incorporating observational noise into the synthetic gravity disturbance data to simulate realistic airborne gravity survey conditions. As in Test 1, the baseline gravity disturbance is generated by forward-modeling the true bathymetry. To simulate data collection errors, we add random Gaussian noise with a standard deviation of 3 mGal (Fig.
For each filter, we computed the RMSE relative to the original noise-free gravity data (Fig.
Errors in the gravity data for Test 2.
The density contrast optimization identified an optimal value of 1471 kg m−3 (Fig.
For uncertainty quantification, we again used a Monte Carlo simulation. This time, in addition to sampling the damping and density contrast values, we also added a new realization of Gaussian noise (3 mGal standard deviation) to the gravity data in each iteration. After noise addition, we applied the 24 km Gaussian filter as in the main inversion. The resulting uncertainty estimate (Fig.
This scenario builds upon Test 1 by simulating a realistic airborne gravity survey, in contrast to the full-resolution synthetic gravity disturbance data used previously. The synthetic airborne survey follows a typical Antarctic design, found from the analysis of many published Antarctic airborne gravity surveys. From this analysis, we use flight lines spaced 10 km apart, orthogonal tie lines every 50 km, and data points sampled every 500 m along-track at a constant altitude of 1 km. The resulting gravity data have a median proximity of 2.3 km (Fig.
To estimate the gravity data uncertainty introduced by this interpolation, we performed a Monte Carlo simulation of the equivalent source method, sampling the interpolation parameters (e.g., source depth and damping; see Appendix
Airborne gravity survey for Test 3.
The density contrast optimization yielded an optimal value of 1464 kg m−3 (Fig.
To assess uncertainty, we repeated the Monte Carlo analysis as in Test 1, this time including variability in three components: the damping parameter, density contrast, and equivalent source interpolation parameters. The resulting uncertainty estimate (Fig.
This fourth scenario extends Test 1 by incorporating a regional component into the gravity disturbance data. The regional component of the gravity field is generally the long-wavelength component created from subsurface geologic density variations. To simulate this, we use crystalline basement topography from the ANTOSTRAT shipborne seismic compilation
The regional gravity was then added to the forward-modeled gravity effect of the true bathymetry to form the synthetic gravity disturbance data (Fig.
Following regional field removal, the residual gravity misfit (Fig.
Gravity anomalies for Test 4: regional gravity component.
Using the true density contrast, the estimated regional component (Fig.
To evaluate uncertainty, we ran the same Monte Carlo simulation as in previous tests. The resulting uncertainty estimate (Fig.
This final scenario combines all components from Tests 1–4 to simulate a realistic inversion scenario, incorporating all expected sources of complexity: (1) unknown true density contrast, which must be estimated, (2) errors in gravity disturbance data, (3) gravity data acquired along airborne flight paths, and (4) a regional gravity component that must be estimated and removed. To generate the gravity disturbance dataset, we combined the full-resolution forward-modeled gravity effect of the true bathymetry with the regional gravity field produced by the basement topography. We then added 1 mGal of pseudo-random noise, sampled the data along simulated flight lines, applied a low-pass filter, and re-interpolated it onto a grid using equivalent sources (Appendix
Errors in the gravity data for Test 5.
As in Test 2, we applied low-pass filtering to reduce short-wavelength gravity noise. Here, the noise was first added to gridded data, which were then sampled along flight paths. We applied a range of filter widths to the flight-line data and calculated the RMSE between the filtered and noise-free signals. The optimal filter width was 24 km (Fig.
With the interpolated gravity data, we followed the same steps as in Tests 3 and 4: we computed the gravity misfit using an incorrect density contrast, estimated and removed the regional component using the Constraint Point Minimization method, then performed damping cross-validation and density contrast optimization. As before, we also repeated the regional separation using the true density contrast to assess the performance of regional field estimation independently from density estimation.
Regional gravity estimation for Test 5;
Using the true density contrast, the estimated regional gravity field (Fig.
We assessed the uncertainty of this inversion using a Monte Carlo simulation that varied four components: damping, density contrast, gravity noise, and interpolation effects (Fig.
To understand the relative contribution of each uncertainty source, we performed additional sensitivity tests, each including only a single parameter (10 runs per test): Damping parameter: RMS 1 m, spatially uniform; Interpolation of flight-line data: RMS 4.8 m, with localized high values near topographic ridges and model edges; Gravity data: RMS 8.5 m, generally spatially uniform; Density contrast: RMS 6.9 m, especially concentrated along topographic ridges.
Among all components, uncertainty from density contrast had the largest effect and dominated the spatial distribution of the total uncertainty (Fig.
Sensitivity analysis for Test 5; realistic scenario.
As shown in Test 4, accurately estimating the regional component of the gravity misfit is a vital part of the inversion process. The reliability of regional field estimation depends on both the variability of the regional field itself and the number and spatial distribution of constraint points within the inversion domain. To test this, we ran an ensemble of 100 synthetic inversions, covering all combinations of 10 levels of regional field variability and 10 varying amounts of constraint points. Regional field variability, quantified as the standard deviation of the regional misfit, varied linearly from
These tests were performed in idealized conditions: full-resolution synthetic gravity data with no added noise or flight line interpolation. To reduce computational cost, we omitted the damping parameter optimization and instead used a constant damping value of 0.025 for all runs. For each inversion, we first estimate the optimal density contrast using 8 trial values ranging from 1400 to 3300 kg m−3, assuming an initial regional field of 0 mGal (Fig.
Ensemble 1; number/proximity of constraint points vs regional gravity field variability.
The starting bathymetry RMSE values across Ensemble 1 ranged from 6 to 46 m, while the inverted bathymetry RMSEs ranged from 2 to 121 m. Corresponding topographic improvements ranged from
Figure
Figure
We used these ensemble results to make best-case predictions of inversion RMSE and topographic improvement for 110 Antarctic ice shelves. Remaining ice shelves were not included because their median constraint proximities and/or estimated regional field variability (calculated as the standard deviation of the topography-corrected gravity disturbance) were outside the limits of the values tested in Ensemble 1. To avoid clutter, only ice shelves larger than 1000 km2 are shown in Fig.
To evaluate the relative importance of the gravity data errors (e.g., noise, leveling artifacts) and gravity data resolution (i.e., flight line spacing), we ran three ensembles of 100 inversions. Each ensemble tested 10 gravity noise levels (0–5 mGal) and 10 numbers of gravity flight lines (3–30 lines in both E–W and N–S directions), corresponding to average line spacings of 6–60 km, or median proximities to gravity data points of 0.8–8 km. All ensembles used a constant along-line observation spacing of 500 m. The three ensembles differ in regional gravity field variability; Ensemble 2 (Fig.
Observed gravity was low-pass filtered with an optimal filter width estimated by the same method as Test 5. Ensemble 2 used the true density contrast, while Ensembles 3 and 4 used the density optimization procedure from Ensemble 1. To reduce computational cost, we applied a fixed damping value of 0.025.
Gravity noise vs quantity of gravity data. 1st column; Ensemble 2 with a no regional field and 2nd column; Ensemble 4 with a high-variability regional field (41 mGal standard deviation).
Inverted bathymetry RMSEs ranged from 3 to 49 m, with topographic improvements (relative to the 26 m RMSE starting model) from
Results for Ensemble 3 were similar to Ensemble 4, and are described in the Appendix
This analysis demonstrates that both reducing gravity noise and increasing flight-line density improve inversion performance. However, their relative benefits diminish in the presence of highly-variable regional fields. In low-regional-field settings, improvements from denser or cleaner data are more pronounced.
Our synthetic inversions and Antarctic ice shelf analysis highlight key factors governing the accuracy of performing gravity inversions for bathymetry.
Introducing 1 mGal (3 mGal before filtering) of random noise (Test 2) degraded bathymetry accuracy by
Simulating an airborne survey (Test 3) added
Point measurements of bathymetry beneath and surrounding an ice shelf (constraint points) underpin every stage of our inversion: initial model creation, regional-field estimation, regularization, and density contrast estimation. Even with sparse, noisy gravity data, well‐distributed constraints keep inversion errors low; conversely, with few constraints, inversions only perform well if there is a regional field with exceptionally low variability. Ensemble 1 shows inversion RMSE decays exponentially with the number of constraints (Fig.
To address this, we identified which Antarctic ice shelves would likely benefit most, or least, from additional bathymetric measurements. The skewness of the constraint proximity values offers insight into the spatial distribution of existing constraints. Low or negative skew suggests that large areas of the shelf are far from existing constraints. Adding a constraint in one of these areas would reduce constraint proximity for a large portion of the ice shelf, thus notably improving inversion performance. High positive skew indicates that only a small portion of the shelf is distant from existing constraints. In these cases, adding a new constraint would have only a minor effect on median constraint proximity and inversion performance. We have calculated the skewness of constraint proximity for all large (
The impact of adding a constraint point (green triangle) on median constraint proximity for two ice shelves; Cosgrove
From this, we infer that ice shelves with a low skewness in constraint proximity would benefit significantly from each new bathymetry measurement (left side of Fig.
Our synthetic tests yield these guidelines: Prioritize filling large bathymetry gaps: add a few well-placed constraint points. If constraints are adequate, focus on acquiring initial gravity lines, even at lower precision. Once a minimal gravity grid is in place, shift to improving gravity data quality through increasing numbers of tie lines, reducing flight speed, or re-flying noisy lines. Inversion errors are dominated by regional‐field misestimation, so additional gravity data or noise reduction have limited impact. Instead, concentrate on bathymetry constraints until median proximity gains plateau, balancing field effort and model improvement.
We recommend using median proximity, rather than point/line spacing or count, to assess and compare spatial coverage of both gravity and bathymetric data. Calculating median constraint proximity for all ice shelves (Fig.
Future work is needed to test the optimal survey designs for constraint points. Measuring constraints on an even grid naturally provides the most uniform spatial coverage, and adding constraints at the locations of maximum data proximity is optimal for reducing median constraint proximity and reducing errors at the locations where they are highest. However, we hypothesize that focusing constraint point placement according to the gravity disturbance data, either at the maxima or minima, or at locations with high gradients, may provide improvement in estimation of the regional gravity field, and thus provide an effective means for reducing inversion errors. In addition to the ice shelves listed in the above section based on their constraint proximity skew, the following shelves have very large median constraint proximities and would benefit greatly from additional constraints: Shackleton, Roi-Baudouin, Wilkins, Ross, and Amery.
Gravity inversion generally outperforms simple interpolation of constraints, except when combining sparse constraints, high gravity noise, and highly-variable regional gravity fields. By mapping predicted RMSE and topographic improvement for each Antarctic ice shelf, we can identify ideal inversion targets (large positive improvement scores) and shelves where inversion may degrade results (large negative scores). Recently,
Our Monte Carlo-based uncertainty estimates match true RMSE within
Our cross-validation procedures consistently recovered optimal values for both the damping parameter and the density contrast (Fig.
Gravity inversion is a viable technique for understanding the shape of the seafloor beneath floating ice shelves. We present an open-source gravity inversion algorithm specifically designed to model bathymetry and its uncertainty. Using synthetic tests based on real bathymetry data from the Ross Sea, Antarctica, we evaluated the algorithm's performance sensitivity to various factors. Our tests show that the estimation of the regional gravity field is a dominant source of inversion error. However, this error can be significantly reduced with sufficient point measurements of bathymetry. Interpolation errors from standard survey flight-line spacing and typical gravity data uncertainty (
In general, inversions consist of three components;
Forward operator ( Physical Earth property model ( Fit function (
Here, a mathematical derivation of a generalized discrete inversion problem is shown. The inversion problem can be framed as finding the set of physical parameters (
This is defined as the
Once the Jacobian is created, a matrix equation of the form
The derivative of the forward operator with respect to the parameter of interest determines the linearity of an inversion. For a gravity inversion attempting to recover rock density, the derivative of gravity with respect to density is constant, and thus the inversion is considered linear
For a geometric gravity inversion, we need to process the gravity data (Fig.
To calculate the gravity effect of the normal Earth at each observation,
The gravity disturbance consists of two components;
This configuration of prisms correctly discretizes the terrain mass effect because the ice surface is equal to the water surface in areas of no ice thickness, and the water surface is equal to the rock surface in areas of no water thickness. Due to this, the overlap between the three sets of prisms, shown in Fig.
Subtracting this terrain mass effect from the gravity disturbance yields the topography-corrected disturbance,
In this gravity inversion, as implemented in
Before the inversion, we need to separate the gravity misfit,
There are several options for discretizing a topographic surface, representing a density contrast between two materials, as a series of adjacent vertical right-rectangular prisms (Fig.
Any forward gravity calculation of a layer of prisms will include gravitational edge effects. These effects are typically a decay in the gravitational field towards the edges of the prism model due to the void space (density of 0) beyond the edge of the model. The magnitude of decay is dependent on both the average thickness and density of prisms near the edge. Thicker and or denser prisms create a larger contrast with the void space, resulting in a greater amount of decay. Therefore, the absolute density method of discretization (Fig.
A buffer zone can be used to reduce the impact of these edge effects by calculating forward gravity confined to an inner region, while the prism model extends beyond. This limits the majority of the edge effects to outside the region of calculation. Too large of a buffer zone will add an unnecessary amount of computation (more prisms), while too small a zone will introduce unacceptably large edge effects to the calculations.
A vertical prism can be approximated as a sector of an annulus (a cylindrical shell). This approximation greatly simplifies the calculation of the vertical derivative of the prisms' gravity. The vertical component of gravity of an entire annulus is defined by
Taking the derivative of this with respect to
During the calculation of cell-wise statistics of the inversion ensembles resulting from the Monte Carlo analysis, the statistics can be weighted by the inverse square of each inverted bathymetry's root mean square difference with the constraint point depths, if they are available (Fig.
In Test 1, two parameters were included in the Monte Carlo analysis: (1) the damping parameter and (2) the density contrast. For the damping parameter, we use a standard deviation (0.24) calculated from half of the standard deviation of the damping values in the cross-validation (Fig.
In Test 2, three parameters were included in the Monte Carlo analysis: (1) the damping parameter, (2) the density contrast, and (3) the gravity data points. For the damping parameter, we use a standard deviation (0.24) calculated from half of the standard deviation of the damping values in the cross-validation (Fig.
In Test 3, three parameters were included in the Monte Carlo analysis: (1) the damping parameter, (2) the density contrast, and (3) the gravity data points. For the damping parameter, we use a standard deviation (0.24) calculated from half of the standard deviation of the damping values in the cross-validation (Fig.
In Test 4, two parameters were included in the Monte Carlo analysis: (1) the damping parameter and (2) the density contrast. For the damping parameter, we use a standard deviation (0.24) calculated from half of the standard deviation of the damping values in the cross-validation (Fig.
In Test 5 three parameters were included in the Monte Carlo analysis; (1) the damping parameter, (2) the density contrast, and (3) the gravity data. The damping parameter was assigned an standard deviation of 0.24, calculated from half the standard deviation of the cross-validation results (Fig.
In Test 3, the full-resolution gravity data is sampled along the paths of an airborne survey. We then use equivalent sources to interpolate this data back to a full grid. We block-mean the data at a 2 km spacing to reduce aliasing effects due to oversampling along flight lines
In Test 4, we introduced a regional component to the observed gravity data. To estimate and remove this component, we use Constraint Point Minimization, which requires interpolation of the gravity data. For this, we use the equivalent source technique and find the optimal source depth with the same cross-validated optimization as above. For this application, we use no damping. We perform 20 trials of source depth values (relative to observation heights) between 100 m and 400 km. We use these fitted sources to predict the gravity data across the entire grid, with a 2 km spacing (Fig.
In Test 5, the same cross-validated optimization as Test 3 was used to determine optimal parameter values for interpolating the flight line data. This resulted in an optimal source depth and damping of 57 km and
Inverted bathymetry RMSEs ranged from 9 to 31 m, with topographic improvements from
Ensembles 2–4; gravity noise vs quantity of gravity data, with varying levels of regional field variability. 1st column; Ensemble 2 with a no regional field, 2nd column; Ensemble 3 with a medium-variability regional field (18 mGal standard deviation), and 3rd column; Ensemble 4 with a high-variability regional field (41 mGal standard deviation). 1st row;
All code used in this analysis is available at
MT, HH, FCT, and KT designed the experiments, and MT carried them out. MT wrote the inversion code, with contributions from FCT. MT prepared the manuscript with contributions from all coauthors.
The contact author has declared that none of the authors has any competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.
The authors would like to acknowledge informative discussions with Jenny Black, Leo Uieda, Santiago Soler, Robin Bell, Andrew Hoffman, and Caitlin Dieck Locke, which helped develop the inversion algorithm or shape the manuscript. We are extremely grateful to the developers and maintainers of open-source software used throughout this research, without which this work would not have been possible.
This research has been supported by the Ministry for Business Innovation and Employment (grant nos. ATA1801 and ASP021-01) and the National Science Foundation (grant no. 2002346).
This paper was edited by Adam Booth and reviewed by three anonymous referees.