The Task Team on Homogenization (TT-HOM) of the Open Panel of CCl Experts on Climate Monitoring and Assessment (OPACE-2) of the Commission on Climatology (CCl) of the World Meteorological Organization (WMO) has published their Guidance on the homogenisation of climate station data.

The guidance report was a bit longish, so at the end we decided that the last chapter on "Future research & collaboration needs" was best deleted. As chair of the task team and as someone who likes tp dream about what others could do in a comfy chair, I wrote most of this chapter and thus we decided to simply make it a blog post for this blog. Enjoy.

## Introduction

This guidance is based on our current best understanding of inhomogeneities and homogenisation. However, writing it also makes clear there is a need for a better understanding of the problems.

A better mathematical understanding of statistical homogenisation is important because that is what most of our work is based on. A stronger mathematical basis is a prerequisite for future methodological improvements.

A stronger focus on a (physical) understanding of inhomogeneities would complement and strengthen the statistical work. This kind of work is often performed at the station or network level, but also needed at larger spatial scales. Much of this work is performed using parallel measurements, but they are typically not internationally shared.

In an observational science the strength of the outcomes depends on a consilience of evidence. Thus having evidence on inhomogeneities from both statistical homogenisation and physical studies strengthens the science.

This chapter will discuss the needs for future research on homogenisation grouped in five kinds of problems. In the first section we will discuss research on improving our physical understanding and physics-based corrections. The next section is about break detection, especially about two fundamental problems in statistical homogenisation: the inhomogeneous-reference problem and the multiple-breakpoint problem.

Next write about computing uncertainties in trends and long-term variability estimates from homogenised data due to remaining inhomogeneities. It may be possible to improve correction methods by treating it as a statistical model selection problem. The last section discusses whether inhomogeneities are stochastic or deterministic and how that may affect homogenisation and especially correction methods for the variability around the long-term mean.

For all the research ideas mentioned below, it is understood that in future we should study more meteorological variables than temperature. In addition, more studies on inhomogeneities across variables could be helpful to understand the causes of inhomogeneities and increase the signal to noise ratio. Homogenisation by national offices has advantages because here all climate elements from one station are stored together. This helps in understanding and identifying breaks. It would help homogenisation science and climate analysis to have a global database for all climate elements, like iCOADS for marine data. A Copernicus project has started working on this for land station data, which is an encouraging development.

## Physical understanding

It is a good scientific practice to perform parallel measurements in order to manage unavoidable changes and to compare the results of statistical homogenisation to the expectations given the cause of the inhomogeneity according to the metadata. This information should also be analysed on continental and global scales to get a better understanding of when historical transitions took place and to guide homogenisation of large-scale (global) datasets. This requires more international sharing of parallel data and standards on the reporting of the size of breaks confirmed by metadata.

The Dutch weather service KNMI published a protocol how to manage possible future changes of the network, who decides what needs to be done in which situation, what kind of studies should be made, where the studies should be published and that the parallel data should be stored in their central database as experimental data. A translation of this report will soon be published by the WMO (Brandsma et al., 2019) and will hopefully inspire other weather services to formalise their network change management.

Next to statistical homogenisation, making and studying parallel measurements, and other physical estimates, can provide a second line of evidence on the magnitude of inhomogeneities. Having multiple lines of evidence provides robustness to observational sciences. Parallel data is especially important for the large historical transitions that are most likely to produce biases in network-wide to global climate datasets. It can validate the results of statistical homogenisation and be used to estimate possibly needed additional adjustments. The Parallel Observations Science Team of the International Surface Temperature Initiative (ISTI-POST) is working on building such a global dataset with parallel measurements.

Parallel data is especially suited to improve our physical understand of the causes of inhomogeneities by studying how the magnitude of the inhomogeneity depends on the weather and on instrumental design characteristics. This understanding is important for more accurate corrections of the distribution, for realistic benchmarking datasets to test our homogenisation methods and to determine which additional parallel experiments are especially useful.

Detailed physical models of the measurement, for example, the flow through the screens, radiative transfer and heat flows, can also help gain a better understanding of the measurement and its error sources. This aids in understanding historical instruments and in designing better future instruments. Physical models will also be paramount for understanding the impact of the surrounding on the measurement — nearby obstacles and surfaces influencing error sources and air flow — to changes in the measurand, such as urbanisation/deforestation or the introduction of irrigation. Land-use changes, especially urbanisation, should be studied together with relocations they may provoke.

## Break detection

Longer climate series typically contain more than one break. This so-called multiple-breakpoint problem is currently an important research topic. A complication of relative homogenisation is that also the reference stations can have inhomogeneities. This so-called inhomogeneous-reference problem is not optimally solved yet. It is also not clear what temporal resolution is best for detection and what the optimal way is to handle the seasonal cycle in the statistical properties of climate data and of many inhomogeneities.

For temperature time series about one break per 15 to 20 years is typical and multiple breaks are thus common. Unfortunately, most statistical detection methods have been developed for one break and for the null hypothesis of white (sometimes red) noise. In case of multiple breaks the statistical test should not only take the noise variance into account, but also the break variance from breaks at other positions. For low signal to noise ratios, the additional break variance can lead to spurious detections and inaccuracies in the break position (Lindau and Venema, 2018a).

To apply single-breakpoint tests on series with multiple breaks, one ad-hoc solution is to first split the series at the most significant break (for example, the standard normalised homogeneity test, SNHT) and investigate the subseries. Such a greedy algorithm does not always find the optimal solution. Another solution is to detect breaks on short windows. The window should be short enough to contain only one break, which reduces power of detection considerably. This method is not used much nowadays.

Multiple breakpoint methods can find an optimal solution and are nowadays numerically feasible. This can be done in a hypothesis testing (MASH) or in a statistical model selection framework. For a certain number of breaks these methods find the break combination that minimize the internal variance, that is variance of the homogeneous subperiods, (or you could also state that the break combination maximizes the variance of the breaks). To find the optimal number of breaks, a penalty is added that increases with the number of breaks. Examples of such methods are PRODIGE (Caussinus & Mestre, 2004) or ACMANT (based on PRODIGE; Domonkos, 2011b). In a similar line of research Lu et al. (2010) solved the multiple breakpoint problem using a minimum description length (MDL) based information criterion as penalty function.

This penalty function of PRODIGE was found to be suboptimal (Lindau and Venema, 2013). It was found that the penalty should be a function of the number of breaks, not fixed per break and that the relation with the length of the series should be reversed. It is not clear yet how sensitive homogenisation methods respond to this, but increasing the penalty per break in case of low SNR to reduce the number of breaks does not make the estimated break signal more accurate (Lindau and Venema, 2018a).

Not only the candidate station, also the reference stations will have inhomogeneities, which complicates homogenisation. Such inhomogeneities can be climatologically especially important when they are due to network-wide technological transitions. An example of such a transition is the current replacement of temperature observations using Stevenson screens by automatic weather stations. Such transitions are important periods as they may cause biases in the network and global average trends and they produce many breaks over a short period.

A related problem is that sometimes all stations in a network have a break at the same date, for example, when a weather service changes the time of observation. Nationally such breaks are corrected using metadata. If this change is unknown in global datasets one can still detect and correct such inhomogeneities statistically by comparison with other nearby networks. That would require an algorithm that additionally knows which stations belong to which network and prioritizes correcting breaks found between stations in different networks. Such algorithms do not exist yet and information on which station belongs to which network for which period is typically not internationally shared.

The influence of inhomogeneities in the reference can be reduced by computing composite references over many stations, removing reference stations with breaks and by performing homogenisation iteratively.

A direct approach to solving this problem would be to simultaneously homogenise multiple stations, also called joint detection. A step in this direction are pairwise homogenisation methods where breaks are detected in the pairs. This requires an additional attribution step, which attributes the breaks to a specific station. Currently this is done by hand (for PRODIGE; Caussinus and Mestre, 2004; Rustemeier et al., 2017) or with ad-hoc rules (by the Pairwise homogenisation algorithm of NOAA; Menne and Williams, 2009).

In the homogenisation method HOMER (Mestre et al., 2013) a first attempt is made to homogenise all pairs simultaneously using a joint detection method from bio-statistics. Feedback from first users suggests that this method should not be used automatically. It should be studied how good this methods works and where the problems come from.

Multiple breakpoint methods are more accurate as single breakpoint methods. This expected higher accuracy is founded on theory (Hawkins, 1972). In addition, in the HOME benchmarking study it was numerically found that modern homogenisation methods, which take the multiple breakpoint and the inhomogeneous reference problems into account, are about a factor two more accurate as traditional methods (Venema et al., 2012).

However, the current version of CLIMATOL applies single-breakpoint detection tests, first SNHT detection on a window then splitting, to achieve results comparable to modern multiple-breakpoint methods with respect to break detection and homogeneity of the data (Killick, 2016). This suggests that the multiple-breakpoint detection principle may not be as important as previously thought and warrants deeper study or the accuracy of CLIMATOL is partly due to an unknown unknown.

The signal to noise ratio is paramount for the reliable detection of breaks. It would thus be valuable to develop statistical methods that explain part of the variance of a difference time series and remove this to see breaks more clearly. Data from (regional) reanalysis could be useful predictors for this.

First methods have been published to detect breaks for daily data (Toreti et al., 2012; Rienzner and Gandolfi, 2013). It has not been studied yet what the optimal resolution for breaks detection is (daily, monthly, annual), nor what the optimal way is to handle the seasonal cycle in the climate data and exploit the seasonal cycle of inhomogeneities. In the daily temperature benchmarking study of Killick (2016) most non-specialised detection methods performed better than the daily detection method MAC-D (Rienzner and Gandolfi, 2013).

The selection of appropriate reference stations is a necessary step for accurate detection and correction. Many different methods and metrics are used for the station selection, but studies on the optimal method are missing. The knowledge of local climatologists which stations have a similar regional climate needs to be made objective so that it can be applied automatically (at larger scales).

For detection a high signal to noise ratio is most important, while for correction it is paramount that all stations are in the same climatic region. Typically the same networks are used for both detection and correction, but it should be investigated whether a smaller network for correction would be beneficial. Also in general, we need more research on understanding the performance of (monthly and daily) correction methods.

## Computing uncertainties

Also after homogenisation uncertainties remain in the data due to various problems: Not all breaks in the candidate station have been and can be detected.

False alarms are an unavoidable trade-off for detecting many real breaks.

Uncertainty in the estimation of correction parameters due to limited data.

Uncertainties in the corrections due to limited information on the break positions.

From validation and benchmarking studies we have a reasonable idea about the remaining uncertainties that one can expect in the homogenised data, at least with respect to changes in the long-term mean temperature. For many other variables and changes in the distribution of (sub-)daily temperature data individual developers have validated their methods, but systematic validation and comparison studies are still missing.

Furthermore, such studies only provide a general uncertainty level, whereas more detailed information for every single station/region and period would be valuable. The uncertainties will strongly depend on the signal to noise ratios, on the statistical properties of the inhomogeneities of the raw data and on the quality and cross-correlations of the reference stations. All of which vary strongly per station, region and period.

Communicating such a complicated errors structure, which is mainly temporal, but also partially spatial, is a problem in itself. Furthermore, not only the uncertainty in the means should be considered, but, especially for daily data, uncertainties in the complete probability density function need to be estimated and communicated. This could be communicated with an ensemble of possible realisations, similar to Brohan et al. (2006).

An analytic understanding of the uncertainties is important, but is often limited to idealised cases. Thus also numerical validation studies, such as the past HOME and upcoming ISTI studies are important for an assessment of homogenisation algorithms under realistic conditions.

Creating validation datasets also help to see the limits of our understanding of the statistical properties of the break signal. This is especially the case for variables other than temperature and for daily and (sub-)daily data. Information is needed on the real break frequencies and size distributions, but also their auto-correlations and cross-correlations, as well as explained in the next section the stochastic nature of breaks in the variability around the mean.

Validation studies focussed on difficult cases would be valuable for a better understanding. For example, sparse networks, isolated island networks, large spatial trend gradients and strong decadal variability in the difference series of nearby stations (for example, due to El Nino in complex mountainous regions).

The advantage of simulated data is that it can create a large number of quite realistic complete networks. For daily data it will remain hard for the years to come to determine how to generate a realistic validation dataset. Thus even if using parallel measurements is mostly limited to one break per test, it does provide the highest degree of realism for this one break.

## Deterministic or stochastic corrections?

Annual and monthly data is normally used to study trends and variability in the mean state of the atmosphere. Consequently, typically only the mean is adjusted by homogenisation. Daily data, on the other hand is used to study climatic changes in weather variability, severe weather and extremes. Consequently, not only the mean should be corrected, but the full probability distribution describing the variability of the weather.

The physics of the problem suggests that many inhomogeneities are caused by stochastic processes. An example affecting many instruments are differences in the response time of instruments, which can lead to differences determined by turbulence. A fast thermometer will on average read higher maximum temperatures than a slow one, but this difference will be variable and sometimes be much higher than the average. In case of errors due to insolation the radiation error will be modulated by clouds. An insufficiently shielded thermometer will need larger corrections on warm days, which will typically be more sunny, but some warm days will be cloudy and not need much correction, while other warm days are sunny and calm and have a dry hot surface. The adjustment of daily data for studies on changes in the variability is thus a distribution problem and not only a regression bias-correction problem. For data assimilation (numerical weather prediction) accurate bias correction (with regression methods) is probably the main concern.

Seen as a variability problem, the correction of daily data is similar to statistical downscaling in many ways. Both methodologies aim to produce bias-corrected data with the right variability, taking into account the local climate and large-scale circulation. One lesson from statistical downscaling is that increasing the variance of a time series deterministically by multiplication with a fraction, called inflation, is the wrong approach and that the variance that could not be explained by regression using predictors should be added stochastically as noise instead (Von Storch, 1999). Maraun (2013) demonstrated that the inflation problem also exists for the deterministic Quantile Matching method, which is also used in daily homogenisation. Current statistical correction methods deterministically change the daily temperature distribution and do not stochastically add noise.

Transferring ideas from downscaling to daily homogenisation is likely fruitful to develop such stochastic variability correction methods. For example, predictor selection methods from downscaling could be useful. Both fields require powerful and robust (time invariant) predictors. Multi-site statistical downscaling techniques aim at reproducing the auto- and cross-correlations between stations (Maraun et al., 2010), which may be interesting for homogenisation as well.

The daily temperature benchmarking study of Rachel Killick (2016) suggests that current daily correction methods are not able to improve the distribution much. There is a pressing need for more research on this topic. However, these methods likely also performed less well because they were used together with detection methods with a much lower hit rate than the comparison methods.

The deterministic correction methods may not lead to severe errors in homogenisation, that should still be studied, but stochastic methods that implement the corrections by adding noise would at least theoretically fit better to the problem. Such stochastic corrections are not trivial and should have the right variability on all temporal and spatial scales.

It should be studied whether it may be better to only detect the dates of break inhomogeneities and perform the analysis on the homogeneous subperiods (removing the need for corrections). The disadvantage of this approach is that most of the trend variance is in the difference in the mean of the HSPs and only a small part is in the trend within the HPSs. In case of trend analysis, this would be similar to the work of the Berkeley Earth Surface Temperature group on the mean temperature signal. Periods with gradual inhomogeneities, e.g., due to urbanisation, would have to be detected and excluded from such an analysis.

An outstanding problem is that current variability correction methods have only been developed for break inhomogeneities, methods for gradual ones are still missing. In homogenisation of the mean of annual and monthly data, gradual inhomogeneities are successfully removed by implementing multiple small breaks in the same direction. However, as daily data is used to study changes in the distribution, this may not be appropriate for daily data as it could produce larger deviations near the breaks. Furthermore, changing the variance in data with a trend can be problematic (Von Storch, 1999).

At the moment most daily correction methods correct the breaks one after another. In monthly homogenisation it is found that correcting all breaks simultaneously (Caussinus and Mestre, 2004) is more accurate (Domonkos et al., 2013). It is thus likely worthwhile to develop multiple breakpoint correction methods for daily data as well.

Finally, current daily correction methods rely on previously detected breaks and assume that the homogeneous subperiods (HSP) are homogeneous (i.e., each segment between breakpoints assume to be homogeneous) . However, these HSP are currently based on detection of breaks in the mean only. Breaks in higher moments may thus still be present in the "homogeneous" sub periods and affect the corrections. If only for this reason, we should also work on detection of breaks in the distribution.

## Correction as model selection problem

The number of degrees of freedom (DOF) of the various correction methods varies widely. From just one degree of freedom for annual corrections of the means, to 12 degrees of freedom for monthly correction of the means, to 40 for decile corrections applied to every season, to a large number of DOF for quantile or percentile matching.

A study using PRODIGE on the HOME benchmark suggested that for typical European networks monthly adjustment are best for temperature; annual corrections are probably less accurate because they fail to account for changes in seasonal cycle due to inhomogeneities. For precipitation annual corrections were most accurate; monthly corrections were likely less accurate because the data was too noisy to estimate the 12 correction constants/degrees of freedom.

What is the best correction method depends on the characteristics of the inhomogeneity. For a calibration problem just the annual mean could be sufficient, for a serious exposure problem (e.g., insolation of the instrument) a seasonal cycle in the monthly corrections may be expected and the full distribution of the daily temperatures may need to be adjusted. The best correction method also depends on the reference. Whether the variables of a certain correction model can be reliably estimated depends on how well-correlated the neighbouring reference stations are.

An entire regional network is typically homogenised with the same correction method, while the optimal correction method will depend on the characteristics of each individual break and on the quality of the reference. These will vary from station to station, from break to break and from period to period. Work on correction methods that objectively select the optimal correction method, e.g., using an information criterion, would be valuable.

In case of (sub-)daily data, the options to select from become even larger. Daily data can be corrected just for inhomogeneities in the mean (e.g., Vincent et al., 2002, where daily temperatures are corrected by incorporating a linear interpolation scheme that preserves the previously defined monthly corrections) or also for the variability around the mean. In between are methods that adjust for the distribution including the seasonal cycle, which dominates the variability and is thus effectively similar to mean adjustments with a seasonal cycle. Correction methods of intermediate complexity with more than one, but less than 10 degrees of freedom would fill a gap and allow for more flexibility in selecting the optimal correction model.

When applying these methods (Della-Marta and Wanner, 2006; Wang et al., 2010; Mestre et al., 2011; Trewin, 2013) the number of quantile bins (categories) needs to be selected as well as whether to use physical weather-dependent predictors and the functional form they are used (Auchmann and BrĂ¶nnimann, 2012). Objective optimal methods for these selections would be valuable.

## Related information

WMO Guidelines on Homogenization (English, French, Spanish)

WMO guidance report: Challenges in the Transition from Conventional to Automatic Meteorological Observing Networks for Long-term Climate Records