Recent applications in Earth sciences require geoid models to be determined with a sub-centimetre internal error. Regional models of the geoid are usually determined using discrete gravity values measured at and/or outside the Earth, and global models of the Earth gravity field and topographic surface. In this article, we review previous studies that (to some extent) discuss the estimation of the geoid internal error, and provide formulations and methodologies required for a comprehensive formal propagation of errors of gravity data and global models through a mathematical model used for regional geoid determination. The mathematical model is based on combining the inverse Poisson integral equation and the Hotine integral transform in the Helmert harmonic space; also called the one-step integration method. Calculations and tests are performed in one of the most challenging test areas ("the Colorado test area") using ground and airborne gravity observations, a global digital terrain model (DTM) for topographic effects on gravity and the geoid, and a global Earth gravitational model (EGM) for the long-wavelength components of gravity and the geoid.There are three main contributors to the total internal error of the geoid height, namely those associated with the EGM (for estimating the long-wavelength geoid height), DTM heights (for evaluation of the topographic effects on observed gravity and the geoid height), and gravity observations (for determining the short -wavelength components of the geoid height). The geoid errors stemming from the EGM formal variances of its spherical harmonic coefficients amount to 0.3 cm of the total internal error budget of the geoid. The mean value of the standard deviation of the geoid height stemming from the topographic effects (due to uncertainties of DTM heights) is 1.0 cm. The observation errors and spatial distribution (resolution) of regional ground and airborne gravity observations, i.e., the design of the project, are the dominant contributors to the internal error of the geoid height. The internal error estimate of the geoid model in the Colorado test area computed on a 1 ' x 1 ' grid is 2.7 cm which agrees with the differences between various geoid models that contributed to the Colorado 1-cm geoid experiment.Determining a regional geoid model with a sub-centimetre internal error in areas of rough topography, such as that of Colorado, requires high-accuracy and high-resolution gravity observations. To assess the accuracy of regional gravity measurements required for sub-centimetre geoid models, we simulate airborne gravity mea-surements in the Colorado test area at a practical lower flight altitude, and improve the spatial distribution of existing ground gravity observations by filling in gaps using synthetic (EGM-based) gravity disturbances. Results show that carrying out airborne gravity surveys with a gentle drape approach at an average flight altitude be-tween 300 and 500 m above the Earth's surface provides airborne gravity measurements with a mean standard deviation of 0.75 mGal at 2.2 km spatial resolution. Thus, a sub-centimetre regional gravimetric geoid model in the Colorado test area would be achievable using the proposed configuration of the ground and airborne gravity observations and more accurate DTMs.

We estimate the uncertainty of the modelled geoid heights based on the standard deviations of the topographic mass density variation. We model the geoid using the one-step integration method considering mass density variations along with their associated error estimates to calculate the direct and indirect topographic density effects on the geoid heights in the Helmert space. We employ the UNB_TopoDensT_2v01 global lateral density model and its standard deviations and test our algorithms in the Auvergne test area, in central France. Our results show that the topographic mass density variations are currently known well enough to model the geoid with sub-centimetre internal error in topographically mild regions such as Auvergne.

In this contribution, we estimate the uncertainty (error) of the input gravity measurements needed for the determination of the geoid with an internal sub-centimetre accuracy. The accuracy of the geoid height is a function of the resolution/accuracy of the input gravity and topographical data, and the methodology used to solve a geodetic boundary value problem. The purpose of this study is to estimate the maximum allowable error in the terrestrial gravity measurements based on a required standard deviation of the error in the geoid heights (e.g., ≤1cm). This is done with an assumption of a known Digital Elevation Model (DEM), and an Earth Gravitational Model (EGM) along with their error estimates. We use the one-step integration method (one-step kernel) for the determination of the geoid. In this method, the anomalous gravity at any surface above the geoid is estimated by integrating over the geoid-level disturbing potentials in harmonic space. By applying the covariance law to the one-step integration method, the error of the gravity measurements at the Earth's surface can be estimated using the expected error of the geoid heights. Taking advantage of the remove-compute-restore technique, we estimate the error of the residual surface gravity measurements using the (known) error estimates of the topographical and EGM corrections. We select the Colorado test area (35°N - 40°N, 250°E - 258°E) to generate a 1¢×1¢ grid of geoid random errors with a standard deviation of 1cm. We use the topographical data from the Shuttle Radar Topography Mission (SRTM) Ver. 3.0. and the global model of DIR_R5 up to degree/order 140 to apply the remove-compute-restore technique. The uncertainty estimate of the SRTM heights and the covariance matrix of the spherical harmonic coefficients of the DIR_R5 are used to calculate the errors of the topographical gravitational attraction and low-degree EGM signals on the geoid heights and surface anomalous gravity data. Our preliminary results show that to achieve a sub-centimetre accuracy in the Colorado area, we require grid surface gravity measurements with a standard deviation of less than 2.5mGal. This result is optimistic as in the geoid determination process, the anomalous gravity data are downward continued from the Earth’s surface to the geoid, whereas this step is not required in our experience. Besides, we assume a constant standard deviation of 1cm for all the errors of the geoid heights, whereas such high accuracy may not be needed in high mountains. We will provide further results for the elevation-dependent geoid error and also investigate the effect of downward continuation on our results.

Each of the even and odd data sets is an independent data set, separately levelled, gridded and low-pass filtered to create even and odd grids. Each data set contains a geological com- ponent and a noise component. The geological component in each data set is identical, i.e. they are both measured over the same survey area and the geological signal is well sampled on each of the odd and even data sets. The noise component is assumed to be white, containing all frequencies in equal pro- portion. Tests with AIRGrav airborne gravity data sets indi- cate that, except for the highest frequencies, which would have been filtered out of realistic gravity grids, the remaining noise is very close to white. Gravity has been measured from aircraft in flight since the late 1950s (Thompson & LaCoste 1960). Recent improvements in GPS processing, and a new gravity instrument, the AIRGrav system (Argyle et al. 2000), have resulted in significantly re- duced noise levels in airborne gravity data. In this paper we present a methodology to quantitatively calculate noise levels of airborne gravity data sets by dividing the flight lines into two equal data sets (the 'even' and 'odd' lines), gridding and filtering the separate data sets, and measuring the difference between the resultant grids. The data is low-pass filtered be- fore the noise level is measured, and noise levels are calculated for specific filter lengths. We also present an example of this noise calculation performed on an AIRGrav data set from the foothills region of Alberta, Canada, along with an interpreta- tion of the data (Peirce et al. 2002; Sander et al. submitted). AIRGrav airborne gravity data is generally acquired along survey lines spaced between 50 and 3000 m, flown in a grid pattern over the survey area. After normal gravity corrections, data is gridded and filtered to remove high frequency GPS and gravity acquisition noise. On many AIRGrav surveys, SGL over-samples the gravity field to increase the accuracy and resolution of the resultant data. Noise from the over-sampled gravity data cancels in a manner similar to the stacking of seis- mic data. The over-sampled gravity data can be used to calcu- late the noise level on a gravity grid by dividing the data set into two independent data sets covering the same area, and calculating the RMS difference between them. As the data sets cover the same area, the geological signal will cancel, leaving only the noise of the two data sets. The RMS noise measured on the difference grids will be twice the noise level of the com- bined grid, as explained below. In this case 'noise' means the errors between lines or indi- vidual readings within the data set. The method would not measure systematic errors common to the entire data set. Sys- tematic errors could occur if the entire data set was levelled to some predetermined value, or if the same erroneous elevation model was used for terrain corrections for both the odd and even data sets.