We use MODIS IST data from MODIS/Terra Sea Ice Extent 5-Min L2 Swath 1km, Version 6 (MOD29), contained within a multilayer Greenland MODIS-based product . MODIS records surface reflectance from 36 spectral bands of different wavelengths – including those used in IST – near daily for the entire Earth. This dataset spans the period 1 March 2000 to 31 December 2019 and has a spatial resolution of 0.78 × 0.78 km. Here we discard the first 10 months of the dataset, up to 1 January 2001, in order to work only with those years for which a full annual cycle is available. To reduce the computational burden of our model, we also subsample the data by taking 1 in every 50 cells in both x and y dimensions for a total of 1139 cells, roughly equally spaced across in latitude and longitude and thus covering the full range of glaciological and climatological settings across the ice sheet.

The IST measurements represent the temperature at the surface of the ice in cloud-free conditions. Clouds (specifically water vapour) can interfere with the measurements, so a cloud mask is used in the MODIS product to remove measurements made in cloudy conditions. As a consequence, our analysis and predictions are valid for clear conditions only. Due to the generally warmer temperatures seen on cloudy days, were the analysis to be interpreted as representative of clear days also, there would be a strong likelihood of over-estimating the magnitude and frequency of melt events .

As a result of cloud masking, areas on the coast and in the north have a higher proportion of missing data than more central areas (Fig. ). We also see that winter months have more missing data on average than summer months because of cloud cover, with a range of 65.1% of data available in December compared to 91.1% of data available in May. This is important to bear in mind when interpreting the predictions made from the statistical models, as the IST distributions will be more heavily weighted towards warmer temperatures. This should not affect our inference with regards to melt, however, as melt temperatures almost exclusively occur in the summer months, which have a much lower proportion of missing data.

To check for the possible presence of longer-term changes in surface conditions in response to changes in cloud cover, the proportion of data missing due to cloudiness is compared between decades (Fig. ). The clear spatial trends seen in the overall proportions of missing data are not present here, though locations in the north and south of the ice sheet have slightly more data in the most recent decade in contrast to locations at mid-latitudes where the reverse is true. However, these trends are not consistent over all locations, and the magnitudes of the changes are relatively minor with a maximum absolute change of 0.060 and a mean absolute change of 0.010. Seasonal differences in cloud cover and thereby data availability also vary between the two decades. January, February, and October show the largest changes, with a maximum absolute change of 0.044 in February. Ten months have an average decrease in data availability between the decades, but the absolute differences are reasonably minor. Furthermore, of the four months with the lowest changes, three (June, July, and August) also see the highest average temperatures, giving further confidence in our cross-decade comparison of temperature quantiles and melt estimates.

To create a statistical model that is parsimonious and applicable at all cells over a large and geographically varied region, we model the IST data using statistical methods that allow us to treat melting in a probabilistic manner. Exploratory data analysis shows that there is no clear quantile in the temperature distribution that can be attributed as the onset of melting (Fig. ). As a result, we model melting ice temperatures and non-melting ice temperatures separately and estimate the probability of melt occurring over a range of temperatures. This approach allows for some uncertainty in the observations from factors such as the precision of the dataset, which has a stated uncertainty of ±1 ∘C. We hereby refer to temperatures associated with melting ice as “melt” temperatures and temperatures associated with non-melting ice as “ice” temperatures.

A key feature of the dataset and a core modelling consideration is the soft upper limit at 0 ∘C. The melting point of the ice acts as a physical upper limit on ISTs, as once the ice exceeds this temperature it melts and may no longer form the surface of the ice sheet. Some sites have measurements above this limit, which arise due to meltwater sitting on top of the ice. However, the ice under the water places a limit on these melt temperatures; hence the distribution of positive temperatures is truncated close to 0 ∘C. This soft upper limit of ISTs causes a significant peak in the distribution centred at approximately -0.5 ∘C, as any ISTs that would exceed 0 ∘C are truncated to small positive values close to 0 ∘C.

The simplest statistical model would be to fit a single distribution to the full dataset, potentially after an initial transformation. This raises two issues. Firstly, a bimodal distribution is clear at all cells with cells that experience melt having the highest mode close to 0 ∘C (Fig. ). For non-melt cells, the location of the higher mode is more variable. This does not appear to be directly attributable to seasonal differences in temperatures, as the shape of seasonal temperature distributions show as much inter-site variability as they do inter-season variability. Fits of unimodal distributions are particularly poor at the tails of the distributions, which is particularly problematic since our interest lies in melt which is directly connected to the upper tail of the temperature distribution.

Given the focus on melt, an alternative option would be to undertake an extreme value analysis of only the highest temperatures at each cell. This would allow the model to focus on the temperatures of highest interest that are the most difficult for more standard models to capture. This also proves problematic though, as in order to fit the model the temperatures must first be identified as extreme using an extreme value threshold, with temperatures above the threshold being classed as extreme and those below being classed as non-extreme. Due to the mode around 0 ∘C, finding a consistent threshold location using quantiles, gradient analysis, or a specific temperature encounters problems due to the large variety of tail shapes at different cells. Each of the above-mentioned threshold types do not work universally across the ice sheet and in many cases provide much worse fits than a distribution applied to the whole range of temperatures.

A consistent model that can be automatically applied at cells across the ice sheet therefore requires a multi-modal distribution or a time-series model to capture seasonal behaviour. The disadvantage of the latter is that it is less able to capture the mode around 0 ∘C and the truncation of the temperatures, which is where our research interest lies. This further motivates a modelling approach that more directly considers the distribution of the specific dataset and allows for multi-modal distributions. The overall distribution shape is broadly similar between sites with the main difference being the proximity of the distribution to 0 ∘C and thereby the amount of truncation in the data. The considerations above around the multi-modality of the dataset and of the nature of melt temperatures around 0 ∘C give us a basic set of assumptions to base our modelling around that allow the model to retain the same underlying structure regardless of the absolute difference in ISTs between cells.

In order to accommodate spatial variability in the temperature distribution, we model IST using a truncated Gaussian mixture model in which components are assigned to model groups of temperatures that we assign to be either ice or melt. For nI ice components and a single melt component, let ϕi be the weight associated with model component i such that for nc=nI+1 total components, ∑i=1ncϕi=1. For each ice component i (and melt component M), let fi(x) be the probability density function of the truncated normal distribution X∼TN(μi,σi2,ai,bi), where μi is the mean, σi is the standard deviation, and ai (bi) is the lower (upper) truncation point. Then the probability density function of ISTs x is 1p(x)=∑i=1nIϕifi(x)+ϕMfM(x). We set the upper and lower truncation points for the ice and melt components at values that bound each measurement type with relative certainty. For the ice components, a=-∞ as there is no hard lower limit on the temperature of ice (aside from absolute zero), and b=0 as, theoretically, ice temperature can not exceed 0 ∘C. This means that there is no limit on how low ice temperatures can go, but they can not exceed 0 ∘C. For the melt component, b=∞ and a=-1.65, so temperatures in the melt component can not go below -1.65 ∘C but are not upper truncated. We take a bound lower than zero here to account for uncertainty in the data and any potential impurities in the ice surface. The theoretical minimum temperature at which saline ice can melt is -1.65 ∘C and thus should be a conservative estimate for this lower bound. Temperatures between -1.65 ∘C and 0 ∘C can be modelled by either of the ice and melt components or both of them as there is uncertainty as to whether they are associated with melting or non-melting ice.

A mixture model was fitted using the expectation–maximisation (EM) algorithm for each sample cell. The algorithm alternates between two main steps: calculating the component probabilities that each observation xi comes from model component k and maximising the expectations of the model parameters using the component probabilities (for full details see Appendix ). We used this method to obtain estimates of μ, σ, and ϕ for each model component at each cell.

We used the Bayesian information criterion (BIC) to assess the most appropriate number of ice and melt components and found that three ice components and one melt component fit the data best. These components may be broadly interpreted as winter, autumn, and spring and the melt season for the three ice components and single melt component respectively.

When modelled with separate Gaussian components, the characteristics of the different modes of the data are much clearer (Fig. ). The melt component at each cell generally has a much lower variance than the ice components due to the soft upper limit of ISTs and the lower truncation point of the model, whereas the ice components have higher variances and more overlap between components. For the sites that experience melt regularly, a substantial proportion of the overall temperature distribution occurs in the overlap between true ice and true melt. A similar result is seen across sites located on or near the coasts, which further validates the decision to use a fixed melt threshold as the melt temperatures – and thereby the melt process – appear to have consistent characteristics across cells.

Using this model, the probability of melt occurring, which we denote by ρ(x), can be quantified as the ratio of the densities of the ice and melt components. For a given IST x, nI ice components, and melt component M, we have 2ρ(x)=fM(x)fM(x)+∑i=jnIfi(x). Consequences of this definition are that for ISTs below -1.65 ∘C, the probability of melt is 0; for ISTs above 0 ∘C, the probability of melt is 1; and between these values the melt probability depends on relative values of the melt and ice components' densities. For cells with very few or no ISTs above -1.65 ∘C, the weight of the melt component may be close to or equal to 0, in which case the probability of melt occurring is effectively zero. Note that there are discontinuities in the model-based estimate of this probability due to the censoring of the mixture components. These discontinuities occur at the edges of the range of interest (-1.65 and 0 ∘C) and are more or less severe depending on the degree of truncation of the ice and melt components.

Using our model, we calculate the expected number of melt days in each year at each sample cell. Let Ny be the number of melt days in year y, then E[Ny]=∑i=1mρ(xi), where ρ(x) is the notation introduced earlier to denote Pr[melt|X=x] and m is the number of observations in year y. The overall annual average is simply the average of the individual annual averages. We then compare our modelled estimates to a simple threshold-based approach to defining melt, i.e. the average number of days per year with temperatures greater than or equal to -1 ∘C (Fig. ).

The majority of the ice sheet – 90.7% of cells from the expected melt from the model, 79.5% from a threshold of -1 ∘C on the data – experiences some degree of melt on average each year, except for sites in the dry-snow zone in the centre and north of the ice sheet . Of the cells that experience melt, most sites (62.2% from the model, 57.3% from the data) on average see less than 2 d of melt per year, which makes up the rest of the dry-snow zone and most of the percolation zone. The areas with the most melt are located around the coast and in the south and west as may be expected. The main discrepancies between the two measures are at coastal cells, particularly on the west and north coasts. Here, the model estimates a larger amount of melt, with a maximum of 14 additional melt days at one specific cell on the edge of the south-east coast compared to the dataset. However, 89% of cells have an absolute difference of less than 2 melt days, showing the broad agreement between the measures at central cells.

To place the methodology and the results within the context of the literature, we compare the estimates of melt found using our method to those estimated using data from the Programme for Monitoring of the Greenland Ice Sheet (PROMICE) AWSs . The characteristics of the datasets differ with known biases between the two ; however previous validation carried out independently for both datasets suggests it is reasonable to consider both datasets representative of the true surface temperatures. With AWSs as a baseline, Fig.  compares the difference between the AWS melt proportions with the same quantity obtained from (a) MODIS IST and (b) the model fitted to MODIS IST. Despite the relatively large distances between some of the comparison locations, there is still reasonable agreement for a substantial number of AWS locations. The differences between the estimates may also be partially due to other factors surrounding the nature of the comparison, such as the reduced number of data available because of the need for common dates, the difference in time resolutions between the datasets in terms of the number of observations averaged to give a single daily observation, and the different measurement uncertainties that lead to different definitions of melt.

We now use the model fit to calculate quantiles of the ISTs at each cell (Fig. ). This gives context to the overall temperature trends observed in the dataset before looking at melt in more detail. We calculate the 90% quantiles to examine the broader trends of high temperatures that are not necessarily melt temperatures, as well as the 10% quantiles for temperatures that are as relatively low as the 90% quantiles are high. The estimated 10% and 90% quantiles broadly follow the same trends as elevation on the ice sheet. The 10% quantiles have a range from -53.84 ∘C in the centre of the ice sheet to a maximum of -15.75 ∘C at the south tip of the ice sheet. As would be expected, cells at higher elevations have a lower 10% and 90% quantile. However, of more interest are the few (30/1139) cells located on the west, east, and southern coasts that have a 90% quantile above 0 ∘C. At these cells, we would expect at least (in some cases more than) 10% of observed temperatures to be above 0 ∘C and thereby melt temperatures.

We also calculate the 1-year return levels of each cell. This is the IST that is on average only exceeded once per year as estimated from each cell's mixture model. The return levels range from a minimum of -7.08 ∘C in the centre to a maximum of 7.24 ∘C on the west coast. Although, as previously discussed, ISTs should not be seen higher than 0 ∘C, these return levels reflect similar temperatures recorded by observations in the dataset and can be plausible temperatures when considering the effect that meltwater on the surface of the ice sheet has on the observations. The rarity of melt in certain central areas can be seen more clearly, as temperatures in many cells (519/1139) on average reach -1.65 ∘C less than once a year. The trends seen in the return levels also broadly agree with those seen in the quantiles and are in reasonable agreement with the elevations and distance to the coast of each cell, with cells at lower altitudes and closer to the coast generally experiencing more melt.

To examine potential changes in melt over time, we fit mixture models at each cell for two separate decades: 2001 to 2009 and 2010 to 2019. Averaging over a decade helps to smooth some of the annual variability and thus highlight any potential differences as a result of climatic change. To assess any changes in melt and high ISTs between the decades, we compare quantiles between the fitted models and the estimated melt probabilities at each cell in each decade.