Furthest-first (FF) inversion is a form of coordinate-descent optimisation. Briefly, the FF inversion procedure works as follows. Cruise weighted mean offsets (ci,s for cruise i, where s refers to the iteration step; Appendix ) are calculated for all cruises, and the cruise with the largest ci,s normalised to its uncertainty is adjusted to exactly eliminate its offset. In other words, the cruise that lies furthest from its neighbours is adjusted first. All ci,s values are then recalculated accounting for the adjustment, and these steps are repeated until some convergence criterion is reached. During the procedure, some cruises may be adjusted multiple times, with each successive adjustment refining the overall adjustment value after adjacent cruises have also been adjusted, while others may never receive any adjustments. Some cruises can be fixed to receive pre-defined (e.g., zero) adjustments, and temporal trends in variables can be selectively and approximately preserved. The following sections describe the procedure in more detail.

For each variable that will receive adjustments, an n×n crossover matrix Cij of offsets between pairs of cruises (Sect. ) is assembled, where n is the total number of cruises in the dataset. Each element of Cij contains the offset between cruises i and j such that its value is positive if the data from cruise i are greater than those from cruise j and negative otherwise. The matrix is therefore skew-symmetric (i.e., Cij=-Cji). The main diagonal and any elements where no offsets could be calculated (Sect. ) are set to zero. The weights assigned to the ci,s are also assembled into an n×n symmetric matrix Wij, again with zeroes on the main diagonal and where no offsets are available.

An n×n adjustment matrix Aij,0 is initialised, containing all zeros. The values in Aij,s are updated on each iteration (s) of the FF routine.

An n×n date-difference matrix Dij of the time interval between each pair of cruises is calculated. Each element of Dij contains the date of cruise i minus the date of cruise j, in decimal years. Dij is therefore skew-symmetric, like Cij.

A length n vector ri of trends (rate of change) in the crossovers is computed. Each element of ri contains the slope of a weighted least squares regression of the corresponding rows of Cij against Dij, weighted by Wij. A threshold should be applied to determine whether a trend can be computed for each cruise, for example based on the number of crossovers, with trends that are not computed assigned as zero. Any calculated trends judged to be spurious (how best to judge this may vary by application) and thus not retained in the inversion can also be assigned as zero in ri.

A trend-adjusted crossover matrix Tij is calculated: 1Tij′=Cij-riDij2Tij=(Tij′-Tij′T)/2 where superscript T denotes the transpose of the matrix. Tij′ is similar to Cij, but it is no longer skew-symmetric. Each element of Tij′ contains the crossover difference between cruises i and j, corrected for the trend in cruise i. As each of cruises i and j has a different set of crossovers, their trends are rarely identical. This asymmetry can lead the adjustments calculated by the iterative solver to drift considerably rather than converge. The second step of the calculation, Eq. (), resolves this by averaging the values for each pair of cruises. Each element of Tij is thus corrected for the average of the trends in cruises i and j, and Tij is skew-symmetric. If it is not desired for the inversion to attempt to preserve any apparent trends, then Cij can be used instead of Tij in all subsequent calculations, and Dij and ri do not need to be computed.

For calculating the uncertainties associated with the weighted mean offsets calculated in the next section, the effective sample size (ki) is computed from the weights matrix : 3ki=(∑jWij)2∑jWij2

The critical t-distribution value for each element of ki, denoted ti, is also computed. This contains values for the inverse of the cumulative distribution function of the Student's t distribution with the degrees of freedom specified by ki and at a desired confidence level (here, 1σ). When multiplied by the standard error of ci,s in the next section (Eq. ), ti gives the 1σ confidence interval for the mean offsets.

Finally, a length n Boolean vector pi may be assembled to identify a set of cruises that are not permitted to be adjusted in the inversion. Cruises that are permitted to be adjusted are assigned as true in pi and those that are not, false. Cruises that are not permitted to be adjusted are still accounted for in the rest of the inversion procedure (e.g., when calculating weighted mean offsets). This can be used during a complete inversion if certain cruises will not be adjusted. In GLODAPv3, it will also be used for the annual minor version releases, where adjustments are calculated for and applied to newly added cruises, but the adjustments for existing cruises in the product are not revisited.

The matrices Tij, Wij and Aij,s (where s denotes the iteration step, starting with s=0) are used to calculate the vector of weighted mean offsets for each cruise, ci,s: 4ci,s=∑j(Tij+Aij,s)Wij∑jWij

Determining the uncertainty in each weighted mean offset (ui,s) then requires several further calculations. First, the residuals between the individual crossovers and the cruise weighted mean offsets are calculated (Rij,s): 5Rij,s=Tij+Aij,s-ci,s The weighted variances (σi,s2) are then computed as 6σi,s2=ki∑jWij(Rij,s)2(ki-1)∑jWij and are converted into 1σ confidence intervals for each offset (ui,s): 7ui,s=σi,sti/ki Cruises that have crossovers with only one other cruise are assigned a default uncertainty value, which could be adapted depending on the analysis. For GLODAPv3, we used double the standard deviation of all ci,s at the corresponding iteration step.

Only the cruise with the largest absolute normalised offset is adjusted during each iteration, as follows. An adjustment vector (ai,s) is created as a copy of ci,s; all elements in ai,s representing cruises that are not permitted to be adjusted (based on pi) are set to zero; and then all remaining elements other than the one with the maximum absolute value after normalising to uncertainty are also set to zero: 8ai,s=ci,swhere (abs(ci,s/ui,s)=max(abs(ci,s/ui,s))) and pi0otherwise The adjustment matrix is updated by adding the adjustment vector to its columns and subtracting it from its rows: 9Aij,s+1=Aij,s+ai,s-aj,s Equations () to () are repeated until some convergence criterion is reached (Sect. ).

After the final round of adjustment, the 1σ confidence intervals are calculated with Eq. () and these can be reported as the cruise-level uncertainties in the final adjusted values. These values can be compared with the pre-inversion adjustment confidence intervals to assess how much the cruise-level uncertainties have been reduced by the adjustment process.

The FF routine can be stopped after a set number of iterations or when some threshold is reached, for example based on the size of the adjustments being calculated (which virtually always decreases from one iteration to the next) or some statistic related to the ci,s values (i.e., cruise weighted mean offsets). Care must be taken if using adjustment size as a convergence criterion, because, while the adjustments added in subsequent iterations might be individually negligible, they can add up to a significant total change (Sect. ).

Two different approaches were developed and tested using simulations and the real GLODAPv3 dataset. First, we ran the inversion with all cruises permitted to be adjusted (i.e., pi=True for all i), for enough iterations (at least 1 million) for all calculated adjustments to reach steady state with respect to continued iterations. This first approach is referred to as FF₁.

Second, we used a two-step approach. The aim here was to apply an approximate minimum threshold to the size of the adjustments that would be calculated. To begin, the FF inversion was run with all cruises permitted to be adjusted but with a cut-off for stopping iterations at 5 % of the threshold. For example, for dissolved inorganic carbon (DIC) and total alkalinity, the target adjustment threshold was 2 µmol kg⁻¹, so the iterations were stopped once suggested absolute adjustments became smaller than 0.1 µmol kg⁻¹. After this point, pi was updated such that any cruise with an absolute adjustment after this set of iterations of less than the total threshold would never be permitted to be adjusted. In the examples given here, the total threshold was kept at 2 µmol kg⁻¹, but its value could be revised at this point for example to account for regional variations . The inversion was then restarted from the beginning with the updated pi and run until steady state. This second, two-step approach was used for the final GLODAPv3 analysis and is referred to as FF₂.

The FF inversion approach has been implemented in a free and open-source Python package called xover https://github.com/mvdh7/xover, last access: 20 October 2026. This is built upon several other Python packages including JAX , NumPy , SciPy and Statsmodels .

For simulated networks without noise (i.e., where each crossover exactly represents the assigned offset between its pair of cruises), FF₁ inversion successfully converges on adjustments that are perfectly correlated with the assigned offsets, with a slope of 1. FF₂ inversion converges on adjustments with a good correlation with the assigned offsets close to the ideal slope of 1, with some scatter that scales with the size of the target adjustment threshold (Sect. ). In both cases, there can be a small constant bias, because the inversion is designed to target internal consistency between cruises and does not explicitly attempt to minimise the average size of adjustments: adding a constant value to the data for all cruises does not affect the differences between them. The bias is not significant relative to the uncertainties in the overall inversion process, especially when using real datasets, and it could be removed by subtracting the average adjustment from all adjustments after inversion. Either way, a small global bias would not be inconsistent with GLODAP's mission to produce a dataset that is internally consistent while making no claims about its absolute accuracy. However, it should be accounted for as a source of uncertainty for the most accurate work, as it would affect the apparent internal consistency of marine carbonate system calculations.

Random noise was also added to the simulated crossover offsets to represent the fact that the data from one particular profile of a cruise may not exactly reflect the overall bias of the cruise. Both FF₁ and FF₂ still worked reliably, calculating adjustments that were well correlated with the assigned offsets and with the scatter about the correlation scaling with the spread of the added random noise.

The GLODAPv3 analysis uses data from deeper than 1500 m, on the assumption that there should generally be no significant natural variability through time in biogeochemical variables there. However, in some regions this may not be accurate e.g.,, and where clear trends are observed in pre-inversion offsets, the manual QC process has previously tried to avoid eliminating these trends with adjustments. Our simulations confirm that without trend preservation (i.e., using Cij instead of Tij; Sect. ), FF₁ does eliminate virtually all temporal trends in cruise offsets. On the other hand, using Tij leads to a good correlation between trends before and after inversion (r2=0.75; RMSD = 0.06 µmol kg⁻¹ a⁻¹; Fig. a). The results for trend-preserving FF₂ are very similar to FF₁, with an r2 of 0.97 between the two approaches (Fig. b).

The correlation between trends before and after inversion is not perfect, but simulation results where the true trends are known shows that this is expected. First, the trends determined from the pre-inversion crossover matrix Cij do not perfectly represent the true assigned trends (r2=0.46; RMSD = 0.093 µmol kg⁻¹ a⁻¹; “before” in Fig. c). The pre-inversion trends are calculated from uncorrected data so the biases in the cruises that the inversion seeks to solve will also cause incorrect trends to be targeted. There is thus uncertainty in the target trend for FF to preserve. Also, trends are averaged across two cruises in Eq. (), so the “target” trend will already not be exactly the same as either cruise's own trend. This means that cruises connected to other cruises with compatible trends will generally have their trends preserved, but not otherwise, which is desirable, because a trend not supported by connected cruises is more likely to be spurious.

Real-world datasets have additional complications. Each cruise spans a different geographic extent and trends might therefore differ across the cruise. The crossovers it has are with different cruises at different spatial points where the trend might be different than the average for the first cruise. Real-world trends also normally display variability and changes over time.

Despite these limitations, our simulations show that the trends after trend-preserving FF more closely resemble the true assigned trends than before inversion. In the example shown in Fig. c, the root-mean-square difference (RMSD) between the apparent trends and the true assigned values is ∼ 20 % lower after trend-preserving inversion than before inversion.

In simulated networks, convergence within well-connected “clusters” of cruises – representing regions or ocean basins within the real dataset – happens relatively quickly (e.g., within ∼ 5000 FF iterations for a GLODAP-like network of ∼ 500 cruises), and then these clusters are drawn together over a much greater number of iterations (Fig. ). The rate at which clusters converge depends on how well-connected they are, with the real GLODAP dataset taking towards 1 000 000 iterations to converge if all cruises are permitted to be adjusted (Fig. ). Essentially, cruises within the better-connected clusters are adjusted so that they are internally consistent within the clusters first, and then these clusters as a whole converge towards the cluster containing the greatest number of cruises. This cluster convergence takes many more iterations, as multiple cruises have to be shifted together.

It is necessary for the crossover matrix to be symmetrical for the solution to converge reliably. Trends can be slightly more accurately preserved by not symmetrising the trend-corrected crossover matrix, i.e., using Tij′ instead of Tij in the inversion. However, using Tij′ generally causes the FF iterations to fail to converge, because for each pair of cruises i and j, the adjustment that should be applied to minimise the offset from the perspective of i is generally inconsistent with that from the perspective of j. This means that there is a continuous drift in adjustments upon continuing iterations, so no usable solution is returned.

We find overall reductions in the cruise-level uncertainties (ui,s; Eq. ) after FF inversion both in simulations and tests with real data. Using the GLODAPv3 DIC dataset as an example, FF₂ inversion with a minimum adjustment cut-off (Sect. ) adjusts 24 % of ∼ 500 cruises and reduces the average cruise-level uncertainty to ∼ 80 % of its pre-inversion value. This reduction in uncertainty is consistent across all cruises, including those that were not permitted to be adjusted. Using FF₁ inversion with all cruises adjusted reduces this uncertainty further to 50 % of pre-inversion. These numbers support the statement that the FF₂ approach used by GLODAP attempts to maximise internal consistency of the data product while still avoiding small or uncertain adjustments.

As discussed in Sect. , for the GLODAPv3 dataset, the WLSQ inversion returned adjustments for some variables with significant regional biases, where roughly consistent adjustments were calculated across entire regions. While the WLSQ result was mathematically valid, adjusting an entire region is not consistent with the GLODAP philosophy, which prefers to make as few adjustments as possible, and it conflicts with our earlier reasoning (Sect. ) that the apparent regional differences are more likely an artefact of random uncertainties in the limited number of cruises that connect between regions than a true bias. We will use the real GLODAPv3 DIC dataset as an example in this section.

For the same dataset, the FF₁ inversion returns a similar set of regional biases (Fig. a) with an almost perfect correlation with the WLSQ adjustments (r2=0.999; Fig. a). This demonstrates that FF does converge on a valid set of adjustments. The mean adjustment for the WLSQ approach is virtually zero, while FF₁ gives a mean adjustment of ∼ 2.4 µmol kg⁻¹. This discrepancy is not unexpected given that the approach does not have an explicit constraint to hold some particular average value constant, as discussed in Sect. . Again, it would be perfectly valid to add or subtract a constant from all calculated adjustments to have an average of zero.

When implemented globally, the two inversion approaches (WLSQ and FF₁) both indicate that there appears to be a bias of around 6 µmol kg⁻¹ between the Atlantic and Pacific regions. WLSQ corrects the bias by adjusting the Pacific downwards and the Atlantic upwards, whereas FF leaves the Pacific roughly at zero and adjusts the Atlantic upwards more. Which of these (or any other) options is correct is a subjective judgement that cannot be determined objectively by any inversion method. Figure  shows that the Pacific contains more cruises than the other regions and that these are all very densely connected. FF₁ therefore adjusts the other cruises towards this more densely packed region, as discussed further in Sect. . This conceptual explanation is provided for context, and we reiterate that GLODAP has never used a global WLSQ nor FF₁.

However, if a cut-off is used to adjust only a subset of the cruises (i.e., FF₂), then a different pattern emerges: the regional biases disappear, and the pattern of adjustments is more random (Fig. b), as we expect. The adjustments are correlated with the FF₁ version, but there are significant differences (Fig. b). This highlights that choosing not to adjust some cruises changes the adjustments that should be used for other cruises, and it is not valid to select only a subset of points to adjust from WLSQ or FF₁. The adjustment calculated for a given cruise depends on if and how the cruises around it are also adjusted. Using FF₂ eliminates this problem with the GLODAP strategy that only became apparent after a global WLSQ was attempted.

In some applications, the FF₁ solution may be more appropriate to apply than FF₂. Determining which option to use requires expert analysis of the results and understanding of the underlying system. In this instance, as discussed in more detail in the GLODAPv3 data description paper , we can see that the different ocean basins are relatively poorly connected together; for example, the Pacific and Indian-Atlantic are connected by a single cruise in the Southern Ocean (Fig. ). This means that entire regions are shifted to satisfy apparent offsets in a very small number of connecting cruises. If these offsets are thought to be meaningful then FF₁ might be appropriate. On the other hand, if they are more dominated by random noise in the individual offsets then using FF₂ is better.

The main motivation for investigating alternative approaches to the WLSQ method used in earlier versions of GLODAP was that the WLSQ results had unusual features that were difficult to interpret such as the apparent regional biases when applied globally. From the user perspective, the calculation proceeds in a single step that returns the full set of calculated adjustments in one go, which means that it is challenging to rationalise why a particular unexpected adjustment has been made and thus to decide whether or not to accept it. The FF approach is simpler to understand, and the solution can be investigated one iteration at a time to see why a particular adjustment has been calculated. The regional patterns in the DIC adjustment discussed in the previous section are a clear example.

The practical difference between WLSQ and FF can be thought of as analogous to the difference between a mean and a median. Consider a cluster of connected cruises where one “bad” cruise has a large true offset but all others are “good” with near-zero offsets. The WLSQ solution treats the cluster together and will calculate a smaller than necessary adjustment for the bad cruise, compensated for by also requiring a small constant adjustment to all of the good cruises. On the other hand, the FF would adjust the bad cruise first, bringing it into line with the good cruises, for which no constant adjustment would be required. However, in the real world we do not necessarily know which cruises are good and which are bad. FF assumes that if a large number of cruises agree with each other and a smaller number disagree, then the majority are correct, which may not always be the case. For example, if lots of cruises were conducted by one particular laboratory and that laboratory does have biased results, then a single cruise passing through the same area with data from an unbiased laboratory would receive an erroneous adjustment. If such a case were identified through independent evidence, both WLSQ and FF could be initialised with a correction to the biased dataset(s), eliminating the problem.

In earlier versions of GLODAP, some cruises had such large offsets that they skewed the results of the WLSQ inversion. This problem was circumvented by iteratively applying manual pre-inversion adjustments to these cruises before running WLSQ. With FF, such pre-inversion adjustments are generally not necessary, because isolated cruises with very large offsets will be adjusted first so the adjustments for the cruises connected to them will not be adversely affected by their initial presence. However, if there is some independent reason (i.e., not as a result of the crossover analysis) why a specific correction is thought to be appropriate for a cruise, then it is still advisable to apply this before running FF. The FF approach also has the capability of preserving temporal trends in the crossovers to some extent, as discussed in the previous section. WLSQ behaves more like the standard FF, actively eliminating any trends. It might be possible to devise an equivalent adaptation for WLSQ, but to the best of our knowledge, this has not yet been done.