we evaluate the significance of jumps in each individual series using the generalized least squares (GLS) approach, with the identified noise model, which provides statistically more efficient error estimators. Finally, we construct a predictive rule to make the attribution decision. The novelties of our approach lie in the use of the GLS estimation in this context, and in the construction of a predictive rule, using test results obtained from real data with a resampling procedure taking into account the small samples of our dataset (here limited to 494 cases). The method is described in detail in Chapter 4. When applied to the set of 81 GNSS stations from the CODE REPRO2015 dataset, with ERA5 reanalysis used as reference and the NGL GNSS dataset for the nearby stations, 62% of the tested change-points were attributed to GNSS, 19% to the reanalysis, and 10% to changes in both GNSS and reanalysis. Detailed analysis of the GLS test results input to the prediction rule helped us to understand the sensitivity of the attribution method and interpret the final result after aggregation (i.e. when predicted results from several nearby stations are combined). Indeed, the prediction results are sensitive to the resampling strategy used for the construction of the predictive rule. We tested several variants, using different significance levels for the tests, balanced or imbalanced samples and different aggregation rules (based on case-specific prior probabilities from the Test Table, Table 4.1, or using the distance to the nearby station). The imbalance of configurations observed or assumed in reality seems better taken into account with a balanced resampling strategy combined with an aggregated rule based on the prior probabilities. Test results are also sensitive to the magnitude of noise in the data which depends on the distance between the main and nearby stations. The noise levels increase substantially with the distance between the main and nearby stations in a sparse network. This high noise can hinder the significance tests in the four non-collocated series, thus impacting the final prediction result. This challenge can be addressed by applying our method within denser networks, such as provided in NGL dataset. The potential of this dataset will be explored in the near future. Some preliminary results are already presented in the Appendix.

## 5.2 Perspectives

This study raises certain questions that could serve as potential avenues for future work aimed at enhancing GNSS data processing and climatic homogenization.

a) In the context of GNSS data processing, segmentation proves to be highly sensitive to small changes in mean of the series. These characteristics are influenced by several key aspects of GNSS data processing, with particular emphasis on factors such as: the a priori ZHD correction, the antena/radome calibration model, the mapping function and the elevation cutoff angle. The consistency and reliability of GNSS products generated through different software packages and incorporating varying processing features can be a subject of scrutiny. This can be addressed by conducting comparative analyses of different GNSS tropospheric products, especially focusing on noise, bias and homogeneity. The assessment of data homogeneity can be accomplished through segmentation (to detect changes in the mean) and attribution methods (to identify unambiguously those of GNSS origin). Afterwards, it may be tempted to optimize the GNSS processing in order to make the products intrinsically more homogeneous. The homogenization tool can thus serve to assess the progress accomplished at the processing level, in addition to completing the final homogenization of the reprocessed data.

b) In the following, we outline ideas to enhance our homogenization method, with a particular emphasis on
improving both the segmentation and the attribution methods.

The first goal in the short term is to test the current attribution method with an extended GNSS dataset from the NGL repro3. With this dataset, we can strengthen the nearby selection rules by selecting nearby series with a smaller percentage of gaps and shortening the distance between the main station and nearby stations. This will reduce the magnitude of the noise in the difference series and improve the power of the test procedure.

Furthermore, as mentioned in section 4.1.5, the synthetic dataset is generated from the test results of real data. With the NGL repro3 dataset, in particular, the larger size of the dataset can improve the prediction rule. Firstly, it can improve the performance of the current decision rule proposed in chapter 4 because with more tested changes there will be fewer repetitions in the synthetic dataset constructed by resampling. Secondly, we could consider a new method to construct this decision rule. For example, rather than doing resampling series by series, we could consider doing resampling test by test (on all the results at the same time), which would be fairer to take into account the dependence between the tests.

However, one critical issue arises with the NGL data regarding the validation of the final results with metadata. The preliminary results presented in A show a drop in the percentage of validated change-points which is explained by the fact that only one half of the analysed stations have metadata reported in the NGL data holding. This could be partly improved by running additional quality control procedures based on the RINEX files as was done by Quarello et al. (2022).

Second, a R package of the attribution method will be made available, enhancing its accessibility to the scientific community.

Other approaches to the segmentation method implemented currently could be tested, such as using nearby GNSS stations as references instead of a reanalysis. When both the main and nearby stations are processed in a consistent manner, the remaining inhomogeneities would mainly originate from station-specific changes, such as instrumental changes or changes in the surrounding environment. Consequently, this approach can facilitate the attribution of change-points and lead to significant improvements in the homogenized series.

The results of heteroscedasticity and autocorrelation in the difference series, as demonstrated in Chapter 4, underscores the need for two primary enhancements to the segmentation method. Firstly, the current practice of modeling heteroscedasticity on a monthly basis within the segmentation method may benefit from being replaced by a more comprehensive continuous function, as exemplified in Figure 4.2. Secondly, the current use of white noise does not take into account the underlying autocorrelation patterns present in the data. A more appropriate model for data autocorrelation (e.g., AR(1), MA(1), or ARMA(1,1)) should be used to improve the segmentation method. Finally, using an appropriate loss function for segmentation inference like Hubert or Biweight loss (instead of log-likelihood) could avoid data pre-processing with respect to outliers.

In the long-term exploration of this topic, conducting an analysis of homogenized GNSS IWV trends presents a compelling opportunity, especially within the context of climate change studies. This analysis provides empirical evidence to address a key question: How will water vapor change in a warmer climate? The
water vapor feedback is frequently estimated using a global climate model (GCM), which strongly depends on parameterization. Homogenized GNSS datasets can be compared and used as a reference for tuning parameters in climate models, aiding in obtaining the "best" estimates of water vapor feedback and its uncertainties.
# Appendix A

## Appendix: Preliminary results with an extended dataset

This appendix presents some preliminary results of the application of the full homogenization process to an extended GNSS dataset of more than 6000 stations extracted from the NGL data holding. At the time of writing, several steps are already completed, including station selection, ZTD data conversion and screening, segmentation, and validation with respect to the NGL metadata. The attribution step is currently pending. After the attribution, the correction of jumps in the GNSS series will be carried out and trends will be estimated. The IWV data before homogenization is already available publicly here https://doi.org/10.25326/518. The homogenized version will be added later.

### ★ Station selection

The NGL repro3 data holding contained 20,747 stations, as of 13 April, 2023. It is updated on a daily basis. Data products are available in the form of time series of daily station coordinates and daily SINEX tropospheric (tropo) files containing the estimated ZTD and gradients parameters with 5-min sampling. Station metadata is provided in the form of a station list, including a 4-character station ID (renamed from their original names when they are already registred from a previous data network/provider), a priori station coordinates, dates of start and end, and number of computed days.

For the purpose of climate trend estimation, a sub-set of stations with more than 10 years of observations and less than 10% of gaps (at the daily sampling rate) was selected. This selection came up with 6197 stations. After checking the station position stability from the coordinate time series, with thresholds 0.001° in latitude and longitude, and 1 m in height, the number is decreased to 6058 stations. The 5-min resolution tropo files have been downloaded for this subset of stations, and ZTD estimates and their formal errors have been extracted are averaged into daily values.

### ★ Data screening

Daily ZTD values were screened for outliers using a new procedure compared to the earlier procedure used with IGS repro1 and CODE REPRO2015. Firstly, the formal errors have been passed through a segmentation algorithm and a post-processing was developed to detect both singular outliers (e.g. one day with a high formal error)
and extended periods (weeks to months) of high formal errors which are symptomatic of degradation of GNSS instrumentation or measurement conditions. In some occasions, long-term drifts (over several years) of formal errors were also discovered and four stations were blacklisted for this reason. This post-processing relies on several conditional tests and thresholds which required some tuning before being applied to the data set. 0.62% of the ZTD values and 6 stations were rejected based on this screening procedure based on the formal errors only.

Secondly, a complementary screening step was applied based on the GNSS - ERA5 IWV difference. The ZTD values were beforehand converted to IWV using the same procedure as for the IGS repro1 and CODE REPRO2015 data sets, except that here ZHD and $T_m$ values computed from the model were first averaged into daily values and applied to the daily GNSS ZTD estimates. The screening method used for the IWV differences also differs from previous procedures in the sense that it is applied on the full time series (instead of yearly segments) which allows to implement a sliding window approach where the individual daily values are compared to a moving median and tested with respect to a robust scale estimator (MAD). This procedure also relies on some parameterization (e.g. window size and test thresholds) that had to be tuned. With the current settings, it rejects an additional 0.89% of daily values and 2 additional stations, leaving 6048 stations available for the application.

## ★ Segmentation

The quality-checked IWV differences were passed through the GNSSseg segmentation method with standard settings (as in Quarello (2020) and Nguyen et al. (2021)). The segmentation detected 15680 change-points in 6048 stations, which represents an average of 2.6 change-points per station. Among these, 611 stations (10.1%) were considered homogeneous (no change-point detected), and 5437 stations (89.9%) had at least one change-point. The maximum number of detected change-points in one series was 16 (station J137).

The post-processing analysis described in Quarello (2020) was applied to detect "clusters" (groups of consecutive segments with significantly higher means than their adjacent segments) using the standard settings (distance between change-points of 80 days and significance level of 0.05). It detected 1427 clusters among which 1005 deemed significant. For each significant cluster, all the change-points pertaining this cluster were replaced by one single change-point. For the non-significant clusters, all the change-points were eliminated. After this step, 13923 change-points were remaining.

## ★ Validation with respect to metadata

The metadata provided by NGL (named "steps" file) contains information on changes in station instrumentation (antenna, receiver, radome) and other notable information (site change, monument change, volcanic eruption) as well as events flagged as "unknown". In total, we regrouped the different types of changes into 11 categories. However, we must emphasize that information is registered by NGL only for 6283 stations out of 20747. Among the 6048 stations that were analysed, only 3309 (49.6%) have metadata reported. The most frequent changes involve antennas, receivers, cutoff changes, and radomes, in decreasing order, which altogether count for 98% of the registered changes.
Among the 13923 detected change-points, 6921 change-points could be confronted to metadata and 1228 (8.8%) were coincident or "validated" with respect to known changes within +/- 62 days (this is the same window as used in Quarello et al. (2022) and Nguyen et al. (2021)). The proportion of validated antenna, receiver, cutoff, and radome changes is 71.7%, 35.7%, 20.4%, and 34.7%, respectively (some changes involve more than one type, so the sum exceeds 100%), which represent again 98% of all validated changes. We note that the percentage of validated changes (8.8%) of all detected change-points is significantly lower than what was found with the IGS repro1 and CODE REPRO2015 data reported in paper No. 1. The reason is that the metadata provided by NGL are available for only one half (49.6%) of the analysed stations. This points to a severe lack of essential information necessary to control the quality and validate the segmentation results of this dataset. Hopefully, the attribution method will be able to achieve more robust conclusions on the significance and origin of the detected change-points.

★ Attribution

At the time of writing, the attribution of the NGL dataset was not yet completed. Several preliminary steps are currently being performed. The first one concerns the selection of the main and nearby stations. Because the ultimate goal is to estimate climatic trends, we decided to select as main stations all those having at least 20 years of IWV data. This selection retains 718 candidate main stations, among which 20 are homogeneous and 698 have at least one change-point.

Then we select for each main station all possible nearby stations within a maximum horizontal distance of 200, 100, or 50 km, and a vertical distance smaller than 500 m. With these distance limits, some of the main stations have no nearby station and their change-points cannot be tested with the attribution method. The number of main stations having at least one nearby stations decreases from 698 to 673, 631, and 567, respectively, for the three maximum distances. Next we checked more precisely how many of the detected change-points in the main stations can actually be tested based on the matching of the time periods with the nearby stations. Therefore, we pre-select the change-points, for each main station, which do not have another change-point closer than 250 "effective" days (by "effective" we mean that only days with data are counted here, i.e. days with missing data are not counted), either within the time series of the main station itself or in the considered nearby station. This step rejects some of the main change-points and rejects also some of the candidate nearby stations when they have (nearly) coincident change-points. This pre-selection gives the following results (this is only for main stations which have at least one change-point):

* With the 200 km distance limit: 632 main stations and 1708 change-points can be tested with the help of 3960 nearby stations (total 44474 main-break-nearby triplets). Among them, 407 main stations can be tested for all their change-points (i.e., can be fully homogenized). The number of nearby stations per change-points is: mean = 26.0, min = 1, max = 152.

* With the 100 km distance limit: 581 main stations and 1455 change-points can be tested with the help of 2467 nearby stations (total 17208 main-break-nearby triplets). Among them, 317 main stations can be tested for all their change-points (i.e., can be fully homogenized). The number of nearby stations per change-points is: mean = 11.8, min = 1, max = 79.

* With the 50 km distance limit: 499 main stations and 1115 change-points can be tested with the help of 1302 nearby stations (total 6722 main-break-nearby triplets). Among them, 202 main stations can be
tested for all their change-points (i.e., can be fully homogenized). The number of nearby stations per change-points is: mean = 6.0, min = 1, max = 50.

These statistics indicate that on average between 6 and 26 nearby stations can be used to test change-points in the main stations, depending on the distance limit, and that a good fraction of the main stations (between 40 and 64%) can be fully homogenized. For those change-points that cannot be tested, a manual validation remains necessary, which can be assisted by the use of metadata when available.

The issue of lacking metadata in the NGL data holding can be partly improved by running additional quality control procedures based on the RINEX files as was done by Quarello et al. (2022). Fortunately, the NGL data holding contains Quality Assurance (QA) files which contain statistics computed from the GipsyX processing outputs, for each station, daily. The analysis of these statistics may be of some utility to complement the existing metadata for the validation. For example, if a change-point detected by the segmentation of the IWV difference series coincides with a shift in one of the data quality metrics provided in the QA files, this may give some credit to consider the change-point are originating the GNSS series. Note that the segmentation of formal errors implemented in the new screening method described above also follows this idea and these results may be used as well, as are change-points detected in the GNSS position time series.

The next step before the attribution is the vertical correction of all main-nearby couples. This accuracy of this correction is also in the process of being improved compared to the version that was used in paper No. 2. The procedure will be the similar and apply a 2-parameter correction model as proposed by Bock 2022, but the coefficients will be computed from ERA5 reanalysis on the native 0.25° grid from the four surrounding columns for each station (as opposed to the use of a global 2.5° grid in paper No. 2). Another improvement will be the use of a sliding window regression with a 1-day increment (i.e. the coefficient will be regressed from ERA5 data in a window for each central day) instead of a monthly regression time step.
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