More Thoughts on AG–SG Comparisons and SG Scale Factor Determinations

Abstract

We revisit a number of details that arise when doing joint AG–SG (absolute gravimeter–superconducting gravimeter) calibrations, focusing on the scale factor determination and the AG mean value that derives from the offset. When fitting SG data to AG data, the choice of which time span to use for the SG data can make a difference, as well as the inclusion of a trend that might be present in the fitting. The SG time delay has only a small effect. We review a number of options discussed recently in the literature on whether drops or sets provide the most accurate scale factor, and how to reject drops and sets to get the most consistent result. Two effects are clearly indicated by our tests, one being to smooth the raw SG 1 s (or similar sampling interval) data for times that coincide with AG drops, the other being a second pass in processing to reject residual outliers after the initial fit. Although drops can usefully provide smaller SG calibration errors compared to using set data, set values are more robust to data problems but one has to use the standard error to avoid large uncertainties. When combining scale factor determinations for the same SG at the same station, the expected gradual reduction of the error with each new experiment is consistent with the method of conflation. This is valid even when the SG data acquisition system is changed, or different AG’s are used. We also find a relationship between the AG mean values obtained from SG to AG fits with the traditional short-term AG (‘site’) measurements usually done with shorter datasets. This involves different zero levels and corrections in the AG versus SG processing. Without using the Micro-g FG5 software it is possible to use the SG-derived corrections for tides, barometric pressure, and polar motion to convert an AG–SG calibration experiment into a site measurement (and vice versa). Finally, we provide a simple method for AG users who do not have the FG5-software to find an internal FG5 parameter that allows us to convert AG values between different transfer heights when there is a change in gradient.

Appendix

1.1 Weighted Mean and Variance

There are two approaches to finding the variance of weighted samples, assuming N samples of a quantity x_i, each assumed to have a Gaussian probability distribution with a standard deviation σ_i. Interpreting the weight of each sample as its inverse variance, w_i = 1/σ ²_i , the mean (x_m) and variance σ² of the resulting combination are given by

$$ x_{\text{m}} = \varSigma_{\text{i}} \left( {w_{\text{i}} x_{\text{i}} } \right) \, / \, V_{1} ;\sigma^{2} = \, 1/V_{1} $$

(A1)

where V₁ = Σ_i (w_i), assuming the data are uncorrelated. Equation (A1) gives the variance of the weighted mean, used in combining different quantities, derived for example from the LSQ inversion or fitting of parameters to data. It is also the error of compound quantities, so that when all weights are equal, σ² = (1/N²) Σ_i (σ ²_i ). A second approach is to compute the unbiased weighted sample variance

$$ \sigma^{2} = \varSigma_{\text{i}} {{\left[ {w_{\text{i}} \left( {x_{\text{i}} - x_{\text{m}} } \right)^{2} } \right]} \mathord{\left/ {\vphantom {{\left[ {w_{\text{i}} \left( {x_{\text{i}} - x_{\text{m}} } \right)^{2} } \right]} {\left[ {V_{1} - \left( {V_{2} /V_{1} } \right)} \right]}}} \right. \kern-0pt} {\left[ {V_{1} - \left( {V_{2} /V_{1} } \right)} \right]}} \, $$

(A2)

where V₂ = Σ_i (w ²_i ), as given for example in the GNU Fortran library (function gsl_stats_wvariance at https://www.gnu.org/software/gsl/manual/gsl-ref.html#Weighted-Samples). The mean x_m remains as in (A1) and the weighted SEM is defined as σ√(V₂)/V₁. Equation (A2) is used to assess the variance of the data about its mean, and is the weighted version of the usual formula for Gaussian mean and variance. In his useful little book, Topping (1979, Section 43) refers to (A1) as measuring internal consistency of the data, (using the errors associated only with each experiment) as opposed the external consistency of the data where the errors are determined by the spread of each experiment about a common mean.

Hill and Miller (2011) give the name conflation to (A1) and argue this is the correct way to combine different experiments to determine the best value of an unchanging physical quantity. They show that conflation is (a) commutative and associative, (b) iterative, and the (c) conflations of normal distributions are normal. Property (b) is useful as new data can be easily combined by adding to the conflations of the previous datasets. A more mathematically oriented justification can be found in Hill (2011).

1.2 AG Transfer Height

AG transfer height and gradient effects were discussed by Niebauer (1989) and then extended by Nagornyi (1995) and Timmen (2003) who provided more instrumental details. From a user’s point of view it is not possible to deal with changing the gradient using only the transfer height adjustment provided in the FG5 manual. Denote this transfer height correction by

$$ \delta g_{1} = \, - \, \left( {{\text{actual}}\;{\text{height}} - {\text{transfer}}\;{\text{height}}} \right) \times {\text{gradient }} = \, - \, \left( {{\text{AH}} - h} \right) \times \Delta g $$

(A3)

where AH is the actual height and h the transfer height. The actual height is the sum of the factory height and the setup height, and these quantities are given in the FG5 project files. But (A3) is insufficient to recover the transfer height correction if the gradient Δg changes. A series of experiments was performed by NGA (National Geospatial-Intelligence Agency) by varying the transfer heights and gradients in the FG5 processing and recording the calculated values from the FG5 merged project files. Starting with g_c(h, Δg) as the calculated value, we first subtract the standard correction (A1), and also the calculated gravity for zero gradient at this transfer height g_c(h, 0), thus

$$ g_{1}^{\prime } (h,\Delta g) \, = \, g_{\text{c}} (h,\Delta g) \, - \delta g_{1} - \, g_{\text{c}} \left( {h,0} \right) $$

(A4)

We established that the left hand side (LHS) is a linear function of the gradient, i.e.,

$$ g_{1}^{\prime } (h,\Delta g) \, = \, D \times \Delta g $$

(A5)

where for NGA’s FG5-107 used at AP, D = 8.0258 cm. Thus

$$ g_{\text{c}} (h,\Delta g) \, - \, g_{\text{c}} \left( {h, \, 0} \right) \, = \, \left( {h - h_{\text{e}} } \right) \times \Delta g $$

(A6)

introducing h_e = (AH − D) = 122.694 cm as an effective height, such that when h = h_e the gradient has the least effect on the gravity value. Because it is not possible to find h_e or D from the project files, Pálinkáš et al. (2010) observed “…Some users of the gravity data have no access to the FG5g-software; they cannot accurately correct for the new gradient without knowledge of the effective position. There is even a risk that they will compute the new transfer correction with respect to the top of the drop, because this is often presented as the instrument’s reference height.”

This was indeed our earlier experience, and we found h_e by adding one additional step, i.e., re-processing the same FG5 data with zero gradient at the same transfer height, g₁′(h, 0), and subtract this to get g₁′(h, Δg) using (A5) to find D. We can then correct for both transfer height and gradient from one AG setup to another using

$$ g_{\text{c}} (h_{2} ,\Delta g_{2} ) \, - \, g_{\text{c}} (h_{1} ,\Delta g_{1} ) \, = \, (\delta g_{1} - \delta g_{2} ) \, + \, D \times (\Delta g_{2} - \Delta g_{1} ) $$

(A7)

Note the distance D in (A5) remains the same when AH changes, and therefore, needs to be determined only once, whereas the effective height h_e depends on AH. In one of our experiments we start with AH = 130.72 cm, a transfer height of 100 cm, and a gradient of − 3.0 μGal/cm but want gravity at a height of 130 cm and a gradient of − 4.42 μGal/cm; using Eq. (A7) gives − 100.37 μGal, but Eq. (A3) gives a value of − 90.0 μGal, a difference of more than 10 μGal. Equation (A5) in fact is entirely consistent with Eq. (3) in Pálinkáš et al. (2010) to which the reader should refer for complete details (Tables 8, 9).

Table 8 Terminology and abbreviations used in the paper

Table 9 Test results for AP and ST scale factors. Shown are mean differences between equivalent calculations (keeping other options constant); values in (μGal/V)