New furthest-first algorithm sharpens the ocean’s carbon record in GLODAPv3

The global ocean has absorbed almost 30 percent of all the carbon dioxide humanity has emitted since 1850, along with roughly 90 percent of the excess heat trapped by greenhouse gases. Tracking how that carbon spreads through the sea, and how the resulting warming, acidification and deoxygenation unfold, depends on observations from hundreds of research cruises collected over many decades. But those measurements come from different laboratories, instruments and methods, each with its own subtle systematic biases. A new technical study published in the journal Ocean Science describes a redesigned mathematical procedure, called furthest-first inversion, that the Global Ocean Data Analysis Project now uses to reconcile those biases in its latest data product, GLODAPv3.

GLODAP, launched in the late 1990s, has one central goal: to produce a dataset of carbon-relevant variables for the world ocean in which every measurement is comparable across time and space, so that long-term trends can be determined reliably. The project’s emphasis is deliberately on internal consistency rather than absolute accuracy, because there is no established way to objectively determine the latter. At the same time, GLODAP aims not to change the regional or global averages of the variables it adjusts. Achieving that balance requires deciding, cruise by cruise, which datasets need small corrective offsets and which can be left untouched.

The quality-control backbone of GLODAP is the crossover analysis. Deep water, typically below 1500 metres, is chosen for comparison because natural seasonal variability there is minimal. When two cruises sampled the same variable within a lateral distance threshold, which has ranged from 100 to 300 kilometres across GLODAP versions, each profile from one cruise is interpolated to a series of depth, temperature or density levels and compared with overlapping profiles from the other. A weighted mean offset and its standard deviation are then computed, with less-scattered levels carrying more influence. If a cruise has at least three overlapping profiles with another, a crossover difference is recorded, and these differences are compiled into a large matrix covering all cruise pairs.

For previous major versions of the product, the offsets in that matrix were reconciled with a weighted least squares inversion, applied separately to different ocean regions. When the team behind GLODAPv3 attempted a global inversion instead, problems emerged. The calculated adjustments for certain variables showed strong spatial patterns: entire regions received roughly uniform corrections. Those patterns were not visible in the raw pre-inversion differences between cruises, had no apparent physical cause, and were not associated with particular countries, laboratories or research groups. Damping the least squares model with a set of pre-defined core cruises reduced the patterns but did not remove them.

A network analysis revealed the likely culprit. GLODAP cruises are densely connected by crossovers within the major ocean basins but only sparsely connected between basins. The apparent differences between basins therefore probably reflect random uncertainties in the few cruise pairs that link regions, uncertainties that do not average out to zero because the connections are so scarce, rather than true regional biases that should be corrected. In one striking example from the real GLODAPv3 dissolved inorganic carbon dataset, the Pacific and Indian-Atlantic clusters are connected by a single cruise in the Southern Ocean, meaning entire regions could be shifted to satisfy apparent offsets in a very small number of connecting measurements.

The least squares approach carried other technical liabilities. Because it optimises everything at once, it is sensitive to outliers: one cruise with a large offset skews the adjustments calculated for every cruise connected to it. The crossover matrix is also highly overdetermined and unevenly distributed, with strong groupings that least squares methods are known to handle poorly. Moreover, the method minimises individual crossover differences rather than the average difference for each cruise, so individual cruises can end up with non-zero average offsets even after adjustment. Compounding matters, expert panels in earlier versions had routinely ignored or modified some calculated adjustments, but each adjustment is only valid in the context of the complete set, so selectively discarding parts of the solution undermines its coherence.

The new furthest-first method, developed by Matthew Humphreys of the NIOZ Royal Netherlands Institute for Sea Research and colleagues, takes a different route. It is a form of coordinate-descent optimisation in which cruises are adjusted one at a time, iteratively, starting with the cruise whose weighted mean offset is largest relative to its uncertainty, in other words the cruise lying furthest from its neighbours. After each adjustment, all offsets are recalculated and the process repeats until a convergence criterion is met. Some cruises may be adjusted several times as their neighbours shift, while others may never be adjusted at all. The method can also fix a chosen subset of cruises so they receive no adjustment while still accounting for their offsets, and it can approximately preserve selected temporal trends in the crossover differences.

To validate the approach, the team built simulated crossover networks with statistical properties matching the real GLODAPv3 network for dissolved inorganic carbon, in which the true offsets and trends are known because they were assigned. Without added noise, the fully permissive version of the method converged on adjustments perfectly correlated with the assigned offsets. With random noise added, both variants still recovered the assigned offsets reliably, with scatter scaling with the noise. Trend preservation worked as intended: without it, the inversion eliminated virtually all temporal trends, but with the trend-corrected formulation the trends before and after inversion correlated strongly, and the root-mean-square difference between apparent and true trends was about 20 percent lower after trend-preserving inversion than before it.

Convergence behaviour in the simulations mirrored the structure of the real network. Well-connected clusters of cruises, representing regions or basins, became internally consistent within roughly 5000 iterations for a GLODAP-like network of about 500 cruises, and then the clusters as a whole drifted together over many more iterations, with the full real dataset requiring towards one million iterations when all cruises were permitted to be adjusted. For the final GLODAPv3 analysis the team used a two-step variant, FF2, in which iterations are first stopped once suggested adjustments fall below 5 percent of a target threshold, for example 0.1 micromoles per kilogram against a 2 micromoles per kilogram threshold for dissolved inorganic carbon and total alkalinity, and any cruise whose accumulated adjustment stays below the full threshold is then permanently excluded before the inversion restarts. Applied to the real dissolved inorganic carbon dataset, FF2 adjusted only 24 percent of roughly 500 cruises while reducing the average cruise-level uncertainty to about 80 percent of its pre-inversion value, a reduction that extended even to cruises that were not adjusted; the fully permissive variant cut uncertainty further, to 50 percent.

When the researchers compared the fully permissive furthest-first solution with the weighted least squares result on the real GLODAPv3 data, the two agreed almost perfectly, with a correlation of 0.999, confirming that the new method converges on a mathematically valid solution. Both approaches indicated an apparent bias of around 6 micromoles per kilogram between the Atlantic and Pacific, but they distributed the correction differently, and the authors stress that choosing among such options is a subjective judgement no inversion can settle objectively. Crucially, the two-step FF2 variant, with its adjustment cut-off, made the regional biases disappear, yielding the more random adjustment pattern expected if inter-basin offsets are largely noise. The method has been implemented in a free, open-source Python package called xover, and its conceptual transparency, allowing each iteration to be inspected, marks a practical advance over the one-step least squares solution. The authors are careful to note, however, that no algorithm can replace expert judgement in deciding convergence criteria, which cruises may be adjusted, and which trends are worth preserving.

Subject of Research: A new furthest-first inversion algorithm for internal consistency adjustments of ocean biogeochemical cruise data in GLODAPv3

Article Title: Furthest-first inversion for internal consistency adjustments in the biogeochemical data product GLODAPv3

Article References: Humphreys, M. P., Lauvset, S. K., Lange, N., Bittig, H. C., Carter, B. R., Hoppema, M., Murata, A., Olsen, A., Tanhua, T., Ulfsbo, A., Velo, A., Woosley, R. J., Azetsu-Scott, K., Müller, J. D., & Pérez, F. F. (2026). Furthest-first inversion for internal consistency adjustments in the biogeochemical data product GLODAPv3. Ocean Science, 22(5), 3403-3414. https://doi.org/10.5194/os-22-3403-2026

Image Credits: AI Generated

DOI: 10.5194/os-22-3403-2026

Keywords: GLODAPv3, ocean carbon, crossover analysis, inversion algorithm, internal consistency, biogeochemistry, anthropogenic CO2, quality control, coordinate descent, dissolved inorganic carbon, ocean acidification, open-source software