possible error models (Zwieback et al. 2012), with the affine model being more sophisticated. Following Su et al. (2014), we propose an affine error model for ...
Error Model Recent advances in the collocation method (Stoffelen 1998, Su et al. 2014, Gruber et al. 2016) permit precise quantification of physical calibration and performance in observations and analyses. The method employs a variety of possible error models (Zwieback et al. 2012), with the affine model being more sophisticated. Following Su et al. (2014), we propose an affine error model for GlobCurrent, with drifters as the calibration reference, and with the GlobCurrent timeseries providing additional information to bound the calibration and performance metrics. The proposed error model is
An Update on GlobCurrent Calibration and Validation Activities Rick Danielson, Ocean and Coastal Remote Sensing, Nansen Environmental and Remote Sensing Center, Johnny Johannessen (NERSC), Marie-Hélène Rio (CLS), Fabrice Collard (ODL), Craig Donlon (ESA), Bertrand Chapron (Ifremer), and Graham Quartly (PML)
Regional Characterizations A division of the global ocean that is informed by documented sources (in particular the RSMAS website) and whose limits follow the CNES-CLS2013 climatology (i.e., a two-decade mean geostrophic current derived from a combined satellite and in situ mean dynamic topography) described by Rio et al. (2014) is adopted for a definition of regions here. The largescale divisions that are relevant to this report are shown by white lines:
Introduction
for either the zonal or meridional component of the 15-m GlobCurrent combined current (NFR), where I is the corresponding drifter estimate. Errors (δ) with different subscripts are taken to be uncorrelated, but all GlobCurrent estimates (NFR) share the same error at observation time (δN). Note that the affine calibration (α and β) varies for the three GlobCurrent estimates because extrapolation involves its own bias. Corresponding (co)variance estimates are
Physical interpretation of an extensive array of observations is the basis of a true/target/best ocean surface current, but further tuning of such estimates to specific applications is also clearly relevant. Since calibration and validation are inseparable (Stoffelen 1998), this update advances toward both a validation of the GlobCurrent combined (geostrophic and Ekman) 15-m surface current, as well as a calibration of the combined current to the movement of surface drifters. The idea is to tune the GlobCurrent product to applications that seek a proxy for drifter information with the coverage and resolution of the GlobCurrent analysis. It is expected that as sophisticated physical interpretations and corresponding gridded products evolve, the need for such a calibration may diminish, even if some applications may continue to benefit from their own (nonlinear) tuning and parameterization.
Agulhas
Identification of Collocations Drifter velocity estimates are first gridded to the 6-h, 0.25O resolution of the GlobCurrent analysis. Well over a million velocity estimates (a) are available from drifters that likely retained their drogues (Rio et al. 2012). Three groups of the most complete timeseries of more than 10 observations at 2o resolution are taken from this set. The nearest 200 collocations to current speed at intervals of 0.01m/s are used to obtain a functional dependence of the performance metrics.
a) Drifters Sep 2012-2014
where σ is truth or error variance. Ordinary and reverse least squares regression provide an initial bound on all parameters: assuming that σI2 = 0, we have βOLS = Cov(I,N) / Var(I) and assuming that σN2 = 0, we have βRLS = Var(N) / Cov(I,N) (Su et al. 2014). It is notable that β is the only unknown, and hence, all triple collocation metrics of interest (e.g., RMSE and correlation to truth or SNR; McColl et al. 2014, Gruber et al. 2016) can also be bounded. Our use of three sources of current information, but with only two distinct resolutions (point observations and gridded analysis) avoids a well known challenge of the collocation method: correlations not captured by the lowest resolution information source are difficult to estimate, and can require iteration (Vogelzang and Stoffelen 2012). However, there are corresponding NFR covariance terms that appear in the equations for the drifter and GlobCurrent error variances. Following McColl et al. (2014), these are
b) Group A
Gulf Stream
c) Group B
d) Group C
Drifters are taken from Sep. 2012 to Dec. 2014 and were not employed in the analysis. Extrapolation provides the other two data values, based on data from outside a typical analysis data window (cf. Rio et al. 2014). As the GlobCurrent analysis is a linear combination of the Ekman and geostrophic components, extrapolations from two different window lengths (6 h and 5 days) are combined: Ekman 6-hourly
In situ Time Ekman geostrophic
daily
In situ geostrophic
Acknowledgements We illustrate below that the forecast and revcast extrapolation error covariances are not needed, however, to reveal a slight underprediction of strong current in the GlobCurrent analysis. That is, ordinary and reverse least squares regression suffices. In other words, a relevant discretization of the collocations is performed here using a range of observed surface current speeds. It is important to accommodate this nonlinearity in mapping calibration and performance. Mapping a nonlinear relationship between two variables requires a large number of collocations for good binwise resolution. Tolman (1998) demonstrates that high bin resolution, or relatively small samples in individual bins, is preferable. Stoffelen (1998) cautions against overly small samples mainly because a sufficient number is required to identify additive and multiplicative bias (i.e., the impact of an approximately Gaussian error distribution). The present study employs bin averaging to explore nonlinearity in the relationship between GlobCurrent and drifter current estimates. The number of available collocations permits good binwise resolution, with O[100] collocations in each individual bin. The calibration and performance metrics that have been developed in the framework of the collocation method (cf. Stoffelen 1998, McColl et al. 2014, Gruber et al. 2016) are applied. The relevant calibration parameters are
Drifter velocity estimates (based on centered differences) capture motion at scales below what GlobCurrent can resolve, so it is not surprising that the least squares bound on weak current is quite broad. Here, we include a slightly more refined “bounded reverse least squares” that is taken from the σI2 equation to the left, and assuming that σ N2 = 0. This estimate makes use of the extrapolated data (ordinary least squares does not). The signal of an underprediction of strong current in the Agulhas is resolved above the noise in the data at current speeds above about 0.5 m/s, where even the bounding RLS estimate of multiplicative bias (βN) is less than one. Both RMSE and correlation with the true meridional current increase with current speed. Additive bias is minor.
Three groups (A, B, C) are combined to yield 6194 collocations for the Madagascar and Agulhas region. This permits good binwise resolution of the current speed dependence of the collocation metrics. Current speeds of 0.10.7 m/s are resolved. As elsewhere, there is a slight underprediction of strong current, and RMSE and analysis sensitivity trend upward for stonger current speed, as might be expected. Only minor differences between the zonal and meridional components are found (not shown).
North Atlantic
Encouragement of the ESA Ocean Heat Flux project team is gratefully acknowledged. This work has been funded by the ESA GlobCurrent and Ocean Heat Flux and FP7 E-GEM projects.
References Gruber, A., C.H. Su, S. Zwieback, W. Crow, W. Dorigo, and W. Wagner, 2016: Recent advances in (soil moisture) triple collocation analysis. Int. J. Appl. Earth Obs. Geoinf., 45, 200-211. McColl, K. A., J. Vogelzang, A. G. Konings, D. Entekhabi, M. Piles, and A. Stoffelen, 2014: Extended triple collocation: Estimating errors and correlation coefficients with respect to an unknown target, Geophys. Res. Lett., 41, 6229–6236, doi:10.1002/2014GL061322. Stoffelen, A., 1998: Toward the true near-surface wind speed: Error modeling and calibration using triple collocation, J. Geophys. Res., 103(C4), 7755– 7766, doi:10.1029/97JC03180. Rio, M.-H., S. Mulet, and N. Picot (2014), Beyond GOCE for the ocean circulation estimate: Synergetic use of altimetry, gravimetry, and in situ data provides new insight into geostrophic and Ekman currents, Geophys. Res. Lett., 41, doi:10.1002/2014GL061773. Su, C.-H., D. Ryu, W. T. Crow, and A. W. Western, 2014: Beyond triple collocation: Applications to soil moisture monitoring, J. Geophys. Res. Atmos., 119, 6419–6439, doi:10.1002/2013JD021043. Tolman, H. L., 1998: Effect of observation errors in linear regression and binaverage analyses. Quart. J. Roy. Meteor. Soc., 124, 897–917. Vogelzang, J. and A. Stoffelen 2012: Triple collocation. EUMETSAT Report. [Accessed September 2015 at https://nwpsaf.eu/deliverables/scatterometer/TripleCollocation_NWPSAF_TR _KN_021_v1_0.pdf. Zwieback, S., K. Scipal, W. Dorigo, and W. Wagner, 2012: Structural and statistical properties of the collocation technique for error characterization, Nonlin. Processes Geophys., 19, 69–80, doi:10.5194/npg-19-69-2012.
The combined groups yield 9298 collocations for the North Atlantic region (Fig. 1). Current speeds of only 0.1-0.4 m/s are resolved, with almost no additive bias. The multiplicative underprediction of strong current appears to be as great as half the speed of collocated drifters. RMSE changes little across the resolved current speed range and possible range in analysis sensitivity is quite broad, presumably owing to the weaker currents of this region.
Conclusions An error model for GlobCurrent and drifters, with additional information and bounds provided by extrapolated data have been proposed. It is notable that this model does not suffer from a neglect of autocorrelated errors, which are explicitly defined (i.e., by δN and σ N2). The resulting bounds on triple collocation metrics are sufficient to provide some instructive evidence of regional differences in calibration and performance of the GlobCurrent analysis. We propose that the simplest approach to calibration is to adopt α and β values nearest to 0 and 1, respectively, for verifying a reduction in RMSE and improvement in skill in trajectory calculations.
