Surface melting is one of the primary drivers of ice shelf collapse in Antarctica and is expected to increase in the future as the global climate continues to warm because there is a statistically significant positive relationship between air temperature and melting. Enhanced surface melt will impact the mass balance of the Antarctic Ice Sheet (AIS) and, through dynamic feedbacks, induce changes in global mean sea level (GMSL). However, the current understanding of surface melt in Antarctica remains limited in terms of the uncertainties in quantifying surface melt and understanding the driving processes of surface melt in past, present and future contexts. Here, we construct a novel grid-cell-level spatially distributed positive degree-day (PDD) model, forced with 2 m air temperature reanalysis data and spatially parameterized by minimizing the error with respect to satellite estimates and surface energy balance (SEB) model outputs on each computing cell over the period 1979 to 2022. We evaluate the PDD model by performing a goodness-of-fit test and cross-validation. We assess the accuracy of our parameterization method, based on the performance of the PDD model when considering all computing cells as a whole, independently of the time window chosen for parameterization. We conduct a sensitivity experiment by adding

Surface melting is common and well-studied over the Greenland Ice Sheet (GrIS; e.g.

Continental-scale spaceborne observations of surface melt are limited to the satellite era (1979–present), meaning that current estimates of Antarctic surface melt are typically derived from surface energy balance (SEB) or positive degree-day (PDD) models. SEB models are employed in regional climate models such as the Regional Atmospheric Climate MOdel (RACMO;

Although PDD models are empirical, they are often sufficient for estimating melt on the catchment scale

However, as the DDF is related to all terms of the SEB

Although PDD schemes have been used in many Antarctic numerical ice sheet models (e.g.

In this study, we focus on constructing a computationally efficient cell-level (spatially variable) PDD model to estimate surface melt in Antarctica through the past 4 decades, by statistically optimizing the parameters of the PDD model individually in each computing cell. We use the European Centre for Medium-Range Weather Forecasts Reanalysis v5 (ECMWF ERA5;

Table of data that we use in this study.

The dataset we use in this study is the ECMWF ERA5 reanalysis

The particular ERA5 product we use in this study is the hourly 2 m air temperature data, which have been evaluated and used previously for studies in Antarctica (e.g.

The number of melt days retrieved from the satellite observations is used to parameterize the threshold temperature (

We also use a more recently developed satellite melt day dataset which uses a similar algorithm as

To parameterize the DDF for the PDD model, we compare our ERA5 forced numerical experiments to the Antarctic surface melt simulations from RACMO2.3p2

RACMO2.3p2 Antarctic surface melt simulations used here cover the time period from January 1979 to February 2021 with monthly temporal resolution and 27

The spatially coarsest dataset used in this study is the ERA5 reanalysis data, which are in 0.25

The research domain and 27 Antarctic drainage basins

Using an empirical relationship between air temperature and melt, temperature index models are the most commonly used method for assessing surface melt of ice and snow due to their simplicity as they are only meteorologically forced by the air temperature

The PDD model calculates the water equivalent of surface snow melt (

To parameterize the threshold temperature (

In order to obtain the optimal

The DDF is a scaling parameter that controls the meltwater production and is related to all terms of the SEB

In order to determine the optimal DDF, we repeat the calculations for the RMSE between the annual melt amount calculated in each DDF experiment and the melt amount from RACMO2.3p2 simulations for each computing cell. Similarly, we define the optimal DDF where the experiment has the minimal RMSE for each computing cell. If there are multiple DDF experiments that have the same minimal RMSE for their computing cell, we calculate the mean of these DDF as the optimal DDF (this only happened for the cells with very low melt amounts).

Limited by the duration of the satellite era and reanalysis data, the time series of annual data for each computing cell is no larger than 45 years with non-normality. We use the two-sample Kolmogorov–Smirnov test (hereafter, two-sample KS test) to evaluate the dissimilarity between the PDD results and RACMO2.3p2 melt volume outputs at a confidence level of 5 %. We define a “same distribution cell” as a cell with no statistically significant evidence from the two-sample KS test for the rejection of the null hypothesis (that the two samples are from the same continuous distribution).

We consider the spatial variability in PDD parameters by parameterizating the model in each computing cell for the whole time period. However, this does not allow us to explore the variability in the PDD parameters in a temporal sense, as

Schematic overview of the time periods for each CV folders and the HIGH/LOW sensitivity experiments. Panel

Periods of training and testing folds for the

To therefore assess the temporal dependency of the PDD parameters, we perform an adjusted three-fold cross-validation (hereafter 3-fold CV). The satellite melt occurrence estimates used in this study cover 38 years (four years have been omitted). Therefore, we sequentially divide the satellite estimates into two 13-year folds and one 12-year fold (Fig.

Although RACMO2.3p2 is suggested to be one of the best models for reconstructing Antarctic climate, a cold bias of

To explore the sensitivity of PDD parameters and model outputs to biases in both the satellite and RACMO2.3p2 products, we perform two sensitivity experiments. In the first sensitivity experiment, we explore the response of

To assess the applicability of our PDD model in simulating melt under warmer climate scenarios, we conduct temperature–melt sensitivity experiments. To do this, we add constant temperature perturbations of

Figure

The probability distribution of

Figure

We also use the same method and data to parameterize a spatially uniform PDD (hereafter, “uni-PDD”) model (one

We evaluate the parameterized dist-PDD and uni-PDD model outputs (melt day and melt amount) for each computing cell by testing the statistical significance of the similarity between the satellite estimates or RACMO2.3p2 simulations and the dist-PDD/uni-PDD model-derived empirical distribution functions. Figure

The two-sample KS test results. The two-sample KS tests are performed individually for each of the 4515 computing cells. The test result “Same” means the tested cell is a “same distribution cell” where there is no statistically significant evidence for the rejection of the null hypothesis that the testing two samples are from the same continuous distribution (Sect. 3.3.1). Otherwise, the cell is a “different distribution cell” (“Different”). Panels

Summary of the statistics for Fig.

Next, we evaluate the parameterized dist-PDD/uni-PDD model outputs for the whole of Antarctica. Firstly, we evaluate the parameterized optimal

Globally, we show that the accuracy of the PDD models in estimating the surface melt days has improved from the uni-PDD model to the dist-PDD model (Table

The computing cells that have relatively large absolute differences in SD are mainly located over the Wilkins Ice Shelf (

Summary of the statistics for Fig.

Secondly, we evaluate the parameterized optimal DDF and the simulated surface melt amount. Similar to the negative biases between the dist-PDD and the satellite estimates for the CMS for the period from 1979/1980 to 1982/1983 (Fig.

Apart from the 1982/1983 event, other negative biases from dist-PDD are also evident in the period from 1991/1992 to 1992/1993 (Fig.

Figure

To evaluate our dist-PDD model in a temporal sense, we perform 3-fold CV for

Panels

Figure

Although we show parameter changes associated with the time windows for the dominant number of the computing cells, these changes diminish when we look at the whole population of the parameters in each member (Fig.

Next, we evaluate each member's parameters on the testing fold. Firstly, we calculate the CMS/annual melt amount for the time windows of the testing folds from the dist-PDD models that are parameterized by the training folds for each

Secondly, we calculate the Spearman's

To further explore this disagreement in the testing fold, we plot the time series of CMS for satellite estimates, CONTROL estimates and

Figure

Panels

Figure

Figure

The PDD model has the notable advantage of high computational efficiency due to its one-dimensional nature and being solely forced by 2 m air temperature. However, in reality the 2 m air temperature is not the sole driver of Antarctic surface melting (Fig.

In addition, the model simply multiplies a scaling number (DDF) by the summation of temperature above a certain threshold (

We have constructed a PDD model with spatially varying parameters (dist-PDD) and with spatially uniform parameters (uni-PDD) based on the temperature–melt relationship (e.g.

As the parameters were parameterized spatially, the dist-PDD is overall in good agreement with the spatial patterns shown by the satellite and RACMO2.3p2 data, with the exception of an underestimation of melt days and amounts in the ice shelves of the western Antarctic Peninsula and an overestimation of melt days on Shackleton Ice Shelf and of melt amount on Amery Ice Shelf. The most inadequate estimation was in 1982/1983, for which we found large dist-PDD underestimation of both the melt days and amount. We suggest that this underestimation corresponds to SAM-influenced climatic conditions and that the dist-PDD lacks the ability to accurately capture melt if it arises from effects such as Föhn winds that are not reflected in the input ERA5 2 m air temperature fields used to force the calculations (e.g.

These limitations aside, we found overall high fidelity of the dist-PDD model, as suggested by the 3-fold CV. Although the dist-PDD parameters vary on the cell level through the different time window chosen for parameterization, the probability distribution for all computing cells changes negligibly and the overall performance of the dist-PDD model when considering all computing cells is consistent. From the sensitivity experiments, we found the changes in the dist-PDD estimates are comparable to the changes in training data (satellite and RACMO2.3p2 data). The correlations between the dist-PDD estimates and training data exhibit stability regardless of the changes in the training data.

The dist-PDD model not only relatively accurately estimates surface melt in Antarctica compared with the satellite estimates and more sophisticated SEB model; it is also highly computationally efficient. These advantages may allow us to use the dist-PDD model to explore Antarctic surface melt in a longer-term context into the future and over periods of the geological past for which neither satellite observations nor SEB components are available. This efficiency also allows our model to be employed at a far higher spatial resolution than regional climate models. However, due to the systematical limitations of the PDD model and the scarcity of Antarctic surface melt data available

The number of melt days and the area of surface melt can be detected using microwave brightness temperature data since 1979 (e.g.

Daily percentage of missing data for satellite estimates. Satellite SMMR and SSM/I cover the period from 1 April 1979 to 31 March 2021. Satellite AMSR-E covers the period from 1 April 2002 to 31 March 2011. Satellite AMSR-2 covers the period from 1 April 2012 to 31 December 2021.

The positive relationship between 2 m air temperature and surface melt on Antarctic ice shelves

Correlation map between the mean DJF ERA5 2 m air temperature and RACMO2.3p2 annual surface melt amount for the period from 1979/1980 to 2019/2020. It is calculated by the Spearman's rank correlation coefficient on each cell. Black dots mark the cells where the correlations are statistically significant (

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Panels

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Trend of the mean DJF ERA5 2 m air temperature on each computing cell during the period 1979/1980–2019/2020. Black dots mark the trends that are statistically significant (

Figure

Both ERA5 and RACMO2.3p2 exhibit similar spatial patterns for the 1982/1983 DJF 2 m air temperature anomaly (Fig.

We then assess the goodness-of-fit of the dist-PDD model after removing the 1982/1983 period for dist-PDD, satellite and RACMO2.3p2. The exclusion of the 1982/1983 period significantly improves the accuracy of the dist-PDD model in comparison to satellite and RACMO2.3p2 (Fig.

On the basis of this additional experimentation, we are able to confidently conclude that our model is accurate for the vast majority of the time series and that any previously apparent bias was almost entirely due to the anomalous conditions of a single year.

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The Spearman's

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The ERA5 reanalysis data are available from

YZ, NRG and AG conceived the study. YZ performed the analysis and prepared the original draft of the paper. GP and MLL provided satellite products. All authors contributed to writing the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been supported by the Antarctic Research Centre, Victoria University of Wellington. Computational resources have been provided by Rāpoi (Victoria University of Wellington's high-performance computing system). We would like to acknowledge the editor, Brice Noël, and the referees, Christoph Kittel and Devon Dunmire.

Yaowen Zheng and Nicholas R. Golledge are supported by the Royal Society of New Zealand (award no. RDF-VUW1501). Nicholas R. Golledge and Alexandra Gossart are supported by the Ministry for Business Innovation and Employment (grant no. ANTA1801; “Antarctic Science Platform”). Nicholas R. Golledge received support from the Ministry for Business Innovation and Employment (grant no. RTUV1705; “NZSeaRise”).

This paper was edited by Brice Noël and reviewed by Christoph Kittel and Devon Dunmire.