Modeling the Surface Mass Balance of a High Arctic Glacier Using the ERA40 Reanalysis

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, F02014, doi:10.1029/2009JF001364, 2010 Click Here for Full Article

Modeling the surface mass balance of a high Arctic glacier using the ERA‐40 reanalysis Cameron J. Rye,1 Neil S. Arnold,1 Ian C. Willis,1 and Jack Kohler2 Received 28 April 2009; revised 16 September 2009; accepted 6 January 2010; published 2 June 2010.

[1] Investigating glacier mass balance via the use of numerical models provides an important insight into glacier‐climate interactions and allows for the assessment of future changes in sea level and local water resources. The application of mass balance models to unmonitored regions has previously been inhibited by the inadequate spatial coverage of local meteorological observations. One way to overcome this problem is through the use of climate reanalysis products. In this paper we evaluate the ability of the European Centre for Medium‐Range Forecasts ERA‐40 reanalysis to drive a surface mass balance model of Midre Lovénbreen, northwest Spitsbergen, Svalbard. The ERA‐40 reanalysis is validated and bias‐corrected against long‐term observations from two nearby in situ weather stations. The model is calibrated over the period of mass balance observations (1968–2001) using a downhill simplex parameter optimization technique to minimize the error between observed and simulated mass balances. The calibrated model is then used to extend the mass balance record of Midre Lovénbreen back to the beginning of the reanalysis in 1958. Overall, the ERA‐40 reanalysis is found to correspond sufficiently well with surface observations to be used for mass balance modeling. When driven using the ERA‐40 reanalysis, the calibrated surface mass balance model performs very well. The area‐averaged cumulative errors for winter, summer, and net balances are all very small over the period 1968–2001 (<0.3 m water equivalent (w.e.)). For the centerline stakes the correlation coefficients between observed and modeled net, winter, and summer balances are 0.83, 0.84, and 0.77, respectively. Using the hindcasted mass balances, a mean cumulative mass loss of −17.8 ± 3.7 m w.e. is estimated over the 44 year period 1958–2001. Citation: Rye, C. J., N. S. Arnold, I. C. Willis, and J. Kohler (2010), Modeling the surface mass balance of a high Arctic glacier using the ERA‐40 reanalysis, J. Geophys. Res., 115, F02014, doi:10.1029/2009JF001364.

1. Introduction Holmgren, 2005]. However, the application of spatially distributed mass balance models to entire regions has been [2] Small glaciers and ice caps have been in general inhibited by the inadequate spatial and temporal coverage of retreat for several decades and currently provide the largest local meteorological records and mass balance observations contribution to sea level rise not attributable to thermal [Barry, 2006]. Regional mass balances are instead often expansion [Kaser et al., 2006; Meier et al., 2007]. Since the derived by the extrapolation of detailed studies on a small 1990s, increases in atmospheric temperatures have lead to a number of well‐documented glaciers [e.g., Dyurgerov and notable increase in the rate of ice loss [Arendt et al., 2002; Meier, 1997a, 1997b]. This has resulted in a need to Abdalati et al., 2004; Bamber et al., 2005], which is expected develop new methods that allow spatial variations across to continue to accelerate under future warming [Meehl et al., regions to be better described and historical records to be 2007]. Numerical models are used to quantify glacier mass derived for locations where meteorological observations balance and investigate the sensitivity of glaciers to both are available [Walsh et al., 2005]. historical and future climatic change. Traditionally, mass [3] One way in which this problem can be approached is balance models have been applied to individual glaciers and through the use of climate reanalysis products. Reanalyses forced using in situ meteorological observations [e.g., Arnold aim to reconstruct detailed, accurate and continuous records of et al., 1996; Klok and Oerlemans, 2002; Hock and past global atmospheric conditions by using a fixed dynamical system to assimilate observations. Current reanalyses 1Scott Polar Research Institute, University of Cambridge, Cambridge, typically have temporal resolutions of 6–12 hours over UK. 2 periods of decades or more and have spatial resolutions of Norwegian Polar Institute, Polar Environmental Centre, Tromsø, – Norway. 1 2 degrees latitude by longitude. The advantage of reanalysis products is that they can be used to investigate spatial Copyright 2010 by the American Geophysical Union. and temporal climatic variations over large, inaccessible 0148‐0227/10/2009JF001364

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atmospheric and oceanic circulation [Isaksson et al., 2005]. Quantification of Svalbard mass balance is clearly important, yet all the ice masses where measurements have been made account for less than 0.5% of the total glaciated area [Hagen et al., 2003]. Climate reanalysis products therefore provide an ideal opportunity to investigate variations in mass balance over the archipelago. In this paper we apply a distributed physically based surface mass balance model to Midre Lovénbreen, Spitsbergen, Svalbard. Midre Lové- nbreen was chosen as a test study site because it is a well‐ researched glacier, with one of the longest continuous mass balance records in the Arctic (1968–present). The model is forced using a full range of meteorological variables from the ERA‐40 reanalysis (air temperature, wind speed, relative humidity, precipitation, global radiation and incoming longwave radiation). We validate and bias‐correct the ERA‐40 reanalysis against meteorological observations from two in situ weather stations at Ny‐Ålesund, about 5 km north of the glacier. The reanalysis data begin on 1 September 1957 and end on 31 August 2002. Annual mass balances are calculated on the basis of the mass balance year from 1 October to 30 September. Therefore the range of mass balance years that can be modeled using the reanalysis is 1958–2001. The model is calibrated over the period of mass balance observations (1968–2001) using a downhill simplex parameter optimization technique. The calibrated model is then used to extend the mass balance record of Midre Lovénbreen back in time from 1968 to the beginning of the ERA‐40 reanalysis in 1958. Figure 1. Map of Midre Lovénbreen, with contours in meters a.s.l. The bold numbers denote NPI stake numbers. 2. Study Site regions, as opposed to the limited at‐a‐point measurements [6] Midre Lovénbreen is a small high Arctic valley glacier obtained from automatic weather stations. This creates the located on the Brøggerhalvøya coast of northwest Spits- potential to investigate glacier‐climate relationships in bergen, Svalbard (Figure 1). The glacier has an area of about unmonitored regions and extend current mass balance records 5km2, with an altitude range of 50–650 m above sea level back in time before ground‐based observations began. [Kohler et al., 2007] and a maximum thickness of around [4] A small number of studies have investigated the use of 180 m [Björnsson et al., 1996]. Four upland cirques com- reanalysis data for modeling glacier mass balance using bine to feed the main tongue, which flows in a generally linear regression models [Rasmussen and Conway, 2004; northward direction. During the 20th century the glacier Rasmussen and Kohler, 2007], precipitation‐temperature‐ snout retreated by over 1000 m and at the same time lost a area‐altitude models [Zhang et al., 2007], degree‐day total mass equivalent to approximately 10,000,000 m3 of ice models [Hanna et al., 2005; Radić and Hock, 2006; Hock et [Hagen et al., 2003]. The glacier has received significant al., 2007], and energy balance models [Reichert et al., 2001; scientific interest owing to it having the second longest Hock et al., 2007]. Reanalysis products have also been used continuous mass balance record in the Arctic (1968– to investigate the relation between large‐scale circulation present). Previous studies include investigations on mass patterns and glacier mass balance [Gardner and Sharp, balance [Kohler et al., 2007], energy balance [Arnold et al., 2007; Shea and Marshall, 2007] and as a tool for statisti- 2006], glacial hydrology [Rippin et al., 2003; Hodson et cally downscaling general circulation model (GCM) pro- al., 2005], ice dynamics [Hambrey et al., 2005; Rippin et jections to derive future glacier volume changes [Reichert et al., 2005], superimposed ice formation [Wadham et al., al., 2001; Radić and Hock, 2006]. 2006; Wright et al., 2007], and bedrock topography [King [5] In the present study we investigate the potential of the et al., 2008]. European Centre for Medium‐Range Forecasts (ECMWF) ‐ ERA 40 reanalysis to be used for mass balance modeling in 3. Data Svalbard. The archipelago of Svalbard is one of the largest Arctic contributors to sea level rise outside the Greenland 3.1. ERA‐40 Reanalysis ice sheet [Dowdeswell et al., 1997]. The total volume of [7] The ERA‐40 reanalysis is a global reanalysis for the land ice in Svalbard is estimated at 7000 km3 [Hagen et al., period mid‐1957 to mid‐2002, with a temporal resolution of 1993], equivalent to a sea level rise of about 0.02 m. The 6 hours [Simmons and Gibson, 2000]. The reanalyses were archipelago is of particular significance with regards to produced using a T159 (∼125 km) spectral resolution and climate change because its location at the northern limit of 60 vertical levels in the atmosphere. Over the reanalysis the North Atlantic Current makes it sensitive to shifts in period, the observing system significantly changed, with the

2of18 F02014 RYE ET AL.: GLACIER MODELING USING ERA‐40 F02014 inclusion of satellite data from 1979 onward together with Table 1. DEM Data Sources increased observations from aircraft, ocean buoys and other Year Source surface platforms [Uppala et al., 2005]. As a result, the accuracy of the ERA‐40 reanalysis is improved from the end 1962 10 m contour map derived from field survey and of the 1970s [Simmons et al., 2004]. oblique terrestrial photogrammetry [Pillewizer, 1962] [8] For this study we used 6 hourly analyses of 2 m air 1977 10 m NPI contour map derived from vertical aerial temperature, wind speed and relative humidity as well as photographs 24 h forecast cumulative totals of precipitation, global 1995 5 m posting DEM from vertical aerial photographs radiation and longwave radiation. The wind speed and rela- 2003 lidar, ground resolution 0.48 m tive humidity data were retrieved for three pressure levels: 1000 mb, 925 mb and 850 mb. All three were used in the validation process (section 4.1) to identify which pressure explicitly added since it was largely absent in this period level best represents the in situ observations. The reanalysis (J. Kohler, unpublished data, 2000–2008). The implications data were retrieved from the ECMWF data server for the of this are discussed in the analysis of the model results. The whole of Svalbard and then bilinearly interpolated from the winter and summer measurements at stake locations are lin- T159 grid to the location of Ny‐Ålesund (78.9 N, 11.9 E, early extrapolated over the surface hypsometry (section 3.4) Figure 1). This allowed the reanalysis data to be validated to produce area‐averaged balances. The observations show against the in situ weather station observations at this that Midre Lovénbreen has experienced a consistently nega- location. tive mass balance, with a mean annual net loss of −0.38 m – [9] Spin‐up effects within the ERA‐40 forecasts are water equivalent over the period 1968 2001. known to adversely affect simulated variables [Uppala et 3.4. Digital Elevation Model al., 2005]. The best approach to removing these effects is to calculate daily totals from the 24 h forecasts that are [13] A digital elevation model (DEM) of the glacier is started every 12 hours [Martin, 2004]. For each 24 h fore- required by the surface mass balance model to spatially cast, the spin‐up is discarded by subtracting the amount of distribute meteorological data and compute topographic precipitation or radiation accumulated in 12 hours from the shading as well as slope and aspect angles. In this study we amount accumulated in 24 hours. Daily totals are then use an annually evolving 20 m DEM of Midre Lovénbreen, derived as the sum of the subdaily totals for the time intervals created by linearly interpolating four DEMs from 1962, 0–12 hours and 12–24 hours. 1977, 1995 and 2003 (Table 1). Hence, we approximate the “ ” “ ” [10] The surface mass balance model is forced using daily conventional mass balance using four reference surface values of the ERA‐40 variables. Daily means of tempera- DEMs [Elsberg et al., 2001]. In the model the DEM is ‐ ture, wind speed and relative humidity were calculated by updated at the end of each mass balance year. The land ice averaging the 6 hourly analyses. Daily cumulative totals of mask (and therefore the position of the glacier snout) is precipitation, global radiation and longwave radiation were updated at the beginning of each DEM epoch (i.e., 1962, derived from the forecast fields, as described above. 1977, 1995 and 2003). When the mass balances are hindcasted to the beginning of the ERA‐40 reanalysis, we use a 3.2. In Situ Validation Data static reference DEM from 1962 back to 1958. Sensitivity [11] In situ meteorological observations from two weather experiments found that this has a negligible impact on stations at Ny‐Ålesund, located about 5 km north of Midre simulated balances because the bulk of the changes in Lovénbreen (Figure 1), were used to validate the ERA‐40 hypsometry occur at the glacier snout, which only accounts reanalysis. Measurements of temperature, wind speed, rel- for a small percentage of the glacier surface area. Overall the DEMs show a reduction in glacier area from 5.9 km2 in ative humidity and precipitation are available three times a 2 day (0600, 1200, and 1800 UTC) for the period 1969– 1962 to 5.1 km in 2003. Full details of the four DEMs and present from a weather station maintained by the Norwegian data collection methods can be found in the work of Kohler Meteorological Institute (DNMI). These data were supple- et al. [2007]. mented with hourly observations of incoming direct, diffuse and longwave radiation available for the period 1992– 4. Methods present from a weather station maintained by the Alfred ‐ Wegener Institute (AWI) [König‐Langlo and Marx, 1997]. 4.1. ERA 40 Validation and Bias Correction [14] The in situ meteorological observations from Ny‐ 3.3. Mass Balance Observations Ålesund were used to validate the bilinearly interpolated [12] Mass balance observations were used as simulation ERA‐40 reanalysis. Daily, monthly, seasonal and annual targets to calibrate the surface mass balance model. The means were validated using Pearson’s product‐moment Norwegian Polar Institute has measured winter and sum- correlation coefficients to determine the association between mer mass balance each year since 1968 for 10 stakes along the ERA‐40 and in situ variables. All available observations the centerline of Midre Lovénbreen [Hagen et al., 2003, over the period of the reanalysis were used in the validation Figure 1]. The balances are derived from the measured process. Temperature, precipitation, wind speed and relative change in exposed stake height and the estimated density of humidity were validated for 1969–2002 using data from the the mass gained or lost. Superimposed ice may not be DNMI weather station, while global and longwave radiation explicitly accounted for in the pre‐2000 data; there are no were validated for 1992–2002 using data from the AWI records indicating that superimposed ice is measured in the weather station. For global radiation, we only included days older data. Post‐2000 we know that superimposed ice is not with solar radiation receipts greater than zero in the analysis.

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The in situ data for wind speed and relative humidity were doing so, we assume that the bias correction remains compared with the three ERA‐40 pressure levels (1000 mb, unchanged pre‐1969 for the DNMI data and pre‐1992 for 925 mb and 850 mb). In order to remove the influence of the the AWI data. This is not an unreasonable assumption given seasonal cycle on the daily and monthly correlations, the that the same data assimilation and modeling system is used ERA‐40 and in situ time series were first deseasonalized by for all years of the reanalysis [Simmons and Gibson, 2000]. subtracting from each day and month, the mean of the Thus, the bias correction scheme should produce an accurate corresponding day and month for the entire period of study. representation of the Ny‐Ålesund climate throughout the [15] After evaluating the association between the two data 1957–2002 period. The quantile mapping technique does sets, we corrected the ERA‐40 reanalysis for model bias. not alter the temporal properties of the ERA‐40 data and Climate model simulations, including reanalysis products, will therefore not affect the temporal correlation between the are frequently biased relative to in situ observations owing reanalysis and the observations. to their parameterization schemes and coarse grid resolutions. These errors have to be corrected before the climate 4.2. Surface Mass Balance Model model output can be used at the subgrid scale in environ- [18] The surface mass balance model consists of two mental models. The most common method of bias correc- coupled components. The first is a surface model that cal- tion involves rescaling the mean and/or variance of the culates the exchange of energy between the glacier surface simulated climate variable to the observations by assuming a and the atmosphere. The second is a subsurface model, probability distribution (see R. L. Wilby et al., Guidelines for use which simulates changes in temperature, density and water of climate scenarios developed from statistical downscaling content in the snow, firn and upper ice layers. The model methods, supporting material to the IPCC, 2004; available at calculates the surface mass balance for each 20 m DEM grid http://www.ipcc-data.org/guidelines/dgm_no2_v1_09_2004. cell of the glacier at a time step of 1 h. The model is forced pdf). However, the modeled and observed distributions may using diurnal ERA‐40 data, which are converted into hourly not be accurately known, of the same type, nor statistically values. Daily temperatures are interpolated by fitting a sine well behaved. To overcome these limitations we use a curve to the mean, maximum and minimum temperatures of quantile mapping technique [Panofsky and Brier, 1968]. the 6 hourly reanalyses. Forecast totals of precipitation are First, empirical cumulative density functions are derived for subdivided equally between 24 hours. Daily solar radiation both the model results and observations. The quantiles of the is scaled using hourly potential clear‐sky radiation (see two distributions are then matched to produce a correction section 4.2.1). Wind speed, relative humidity and longwave function that is used to replace each simulated quantile with radiation are not interpolated and instead the daily values the observed quantile that has the same occurrence proba- are assumed constant for each hour. The energy balance bility. This method has been previously employed to bias‐ and subsurface models are described in sections 4.2.1 and correct a variety of climate model variables, on both 4.2.2. monthly and daily timescales [e.g., Leung et al., 1999; Wood 4.2.1. Surface Energy Balance Model et al., 2002, 2004; Boé et al., 2007; Déqué, 2007; Hashino [19] The energy balance model is based on that developed et al., 2007]. by Arnold et al. [1996] and subsequently updated by Brock [16] In this study, the ERA‐40 data were bias‐corrected et al. [2000], Arnold [2005], and Arnold et al. [2006]. The on a daily timescale. For practical reasons, we generated a model determines the total surface energy balance from five correction function consisting of the 999 quantiles (i.e., main components: permillages) of the ERA‐40 and observed distributions.

Values between two quantiles were linearly interpolated, QM ¼ Q* þ L* þ QS þ QL þ QG; ð1Þ while data outside the quantile range were extrapolated by assuming a constant bias correction. For example, if the where QM is the energy available for melt, Q* is net ‐ uppermost quantile in the ERA 40 temperature distribution shortwave radiation, L* is the net longwave radiation, QS is is corrected by +2°C, all temperature values that fall above the sensible heat flux, QL is the latent heat flux and QG is the this quantile are also corrected by +2°C. If a variable is ground heat flux in the snow or ice. All units are in W m−2. bounded, then the correction cannot exceed the bound; for The model computes these fluxes on an hourly time step example, precipitation cannot be extrapolated beyond the from measurements of air temperature, wind speed, relative lower bound of zero. An alternative approach would be to fit humidity, incoming shortwave radiation (direct and diffuse) theoretical probability distributions to extrapolate the data; and incoming longwave radiation. In what follows we however this is less robust and can produce unrealistic describe the calculation of each of the energy balance com- extreme values at the ends of the ERA‐40 distributions. ponents using the ERA‐40 reanalysis. [17] Figure 2 shows the simulated and observed empirical [20] The net solar radiation flux (Q*) for each DEM cell cumulative density functions for each of the variables used and every hour is calculated from downward instantaneous ‐ in this study. The ERA 40 global and longwave radiation fluxes of direct (Qidir′ ) and diffuse (Qidif′ ) shortwave radiation show minimal biases and were therefore used uncorrected in and the surface albedo (a): the surface mass balance model. Temperature, precipitation, 1 0 0 : 2 wind speed and relative humidity all display unambiguous Q* ¼ð ÞðQidir þ Qidif Þ ð Þ biases and were corrected using the quantile mapping scheme. We derived the correction function for the valida- The direct and diffuse fluxes for each grid cell are derived ‐ ′ tion period (1969–2002 for the DNMI data and 1992–2002 from ERA 40 global radiation (QG) in three steps. First, for the AWI data). The bias correction was then applied to daily (24 h forecast) totals of ERA‐40 global radiation are ′ ′ the entire period of the ERA‐40 reanalysis (1957–2002). In separated into direct (Qdir) and diffuse (Qdif) fractions using

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Figure 2. Empirical cumulative density functions of (a) observed and ERA‐40 temperature, (b) precipitation, (c) wind speed, (d) relative humidity, (e) global radiation, and (f) downward longwave radiation. an empirical relation. Second, the daily totals are converted analysis. The measured global radiation was computed as into hourly values by scaling the data using hourly potential the sum of measured diffuse and direct (normalized for the clear‐sky solar radiation. Third, the hourly fluxes are mod- horizontal) radiation components. Daily mean theoretical ified for each DEM cell depending on the local terrain top‐of‐atmosphere radiation was calculated using standard parameters, namely slope, aspect and topographic shading. methods [Sellers, 1965]. The data were preprocessed to [21] The ERA‐40 global radiation is separated into direct remove global or top‐of‐atmosphere daily means less than and diffuse components for each day on the basis of the ratio 20 W m−2 because the ratios are expected to lose accuracy of ERA‐40 surface global radiation QG′ to theoretical top‐of‐ under decreasing radiation amounts [Hock and Holmgren, atmosphere radiation QTOA′ [Hock and Holmgren, 2005]. 2005]. A cubic polynomial was then fitted to the data by This ratio is essentially a measure of cloudiness. As the ratio the least squares method (Figure 3): QG′ /QTOA′ decreases (and cloudiness increases), the diffuse 8 9 0 < 0:15 x 0:75 = proportion of global radiation Qdif′ /QG′ increases. An Q ′ ′ dif ¼ 0:77 þ 2:5x 7:82x2 þ 4:68x3 0:15 < x < 0:8 ; empirical relationship was derived between QG/QTOA and Q0 : ; G 1:0 x 0:20 Qdif′ /QG′ using daily means of diffuse and global radiation measured at the AWI Ny‐Ålesund weather station. All ð3Þ available data over the period 1992–2007 were used in the

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received at each DEM cell is computed by first converting the ERA‐40 horizontal flux into the equivalent radiation received by a normal surface (Qndir′ ), by dividing the direct radiation by the cosine of the solar zenith angle. The normal radiation is then modified for the local aspect and slope angles:

0 0 sin cos 0 cos sin 0 cos 0 ; 5 Qidir ¼ Qndir½Z Z þ Z Z ðA A Þ ð Þ where Z is the angle of the Sun above the horizon, Z′ is the DEM cell surface slope, A is the solar azimuth and A′ is the azimuth surface slope. The azimuth is defined as degrees from due south, positive to the west and negative to the east. The instantaneous direct radiation is supplemented by the instantaneous diffuse flux Qidif′ , calculated by multiplying the ERA‐40 horizontal diffuse radiation by the DEM cell Figure 3. Ratio of diffuse radiation to global radiation ′ ′ ‐ ‐ skyview factor and adding reflected radiation from the (Qdif/QG) versus the ratio of global radiation to top of atmo- surrounding topography [Arnold et al., 2006]: sphere radiation (QG′ /QTOA′ )atNy‐Ålesund. Data are daily means over the period 1992–2002. 0 0 1 0 ; 6 Qidif ¼ fsQdif þ tð fsÞQG ð Þ

where f is the skyview factor computed following Oke where x is the ratio of global to top‐of‐atmosphere radiation s [1987] and a is the mean albedo of the visible terrain, Q′ /Q′ . The upper and lower limits of 0.75 and 0.20 were t G TOA taken as 0.25 following Arnold et al. [2006]. arbitrarily selected from the graph (Figure 3). Some scatter [24] In previous versions of the model, the surface albedo exists in the data because the relationship varies with cloud a was calculated using empirical relations derived from type. The relationship is in good agreement with that field observations [Arnold et al., 1996, 2006]. In this study derived by Hock and Holmgren [2005] for Kiruna, Sweden, we calculate the albedo internally as a linear function of the also shown in Figure 3. For each day, equation 3 is used to density of the top 10 cm of the subsurface grid (r ), similar derive the diffuse fraction of ERA‐40 daily global radiation, top to that used by Greuell and Konzelmann [1994]: which is then subtracted from global radiation to obtain the direct component. All fluxes are computed for a horizontal ð top iceÞ ; 7 surface because the incidence angle of ERA‐40 global ¼ ice þð snow iceÞ ð Þ ðsnow iceÞ radiation is the mean cosine of the zenith angle of each 24 h forecast, which is assumed to be horizontal. where asnow, rsnow, aice and rice denote fresh snow albedo, [22] The daily ERA‐40 radiation totals are converted into fresh snow density, ice albedo and ice density, respectively. hourly values using the mean ratio of ERA‐40 radiation to In this parameterization, changes in surface density repre- clear‐sky radiation for each day. This ratio is then scaled for sent changes in grain size and surface roughness that affect each hour using potential clear‐sky radiation. This approach the surface albedo [Greuell and Konzelmann, 1994]. The assumes that the ratio of ERA‐40 to clear‐sky radiation (and model albedo decreases as the surface density increases therefore cloudiness) is constant for each day. For every owing to melting and refreezing, while new snowfall con- hour ERA‐40 incoming direct solar radiation at horizontal versely reduces the density of the upper model layers and incidence (Q′ ) is calculated from: dir increases the albedo. If no snow cover exists, then a = aice. There are no long‐term field observations over the period of 0 0 Qs 0 a a r Q ¼ Q ; ð4Þ study to constrain the values of snow, ice and snow. These dir 0 c Qsc parameters are instead assumed constant in time and space 0 0 and are optimized during model calibration (section 4.3). = ‐ ‐ −3 where Qs Qsc is the ratio of daily mean ERA 40 to clear For the density of ice, a value of 0.91 g cm is assumed ′ ‐ sky direct solar radiation and Qc is the clear sky direct solar [Paterson, 2000]. radiation for the given hour, all at horizontal incidence. The [25] The net longwave radiation (L*) is the sum of ‐ ERA 40 diffuse component for each hour is also computed incoming fluxes received from the sky and surrounding using the same approach. Both direct and diffuse potential terrain (Lin) and outgoing radiation emitted by the glacier clear‐sky radiation are calculated using standard solar surface (Lout). We do not convert the daily totals of ERA‐40 equations [Iqbal, 1984]. longwave radiation into hourly values, as with solar radia- ‐ [23] Finally, the hourly ERA 40 direct and diffuse hori- tion, but instead assume that the daily mean longwave zontal radiation components are modified for the local radiation is representative of each hour. The longwave conditions at each DEM cell. Solar positional parameters are radiation received by each DEM cell from the sky is com- computed using standard astronomical theory [e.g., puted by first correcting the ERA‐40 longwave radiation for Walraven, 1978]. Terrain slope, aspect and hourly topo- the elevation of each cell using the Stefan‐Boltzmann graphic shading are derived for each DEM cell using the equation and then for the sky‐view factor. The longwave method of Arnold et al. [1996]. Shaded grid cells only sky radiation is supplemented with the longwave radiation “ ‐ ” receive diffuse radiation, while in Sun grid cells receive emitted from the surrounding terrain. This is calculated both direct and diffuse. The instantaneous direct radiation

6of18 F02014 RYE ET AL.: GLACIER MODELING USING ERA‐40 F02014 using the Stefan‐Boltzmann equation and the mean surface the snowpack accumulates, cells are removed from the temperature of the visible terrain, which is approximated by bottom of the grid to maintain a constant thickness of 25 m the lapse rate corrected air temperature of each grid cell of snow/ice. [Arnold et al., 2006]. Thus: [28] To maintain numerical stability, the model is run using a time step of 15 min. The model is driven using the 0 1 4 ; 8 Lin ¼ Lincfs þð fsÞ Tterr ð Þ hourly energy available for melt computed by the surface energy balance model (assuming a constant rate of melt ′ ‐ where Linc is the elevation corrected incoming longwave during one hour). Of this energy, the longwave and turbu- radiation, s is the Stefan‐Boltzman constant (5.67 × 10−8 W −2 −4 lent flux components are entirely absorbed by the uppermost m K ), and Tterr is the mean surface temperature of the cell. Incoming shortwave radiation with wavelengths greater visible terrain. The outgoing longwave radiation emitted than 0.8 mm (36%) is also absorbed in the top cell, while the from the glacier surface is computed as a function of surface remainder penetrates beyond the surface and is absorbed in ‐ temperature (Ts) and the Stefan Boltzmann constant: underlying cells according to Beer’s law. For each 15 min time step, a one‐dimensional thermodynamic equation is L T 4: 9 out ¼ s ð Þ then solved over the gird. A density‐dependent effective thermal conductivity [Sturm et al., 1997] is used to describe In the previous version of the model, Arnold et al. [2006] the potential for heat transfer in each snow cell, while for ice calculated the surface temperature using a simple two‐layer cells thermal conductivity is computed as a function of subsurface model based on that of Klok and Oerlemans temperature following Yen [1981]. If the temperature of the [2002]. Here the surface temperature is taken as the temper- uppermost grid cell exceeds 0°C, the temperature of the cell ature of the uppermost grid cell in the subsurface model is set to 0°C and surplus energy is used for melting the (section 4.2.2). surface. Melting also occurs in underlying layers owing to [26] The sensible (Q ) and latent (Q ) turbulent heat fluxes s L the penetration of shortwave radiation. Meltwater is able to are calculated for each DEM cell and every hour using the percolate downward through the grid. Refreezing occurs in bulk aerodynamic method [Munro, 1990], which requires cells where the temperature is below 0°C and the density is inputs of air temperature, air pressure, wind speed and rela- less than the density of ice. The underlying cell receives any tive humidity. The air temperature of each DEM cell is cal- residual meltwater if either of these conditions is not met, or culated using the calibrated model atmospheric lapse rate − if there is excess meltwater after refreezing. Meltwater (see section 4.3). For pressure, a lapse rate of 10 kPa km 1 is percolates until it reaches the snow/ice interface. At this used. ERA‐40 wind speed and relative humidity are assumed impermeable layer, superimposed ice is formed using the spatially invariant over the glacier surface. This method also approach of Wakahama et al. [1976]. If the rate at which requires knowledge of the surface roughness length. The meltwater reaches the snow/ice interface exceeds the rate of lengths for snow (0.00022 m) and ice (0.00066 m) for Midre superimposed ice formation, then excess water will form Lovénbreen are taken as the mean values from Arnold and runoff, resulting in mass loss from the glacier. Rees [2003]. [29] Densification of snow and firn in the model is driven 4.2.2. Subsurface Model entirely by melting and refreezing. Greuell and Konzelmann [27] The subsurface model is based on that developed by [1994] also accounted for densification owing to settling and Bassford [2002] and updated by Wright [2005] and is a packing when applying their model to the Greenland ice simplified version of the approach applied to the Greenland sheet. However, the formation of ice lenses is prevalent on ice sheet by Greuell and Konzelmann [1994]. For each Midre Lovénbreen [Wright, 2005] and therefore meltwater DEM cell, the model calculates temperature, density and refreezing will dominate the densification process in the water content on a one‐dimensional grid extending verti- upper part of the snow or firn [Paterson, 2000]. As a result, cally from the surface into the glacier. The grid consists of a we assume that the role of settling and packing is negligible. maximum of 500 cells, split into two interconnecting parts, Snow cells within the model are converted to ice once the one for the snowpack and a second for the underlying − density exceeds that of superimposed ice [0.8 g cm 3, continuous glacier ice. The later part has an initial depth of Wright, 2005]. 20 m and consists of 25 cells that increase in size from 5 cm 4.2.3. Role of Superimposed Ice and Internal at the snow/ice interface to a maximum of 200 cm at the Accumulation base of the grid. The size of each ice cell was derived using [30] As mentioned above, superimposed ice may not have a grid stretching function [Tannehill et al., 1997]. The grid been fully incorporated into the mass balance observations. is designed to extend to a depth where the annual temper- The mass balance is calculated by multiplying the change in ature oscillation can no longer be detected [Greuell and surface height by the density of the mass gained or lost. Konzelmann, 1994]. The remaining cells are reserved for Superimposed ice is not explicitly accounted for in the the snowpack, which rises and falls depending on accu- density calculations. Instead, a constant density of 0.55 g mulation and melting. Each snow cell has an initial thick- − m 3 is assumed for all unmelted snow, firn, or superimposed ness of 5 cm. If the uppermost snow cell falls below 0 cm ice remaining at the end of each summer [Kohler et al., owing to melting then it is removed from the grid, while if it 2002]. Given the lack of data on superimposed ice accu- grows above 10 cm owing to accumulation it is split into mulation prior to 2000, it is difficult to reliability correct the two new cells. The previous versions of the model neglected observed balances. Instead, we account for superimposed firn accumulation by removing any unmelted snow at the ice in the model simulations and assess the implications of end of each mass balance year. In this study we adapt the this by performing a sensitivity experiment. model to allow for the buildup of a multiyear snowpack. As

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Table 2. Parameters and Ranges Used to Calibrate the Model objective functions that assess the goodness‐of‐fit between model simulations and observed data. In this study we use an Parameter Range of Values Units automated search procedure to optimize six parameter values Precipitation gradient 0–100a % per 100 m within ranges on the basis of previously published values – b Fresh snow albedo 0.65 0.95 (Table 2). The calibration is performed to minimize the Ice albedo 0.4–0.55b Fresh snow density 0–0.4c gcm−3 difference between observed and modeled mass balances Winter lapse rate 0–10d °C km−1 over the period 1968–2001. The parameters chosen are those Summer lapse rate 0–10d °C km−1 that are considered most critical for the performance of the aReported orographic precipitation rates are 5–25% [Hagen and Liestøl, model with respect to the simulated mass balance. Winter 1990; Bruland and Hagen, 2002], but a wider range is allowed to account and summer lapse rates are calibrated independently because for other factors, such as snow redistribution. temperature gradients are often observed to weaken during bOn the basis of values reported for a number of high Arctic locations the summer when the glacier surface is melting [e.g., Greuell [Greuell and Konzelmann, 1994; Knap and Oerlemans, 1996; Zuo and and Böhm, 1998]. We do not consider negative lapse rates Oerlemans, 1996; Arendt, 1999; Gerland et al., 1999]. cOn the basis of values from Paterson [2000]. since although temperature inversions are common in the dLapse rates are varied within physically realistic bounds. Arctic, mean winter and summer lapse rates tend to be positive [e.g., Gardner et al., 2009]. In sections 4.3.1 and [31] A second unknown quantity is internal accumulation, 4.3.2 we describe the search procedure and objective func- which occurs in the firn zone as a result of the refreezing of tions used to constrain the parameter values. meltwater or rain that percolates below the summer surface 4.3.1. Automated Search Procedure of the previous year. In standard mass balance measuring [35] The model is calibrated using the Nelder and Mead techniques, only the balance of the snow layer from the [1965] downhill simplex nonlinear optimization algorithm. current year is considered [Østrem and Brugman, 1991]. The method minimizes an objective function in a multidi- Consequently, all meltwater that percolates into underlying mensional parameter space and has the advantage that it firn is assumed lost through ablation. To ensure that model does not require the calculation of derivatives, only function results are consistent with mass balance measurements, we evaluations. For a detailed description of the method, the assume that all water reaching the base of the snowpack of reader is referred to Press et al. [1992]. We first calibrate the current year forms runoff. The implications of this are the winter balance (1 October to 30 April) by optimizing also tested by means of a sensitivity experiment. the winter precipitation gradient and winter lapse rate. 4.2.4. Accumulation The remaining parameters are then optimized for the net [32] ERA‐40 precipitation is distributed over the glacier balance (1 October to 30 September). The downhill simplex surface using an elevation‐dependent precipitation gradient. approach is often able to find the optimal solution with little Estimates of the orographic precipitation gradient in the difficulty, but if the algorithm is applied to a “rough” response region around Ny‐Ålesund vary widely between 5% and surface, the solution may converge on a local (rather than 25% per 100 m [Hagen and Liestøl, 1990; Bruland and global) minimum. To overcome this problem, we perform Hagen, 2002]. Linear elevation‐accumulation gradients multiple simulations with different starting points within the derived from winter balance measurements on Midre parameter space. The global solution is then selected as the Lovénbreen show significant interannual variations [Kohler solution with the smallest objective function value. Solutions et al., 2002], suggesting that the gradient in precipitation that converge outside the specified parameter space (Table 2) also exhibits substantial year‐to‐year changes. Owing to this are discarded. uncertainty, we treat the precipitation gradient as a tunable 4.3.2. Objective Functions model parameter. The fractions of precipitation falling as [36] Most mass balance models are calibrated to get a good rain and snow in each DEM cell are calculated as a function fit between observed and simulated variables by attempting to of air temperature using a threshold of 1.62°C [Førland and minimize a single measure of model performance [e.g., Hanssen‐Bauer, 2003]. Oerlemans and Fortuin, 1992; Fleming et al., 1997; 4.2.5. Initial Conditions Braithwaite and Zhang, 1999; Hock et al., 2007]. However, [33] Model simulations begin on the first day of the mass experience with environmental models has shown that cali- balance year on 1 October. Each DEM cell is initialized as bration using a single objective function frequently produces bare ice and all subsurface cells are set to the mean annual unrealistic results [Gupta et al., 1998; Yapo et al., 1998]. This air temperature of the ERA‐40 reanalysis. The model is then is because all models are simplifications of reality and spun‐up over 20 pseudoannual cycles using the climatology therefore calibration using a single objective often constrains of the first year of the model run. This allows the mass the model parameters to fit certain characteristics of the balance and subsurface temperature profiles to attain a observed data, but neglects the remaining features. One way steady state, and facilitates the creation of a permanent firn to overcome this problem is to adopt a multiple‐objective zone on the upper part of the glacier. The initial conditions calibration approach that constrains the model parameters do not influence the steady state attained, only the time using multiple goodness‐of‐fit measures to assess different taken to reach a stable solution. aspects of model performance. A multiobjective calibration problem can be formulated as: 4.3. Calibration Minimizefgf1ðÞ; f2ðÞ; :::; f ðÞ ; ð10Þ [34] Model calibration involves constraining model para- n meters such that the simulated results are consistent with where f (), f (),…, f () are the n objective functions to be the available observed data. The parameters are adjusted 1 2 n simultaneously minimized with respect to the parameters of according to a specified search scheme and one or more

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Figure 4. (a) Schematic diagram of a bi‐objective function search space, where F1 and F2 are the two objectives to be minimized. The minimum and maximumvaluesofeachfunctionaredenotedbythe min superscripts min and max. The Pareto optimal set of solutions occurs along a front bounded by F1 min and F2 . When both objectives are considered together, the Pareto optimal solutions are superior to all other solutions within the function search space. (b) The compromise solution, which equally minimizes both objectives. The utopia point represents the point where each objective function has a minimum value simultaneously. the model [Yapo et al., 1998]. The solution to equation 10 is a square error (RMSE) because it is less influenced by outliers set of Pareto optimal solutions that result from various and therefore provides a more robust indicator of the sizes of compromises between the chosen objectives. Pareto optimal typical errors. We also define a third objective function, the solutions are superior to other solutions when all objectives absolute value of the area‐averaged cumulative error (CE), are considered together, but inferior with respect to one or to assess the ability of the model to track the long‐term mass more of the individual objectives [Pareto, 1906]. This con- balance trend: cept is illustrated in Figure 4a. All Pareto solutions are con- Xn sidered equal. It is then up to the modeler to decide the f3 b ; bmod; ; 13 solutions that are required for the question in hand. ð Þ¼ obs y y ð Þ y¼1 [37] A distributed surface mass balance model should ideally be calibrated using at least one objective function where bobs; y and bmod; y are the observed and modeled area‐ that incorporates spatial observations, such as variations in averaged mass balances for year y. albedo or snow depth across the glacier [e.g., Schuler et al., [38] To evaluate whether multiobjective calibration is a ‐ 2007]. In the present study, long term spatial data sets suitable approach for optimizing the performance of the suitable for calibration are not available and therefore the mass balance model, a preliminary single‐objective cali- model can only be optimized against the available mass bration, with respect to the MAE at stake locations (f1 ()), balance measurements. The specific balances measured at was undertaken. The model performance was then assessed stake locations are the main test of model performance. We with respect to the f () and f () objectives. The initial ’ 2 3 define two objective functions to evaluate the model s results (Table 3) revealed multiple minima with different ability to reproduce these data: parameter combinations that perform equally well with Xn respect to the MAE at stake locations. Thus, on the basis of 1 f1ðÞ¼ b ; bmod; ; ð11Þ only one performance criterion, the best parameter set can- n obs j j j¼1 not be established. In addition, although the parameter combinations are able to capture the observed stake data Xn well, they are unable to minimize the f2 ( ) and f3 ( ) ob- 1 f2ðÞ¼ grad ; gradmod; ; ð12Þ jectives (Table 3). The MAE of the net mass balance gradient n obs y y y¼1 in Table 3 ranges from 0.098 to 0.131 m w.e. per 100 m, which corresponds to an underestimation of the mean net where n is the number of observations, bobs,j and bmod, j are mass balance gradient of 18 to 39%. The area‐averaged net the observed and modeled specific balances for stake j, and mass balance CE is underestimated by ∼2 m w.e., which is an gradobs,y and gradmod, y are the observed and modeled mass order of magnitude larger than the error at the stake locations. balance gradients for year y. The f1() objective function is Given the magnitude of these errors, we conclude that a the mean absolute error (MAE) between observed and single‐objective calibration is unacceptable, confirming that modeled balances at stake locations, while f2() is the MAE a multiobjective approach is required. between the observed and modeled mass balance gradient. [39] To solve a multiobjective calibration problem The MAE is preferred over the commonly used root mean (equation 10), the individual objective functions are typically

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Table 3. Best Optimized Parameter Combinations for the Modeled Net Mass Balance With Respect to the MAE at Stake Locations f1 ()a

f1 () f2 () f3 () Precipitation Gradient Winter Lapse Rate Summer Lapse Rate Snow Density Snow Ice (m w.e.) (m w.e. per 100 m) (m w.e.) (% 100 per m) (°C km−1) (°C km−1) (g cm−3) Albedo Albedo 0.24 0.098 2.86 46 6.6 5.8 0.26 0.68 0.49 0.24 0.111 1.93 46 6.6 5.1 0.34 0.66 0.50 0.24 0.131 1.80 46 6.6 4.0 0.38 0.66 0.52 0.24 0.103 2.61 46 6.6 5.0 0.28 0.69 0.48 0.24 0.126 1.65 46 6.6 3.2 0.40 0.69 0.49

a Also shown are the MAE in the mass balance gradient ( f2 ()) and the area‐averaged CE ( f3 ()).

converted into an aggregate objective function (fagg ()). This elevation of each stake. For computational efficiency, we do can be achieved by applying a weight w to each of the n not seek the entire Pareto optimal set. Instead, we apply objectives. The optimization problem thus becomes: equal weights to each objective function to obtain a subset of solutions in the region of the Pareto optimal compromise Minimize ; :::; : 14 faggð Þ¼ ½w1f1ð Þþw2f2ð Þ wnfnð Þ ð Þ solution. The winter balance compromise solution is first established and then used in the calibration of the net bal- The aggregate objective function can then be solved using a ance to derive the net compromise solution. The winter and ‐ single objective optimization technique, such as the down- net compromise solutions are then used for further analysis hill simplex algorithm. The optimal Pareto set of solutions of the model results and to extend the mass balance time is obtained by repeating the optimization process using series back to the beginning of the ERA‐40 reanalysis in different weights. For example, equal weights can be used 1958. to obtain the compromise solution, which simultaneously optimizes all objectives (Figure 4b). In practical situations, computing the entire Pareto set is relatively time con- 5. Results and Discussion suming and therefore often only a subset of the optimal solutions is sought for a particular problem. [42] Unless otherwise stated, all correlation coefficients presented herein are significant at the 5% level. [40] In addition to aggregating the objective functions, they also have to be normalized so that their values are 5.1. ERA‐40 Validation approximately the same magnitude. This ensures that the [43] Daily, monthly and annual ERA‐40 temperature calibration procedure is not biased toward any single means correlate well with the surface observations, yielding objective. A common method of normalization is to scale coefficients of r = 0.93 in all cases. Seasonally, the corre- each function between specified upper and lower bounds: lations are higher during the autumn (September, October min November (SON)), winter (December, January, February 0 fnð Þfn ; 15 fn ð Þ¼ max min ð Þ (DJF)) and spring (March, April, May (MAM)), compared fn fn to the summer (June, July and August (JJA)): rSON = 0.97, max rDJF = 0.96, rMAM = 0.93, rJJA = 0.58. The lower correlation where f n′ () is the normalized objective function, and fn min during the summer is not unexpected given the lower vari- and fn are the arbitrarily estimated maximum and mini- min ability throughout the melt season when temperatures linger mum objective function values. For this study, fn is set to zero and f max is taken as the objective function value close to 0°C. During the other seasons, temperature fluc- n ‐ corresponding to the start point of the downhill simplex tuations are significantly larger and the ERA 40 reanalysis algorithm. Thus, each normalized objective function is is able to capture the observed temporal variability much initialized with a value of 1 and is then equally minimized better. Similar results showing a reduction in summer cor- ‐ toward 0. After a solution is established, the normalized relations in the ERA 40 reanalysis have also been found values are converted back into their original units of over the Greenland ice sheet and Arctic Ocean (see NOAA, measurement. Arctic regional reanalysis, Arctic Project report, 2004; available at http://www.arctic.noaa.gov/aro/reports/fy2004/ [41] The normalized f1 (), f2 () and f3 () objectives are aggregated into a single objective function and then mini- arctic_reanalysis_progress_report.pdf). Further analysis of ‐ mized using the downhill simplex algorithm, as described the results for Ny Ålesund reveals that the summer corre- above. The calibration is performed for all years of the mass lation is unduly influenced by 3 outlying years (1972, 1973 balance record between 1968 and 2001. Mass balance and 2002), which if removed, result in an increased coeffi- measurements are not available for all 10 stakes every year cient of 0.74. This indicates that for the majority of years the ‐ and in 1986 no stake observations are available, so we only ERA 40 reanalysis simulates summer temperatures reason- calibrate against the stakes where data are available. The ably well. precise locations on the DEM of each mass balance stake are [44] Analysis of the daily, monthly, and annual precipi- unknown for large parts of the record. There is a reasonably tation data yields correlation coefficients of 0.53, 0.64, and consistent linear relation between mass balance and eleva- 0.67, respectively. Similar to temperature, the variability in tion along the glacier centerline [Kohler et al., 2002], so we precipitation is best captured during the autumn (rSON = derive modeled balances for the stake locations by linearly 0.64), winter (rDJF = 0.57) and spring (rMAM = 0.60) com- interpolating the modeled centerline balances to the reported pared to the summer (rJJA = 0.45). As expected, these correlations are lower than those for temperature. It is difficult

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Table 4. Best Optimized Parameter Combinations for the Modeled Net Mass Balance With Respect to All Three Objective Functions

f1 () f2 () f3 () Precipitation Gradient Winter Lapse Rate Summer Lapse Rate Snow Density Snow Ice Set (m w.e.) (m w.e. per 100 m) (m w.e.) (% per 100 m) (°C km−1) (°C km−1) (g cm−3) Albedo Albedo 1 0.25 0.087 0.00033 45 7.5 5.5 0.30 0.66 0.44 2 0.26 0.095 0.00004 45 7.5 4.3 0.36 0.67 0.43 3 0.25 0.096 0.00124 45 7.5 4.4 0.40 0.67 0.44 4 0.26 0.092 0.00642 45 7.5 3.3 0.29 0.72 0.37 5 0.25 0.104 0.03648 45 7.5 4.1 0.24 0.69 0.47 to fully assess the quality of the ERA‐40 precipitation Seasonally, correlations are generally weaker, with coeffi- because the Nordic precipitation gauge used at Ny‐Ålesund cients of rSON = 0.51, rDJF = 0.66, rMAM = 0.34 and rJJA = is known to exhibit substantial errors in measuring snowfall 0.17 (insignificant). Of all the climate variables, relative [Førland et al., 1996]. These errors are greatest under con- humidity displays the weakest relationship with the ob- ditions of low temperature and high wind speed. Attempts servations for daily (r = 0.41), monthly (r = 0.35) and annual have been made to produce generic corrections to account for (r = 0.24, insignificant) means. All seasonal correlations are these errors [Førland and Hanssen‐Bauer, 2000], but a also weaker (r < 0.40). The surface mass balance model is significant degree of uncertainty still exists owing to the relatively insensitive to variations in both wind speed and large variability of both temperature and wind speed. As relative humidity. As a result, the lower correlations for these discussed below, despite the lower correlations, the model is variables will not influence the model results in a major way. still able to reproduce the winter balance observations of [47] Overall, the above analysis indicates that the ERA‐40 Midre Lovénbreen very well. reanalysis corresponds sufficiently well with observations at [45] The fluxes of global and longwave radiation received Ny‐Ålesund to be used for mass balance modeling in the by the glacier surface provide the main source of energy for region. melt (see summary table in the work of Willis et al. [2002]) and therefore it is important that the ERA‐40 reanalysis can 5.2. Model Calibration reproduce these variables correctly. The correlation analysis [48] Table 4 shows the best solutions found for the net for global radiation yields coefficients of 0.61, 0.66 and 0.63 mass balance within the region of the Pareto optimal com- for daily, monthly and annual means, respectively. The promise solution, derived by minimizing the equally values for longwave radiation are stronger in all cases, with weighted aggregated objective function using the downhill coefficients of 0.84, 0.90 and 0.87, respectively. Climate simplex algorithm. Given that multiple start points within the reanalyses are known to have problems with accurately parameter space were used in the calibration process, we are simulating the Arctic radiative budget owing to uncertainties confident that approximate global solutions have been found. in modeling the formation and evolution of cloud cover Compared to the single‐objective calibration (Table 3), all of [e.g., Walsh et al., 2009]. This may partly explain the lower the solutions in Table 4 show improvements with respect to correlations for global radiation. But, despite such un- the f2 () and f3 () criteria. Including the f2 () objective certainties, the ERA‐40 reanalysis does capture longwave function in the calibration process decreases the MAE of the radiation surprisingly well. The short temporal coverage of net mass balance gradient to within 0.087–0.104 m w.e. per observations (1992–2002) should also be borne in mind. The 100 m (corresponding to an underestimation of the mean net ERA‐40 reanalysis reproduces seasonal variations in global mass balance gradient by 8–18%), while the addition of the radiation reasonably well during the summer (rJJA = 0.71) f3 () objective function reduces the area‐averaged net mass and autumn (rSON = 0.68), although the correlation is lower balance CE by more than one order of magnitude. At the same during the spring (rMAM = 0.29, insignificant). This is time, the MAE at stake locations increases by ∼0.02 m w.e., encouraging given that the greatest melt will often occur indicating that only a small compromise in the f1 () during the summer and autumn months. For longwave objective function is required to satisfy the f2 () and f3 () radiation seasonal means also correspond well, with all criteria. The MAE at stake locations and the area‐averaged coefficients greater than r = 0.67. Overall the ERA‐40 CE are both reduced to negligible values; however reason- reanalysis radiative fluxes are more than adequate, especially able errors still exist in the MAE of the net mass balance when the small biases (Figure 3) are also considered. The gradient. This implies that there is a trade‐off in the f2 () mean errors in ERA‐40 mean annual global radiation and objective function, which may result from simplifications in mean annual longwave radiation receipts are only 5.5% and the way the model describes the mass balance gradient. This 2.1%, respectively, over the period of study. is considered further in the subsequent analysis of model [46] For wind speed and relative humidity, the in situ data performance (section 5.3). ‐ were compared with three ERA 40 pressure levels (1000, [49] For the winter balance, the majority of start points 925 and 850 mb). The best correlations were found at 925 mb within the parameter space were found to converge on a for wind speed and at 1000 mb for relative humidity. These winter lapse rate of ∼6.5–7.5°C km−1 and a precipitation pressure levels were therefore used in further analysis. Val- gradient of ∼45% per 100 m. The convergence of solutions idating reanalysis wind speeds in mountainous terrain is in the same region of the parameter space gives high con- challenging given the influence of local topography. Even fidence in the description of winter accumulation by the so, the ERA‐40 reanalysis captures local‐scale observations model and in the optimized parameter values. The downhill at Ny‐Ålesund surprisingly well, with daily, monthly and simplex algorithm was able to find a global solution easily annual correlations of r = 0.64, 0.58 and 0.49, respectively. for the winter balance because the parameter search space

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results correspond well with recent lapse rate observations on the neighboring glacier Kongsvegen, which indicate mean winter (September–June) and summer (July–August) lapse rates of ∼7°C km−1 and ∼3°C km−1, respectively [Erath, 2005]. All the solutions found using the simplex algorithm were well within the 0–10°C km−1 range specified by the parameter space (Table 2). The algorithm never tended toward negative lapse rates, even when it was initi- ated near 0°C km−1. This validates our initial assumption that the mean winter and summer lapse rates are positive. The winter precipitation gradient of 45% per 100 m is substantially larger than orographic precipitation rates of 5– 25% per 100 m reported in the literature [Hagen and Liestøl, 1990; Bruland and Hagen, 2002], but is not unexpected given that it will also account for other factors such as snow redistribution. Recent observations of fresh snow densities on Midre Lovénbreen indicate a mean value of 0.23 g m−3 [Wright, 2005], which is smaller than the optimized densities in Table 4. This is probably because the model does not Figure 5. Scatter diagram of the objective function values account for densification owing to settling and packing and of the optimized parameter combinations in Table 4. Each therefore slightly larger initial snow densities are required to point is denoted by its parameter set number corresponding compensate. Both the snow and ice albedo values are as to those listed in Table 4. expected given the range of values prescribed in the calibration process (Table 2). only has two dimensions and consequently there are few [52] In section 5.3 the model performance is further local minima to navigate. Overall, the best solution that evaluated using the compromise solution, which provides equally minimizes all three objective functions has a winter the best overall fit to all three objective functions. Figure 5 lapse rate of 7.5°C km−1 and a precipitation gradient of 45% shows a scatterplot of the three‐dimensional objective per 100 m. This winter balance compromise solution was function space of the solutions in Table 4. It can be seen that subsequently used in the optimization of the net balance parameter set 1 equally minimizes all three objectives and is parameters (Table 4). therefore selected as the compromise solution for further [50] For the net balance, the dimensionality of the solution analysis. search space is larger and therefore the response surface of the aggregate objective function contains more complexi- 5.3. Model Verification ties. As a result, a larger range of parameter values exists [53] The calibrated compromise solution is verified by between the different optimal solutions (Table 4). This comparing the model results to observations. The specific implies that if two or more calibrated versions of the same balances measured at stake locations are the main test of model have similar values of the aggregate objective func- model performance. We also evaluate the simulated area‐ tion, it does not necessarily mean that they will all behave in averaged mass balances and the mean mass balance gradient. the same manner. This can be further investigated by Scatter diagrams of observed and modeled stake balances ‐ examining the various parameter trade offs required to are shown in Figure 6, along with plots of the observed and produce the optimal solutions for the net balance. For modeled area‐averaged mass balance time series and the example, parameter sets with higher snow albedos require corresponding cumulative change. The area‐averaged net lower lapse rates and lower ice albedos to compensate for balances are plotted relative to estimated error bounds of the the reduced snowmelt caused by the higher reflectivity of observed data. The estimated uncertainty in net mass bal- the snow surface (e.g., parameter set 4 in Table 4). Con- ance measurements is taken as ±0.25 m w.e. [Kohler et al., versely, lower snow albedos are balanced by higher lapse 2007; Rasmussen and Kohler, 2007]. For the cumulative rates and higher ice albedos (e.g., parameter set 1 in Table 4). balance, if the errors for each year are assumed random and Obviously models with parameter sets 4 and 1 will not therefore uncorrelated, the standard deviation (SD) for the behave in the same way, despite producing similar error cumulative period T can be computed using the root‐sum‐ magnitudes. This emphasizes the fact that without further square method: information on which parameter values are favorable, it is vffiffiffiffiffiffiffiffiffiffiffiffiffiffi u not possible to obtain a unique parameter set and conse- uXT t Á2; 16 quently all Pareto optimal solutions are considered equal. SDT ¼ j ð Þ One way to further reduce parameter uncertainty would be j¼1 to include additional objectives in the calibration process, such as observations of albedo or the rate of snow line where Dj is the standard deviation for year j. The net mass retreat [e.g., Schuler et al., 2007]. balances in Figure 6a are plotted with error margins of [51] As expected, optimal solutions are found that have ±2 SD, but the uncertainty may be larger if there are sys- higher lapse rates during the winter (7.5°C km−1) compared tematic errors in the mass balance measurements. −1 to the summer (3.3–5.5°C km ) owing to the weakening of [54] Overall the calibrated compromise solution performs temperature gradients over melting snow/ice surfaces. These very well at stake locations. The correlation coefficients

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Figure 6. Evaluation of the compromise solution using scatter diagrams of observed versus modeled stake balances (left panels), time series of area‐averaged mass balance (center panels), and area‐averaged cumulative mass balance (right panels). (a) Net mass balance, (b) winter mass balance, and (c) summer mass balance. Note the different scales on the cumulative plots. The observed area‐averaged net mass balances are plotted with estimated error bounds of ±2 SD (see section 5.3 for details). For 1986 the area‐averaged observed mass balance reconstructed in the work of Kohler et al. [2002] is used since no stake observations exist for this year. between observed and modeled net, winter and summer observations, confirming the good performance of the model balances are 0.83, 0.84, and 0.77, respectively (Figure 6). (Figure 6a). The modeled stake errors are larger during the summer [55] The calibrated model performs well with respect to the (MAE = 0.23 m w.e.) compared to the winter (MAE = area‐averaged cumulative mass balance. The model gener- 0.097 m w.e.). The overall MAE of the net mass balance is ally underestimates cumulative errors, with small under‐ 0.25 m w.e. As expected, the correlations between the predictions in the winter and summer balances aggregating to observed and modeled area‐averaged mass balances are larger underestimations in the net balance. At the end of the lower, with coefficients of 0.59, 0.73 and 0.58 for the net, study period, the cumulative errors for winter, summer and winter and summer balances, respectively. However, the net balances are all very small (<0.3 m w.e.). This is not area‐averaged errors are smaller for the net (MAE = 0.21 m surprising given that the model is optimized to minimize the w.e.), winter (MAE = 0.94 m w.e.) and summer (MAE = total error. However, the differences vary over time, reaching 0.19 m w.e.) balances compared to the errors at stake a maximum error of ∼1.4 m w.e. in 1991. On the whole, the locations. This indicates that although the degree of asso- cumulative errors are well behaved, with a mean of 0.50 m ciation between the observed and modeled area‐averaged w.e. and a standard deviation of 0.50 m w.e. As a result, the mass balances is lower than that at the stake locations, this modeled cumulative net mass balance captures the long‐term does not impact on the model performance with respect to trend and is well within the estimated ±2 SD observed error the magnitude of typical errors. In addition, the modeled bounds. area‐averaged net mass balance time series falls within two [56] The MAE of the net mass balance gradient for standard deviations of the estimated error bounds of the the compromise solution is 0.087 m w.e. per 100 m

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0.30, 0.12 and 0.28 m w.e., respectively. The corresponding area‐averaged values for the net, winter and summer are 0.26, 0.12 and 0.25 m w.e., respectively. Overall these values indicate that the model has good predictive power, particu- larly during the winter. The standard deviations of errors of the net balance are not notably larger than the estimated error of ±0.25 m w.e. in the observations, also indicating the good explanatory power of the model. [58] One of the main sources of error in the model is the uncertainty arising from model calibration. The calibration procedure used in this study is designed to select parameter values that minimize errors between the model results and observations. However, uncertainties still arise because the model parameters are optimized to the mean conditions over the study period. When applying a generalized parameter set to all years, larger model errors are to be expected for years when the observations have larger deviations from the mean conditions. The mass balance observations show that the winter balance has remained relatively stable since 1968, while the summer balance has experienced substantial year‐ to‐year variability [Kohler et al., 2002]. As a result, model errors are larger during the summer when the observed variability is also larger. The modeled summer balance would also be expected to perform slightly worse than the net and winter balances given that the parameters are not directly optimized to fit summer melt. Instead, the optimization of the summer balance is implicit in the calibration of the model to fit the net observations. Part of the uncertainty in simulated balances may also arise from uncertainties in the ERA‐40 reanalysis. The overall fit between model results and observations could be improved by calibrating the model on a decade‐by‐decade or year‐by‐year basis. This would enable the model parameters to be optimized to better account for variability in the observations. However, without a generalized parameter set for all years, the predictive power of the model for hindcasting or forecasting mass balances would be substantially reduced. [59] We also test for systematic bias in the model results by constructing empirical cumulative density functions of the observed and modeled mass balances at the stake locations for the net, winter and summer periods (Figure 7). For the net and winter balances, the systematic bias is negligible. In contrast, a clear bias exists in the summer balance, with the slight overestimation of large and underestimation of Figure 7. Empirical cumulative density plots of the small melt values. This is also shown in the scatterplot in observed and modeled mass balances at stake locations for Figure 6c, which has a slope of less than 1. Glacier melt (a) net, (b) winter, and (c) summer balances. (and therefore model bias) is elevation‐dependent, which implies that the systematic bias in the summer balance arises (corresponding to an underestimation of the mean net mass from errors in the mass balance gradient. As discussed balance gradient by 8%). The main source of this error is above, the MAE of the mass balance gradient is compro- the summer balance, with a MAE of 0.083 m w.e. per 100 m mised to satisfy the f1 () and f3 () objective functions, (14% underestimation of the mean summer gradient), while which results in an underestimation of the observed mean the winter gradient is reproduced very well, with a MAE summer gradient by 14%. This may be a result of some of the of only 0.029 (3% underestimation of the mean winter simplifying assumptions in the model structure. The gradient gradient). in summer balance is controlled largely by the summer lapse [57] The confidence that can be placed in model simula- rate and surface albedo. In the model, the albedo of snow tions depends largely on uncertainty remaining after the varies temporally and spatially as the surface snow density model has been calibrated. Random errors owing to un- varies, while the ice albedo is assumed spatially constant and certainties in the input data, validation data and estimated is calibrated to give a glacier‐average compromise value. parameter values can be assessed using the standard devia- However, studies have shown that the surface albedo of a tion of model errors. The standard deviations of model errors glacier tends to be lower at lower elevations, whether the at stake locations for the net, winter and summer balances are surface is snow or ice. If at low elevations the “true” albedo is

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imposed ice formation and exclude internal accumulation. The importance of these two components is assessed by means of two sensitivity experiments. In the first experiment, the model is run without superimposed ice formation and all meltwater reaching the snow/ice interface is instantly converted into runoff. In the second experiment, meltwater is allowed to percolate and refreeze in firn layers beyond the summer surface of the previous year, thus forming internal accumulation. In both experiments the compromise parameter set is used and all other factors are kept constant. The sensitivity is assessed by the percentage change in the area‐ averaged cumulative mass balance over the study period 1968–2001. [62] The model was found to be relatively insensitive to both superimposed ice and internal accumulation. Neglect- ing superimposed ice results in an increase in cumulative – – Figure 8. Observed (1968 2001) and reconstructed (1958 mass loss by 4%, while including internal accumulation 1967) mass balances for Midre Lovénbreen. The dashed makes the cumulative balance more positive by 2%. The ‐ lines indicated ±2 SD error bounds. For 1986 the area small sensitivity of the model to both these components averaged observed mass balance reconstructed in the work arises because the equilibrium line altitude of Midre Lové- of Kohler et al. [2002] is used since no stake observations nbreen is approximately 400 m and therefore the super- exist for this year. imposed ice and firn zones only comprise a small percentage of the total glacier area (Figure 1). Thus while superimposed lower than the model value, and vice versa at high elevations, ice and internal accumulation may be of local importance, this would at least partly explain the larger observed mean they have minimal impact on the area‐averaged mass bal- summer mass balance gradient. Similarly, lapse rates have ance. As a result, any bias resulting from the inclusion or been shown to be lower over melting ice surfaces in the exclusion of superimposed ice and internal accumulation ablation zone than the accumulation zone [e.g., Greuell and will not unduly influence the model results or calibration of Böhm, 1998]. If the “true” lapse rate on the glacier snout is parameter values. smaller than in the mean value used in the model, and vice – versa in the accumulation zone, this also could explain the 5.5. Midre Lovénbreen Mass Balance 1958 2002 larger observed gradient. One way to overcome this would [63] Figure 8 shows the observed (1968–2001) and re- be to introduce more sophisticated, possibly empirically constructed (1958–1967) cumulative mass balances for based, parameterizations of ice albedo and the lapse rate. But Midre Lovénbreen. Estimated error bounds are computed this would increase the dimensionality of the parameter using the root‐sum‐square method (equation 16). For space to be searched and there would be no guarantee that the observed years, the error is given by Dobs = ±0.25m, as optimized parameter set would be any more applicable on described above. For reconstructed years, uncertainty is other glaciers than the simple, single optimized values used calculated as: here. The source of systematic bias could also originate in the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Á Á2 Á2 ; 17 observations. It is difficult to assess if systematic errors ¼ obs þ mod ð Þ actually exist in the stake measurements, but two likely sources of bias are superimposed ice and internal accumu- where Dmod is the model uncertainty (±0.26 m. w.e.), taken lation. The relative importance of these two components is as the standard deviation of errors in the area‐averaged net discussed in section 5.4. annual balance over the period of observations (1968– [60] The overall aims of this study are not to simulate as 2001). The hindcasted time series shows a progressively accurately as possible the mass balance or spatial variations decreasing mass balance since 1958, with an area‐averaged in any given year, but rather to simulate as accurately as cumulative loss (1958–2001) of −17.8 ± 3.7 m w.e. The net ‐ possible the long term net mass balance of the glacier, and balances are consistently negative, with only 3 positive to evaluate the use of reanalysis products for glacier mass balance years over the 44 year period. However, there are no balance modeling. In this respect, the model performs very statistically significant trends in net, winter or summer mass well, explaining a large proportion of the variability in the balance. net mass balance. Overall, the spread of errors is reasonably small, indicating the model has good predictive power for hindcasting or forecasting mass balances, assuming that the calibrated model parameters are valid for different time 6. Conclusions periods. [64] A surface mass balance model of the glacier Midre Lovénbreen, Svalbard, has been driven using a full range of 5.4. Role of Superimposed Ice and Internal meteorological variables from the ERA‐40 reanalysis. The Accumulation model has been calibrated via an automated multiobjective [61] As discussed above, superimposed ice and internal parameter optimization technique and used to reconstruct the accumulation may not be fully accounted for in the ob- mass balance time series of the glacier back to the beginning servations. In the model simulations we include super- of the reanalysis. Our main findings are as follows.

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[65] 1. The ERA‐40 reanalysis corresponds sufficiently Svalbard to investigate previously unmonitored glaciers and well with surface observations to be used for mass balance hindcast existing mass balance records. However, further modeling in the region around Ny‐Ålesund. Temperature, investigation is required to find out if model parameters global radiation and longwave radiation are all captured calibrated for an individual glacier can be transferred to a very well by the reanalysis, although temperature correla- wider region. In addition, any use of the ERA‐40 reanalysis tions weaken during the summer owing to a reduction in would be based on the assumption that the bias correction temperature variability. As expected, the correlation derived for Ny‐Ålesund is applicable to a wider region. between ERA‐40 and observed precipitation is not as Nonetheless, the use of the ERA‐40 reanalysis to drive a strong. This could in part be due to inaccuracies in the rain distributed glacier mass balance model provides large gauge measurements at Ny‐Ålesund. Nonetheless, the cali- potential for investigating the complex regional interactions brated model is still able to reproduce the winter balance between glaciers, climate and topography. very well. ERA‐40 wind speed and relative humidity have the weakest correlations. However, the surface mass balance model is relatively insensitive to these variables and there- [70] Acknowledgments. C.J.R. was funded by UK Natural Environ- mental Research Council studentship grant NER/S/A/2006/14106. N.S.A. fore any disparity with observations will have limited effect and I.C.W. were funded by the University of Cambridge B.B. Roberts Fund on the model results. and the Scandinavian Studies Fund. We thank the Norwegian Meteorolog- [66] 2. Calibration of the surface mass balance model ical Institute and the Alfred Wegener Institute for the provision of meteorological data. The ERA‐40 data were obtained from the ECMWF data found that no single satisfactory solution is possible when server. The lidar DEMs were produced by the UK NERC Airborne optimizing with respect to a single objective that minimizes Research and Survey Facility. Figure 1 is based on digital spatial data error at stake locations. Instead, model calibration has to be licensed from the Natural Environment Research Council (copyright NERC considered as a multiobjective problem, with a set of equally 2003). We would also like to thank Dirk van As and two anonymous reviewers for their considered and constructive comments on the original plausible Pareto optimal solutions. The model was cali- version of the paper. brated using three objective functions that describe key features of the observed data: the MAE of the specific balances at stake locations, the MAE of the mass balance References gradient, and the absolute value of the area‐averaged CE. ‐ Abdalati, W., W. Krabill, E. Frederick, S. Manizade, C. Martin, J. Sonntag, Finding the whole Pareto set is time consuming; therefore R. Swift, R. Thomas, J. Yungel, and R. Koerner (2004), Elevation we derive only the compromise solution, which equally changes of ice caps in the Canadian Arctic Archipelago, J. Geophys. minimizes all three objectives. The multiobjective approach Res., 109, F04007, doi:10.1029/2003JF000045. ‐ Arendt, A. (1999), Approaches to modelling the surface albedo of a high Arc- offers substantial benefits over the single objective calibra- tic glacier, Geogr. Ann., Ser. A, Phys. Geogr., 81, 477–487, doi:10.1111/ tion. Including the MAE of the mass balance gradient and j.0435-3676.1999.00077.x. the area‐averaged CE in the calibration process improves Arendt, A. A., K. A. Echelmeyer, W. D. Harrison, C. S. Lingle, and V. B. the modeled mass balances with respect to these objectives Valentine (2002), Rapid wastage of Alaska glaciers and their contribution to rising sea level, Science, 297, 382–386, doi:10.1126/science.1072497. and has little impact on the MAE at stake locations. Arnold, N. (2005), Investigating the sensitivity of glacier mass‐balance/ [67] 3. Overall the calibrated compromise solution per- elevation profiles to changing meteorological conditions: Model experi- forms very well at stake locations. The correlation coeffi- ments for haut glacier D’Arolla, Valais, Switzerland, Arct. Antarct. Alp. Res., 37, 139–145, doi:10.1657/1523-0430(2005)1037[0139:ITSOGE] cients between observed and modeled net, winter and 1652.1650.CO;1652. summer balances are 0.83, 0.84, and 0.77, respectively. The Arnold, N. S., and W. G. Rees (2003), Self‐similarity in glacier surface char- correlations between observed and modeled area‐averaged acteristics, J. Glaciol., 49, 547–554, doi:10.3189/172756503781830368. Arnold, N. S., I. C. Willis, M. J. Sharp, K. S. Richards, and W. J. Lawson balances are lower, with coefficients of 0.59, 0.73 and 0.58 (1996), A distributed surface energy‐balance model for a small valley for the net, winter and summer balances, respectively. glacier: 1. Development and testing for Haut Glacier d’Arolla, Valais However, this does not impact on the magnitude of typical Switzerland, J. Glaciol., 42,77–89. errors and the model is able to reproduce the area‐averaged Arnold, N. S., W. G. Rees, A. J. Hodson, and J. Kohler (2006), Topo- graphic controls on the surface energy balance of a high Arctic valley cumulative mass loss very well. One of the main sources of glacier, J. Geophys. Res., 111, F02011, doi:10.1029/2005JF000426. error in the model is the summer mass balance gradient, Bamber, J. L., W. Krabill, V. Raper, J. A. Dowdeswell, and J. Oerlemans which is underestimated on average by 14%. This may be (2005), Elevation changes measured on Svalbard glaciers and ice caps from airborne laser data, Ann. Glaciol., 42, 202–208, doi:10.3189/ due to simplifying assumptions in the model structure that 172756405781813131. relate to the way the model describes the surface albedo and Barry, R. G. (2006), The status of research on glaciers and global glacier lapse rate. On the whole, the predictive power of the model recession: A review, Prog. Phys. Geogr., 30, 285–306, doi:10.1191/ is very good. The standard deviations of stake errors for the 0309133306pp0309133478ra. Bassford, R. P. (2002), Geophysical and numerical modelling investiga- net, winter and summer are 0.30, 0.12 and 0.28 m w.e., tions of the ice caps on Severnaya Zemlya, Ph.D. thesis, Univ. of Bristol, respectively, while for the area‐averaged balances they are Bristol, U. K. 0.26, 0.12 and 0.25 m w.e., respectively. Björnsson, H., Y. Gjessing, S. E. Hamran, J. O. Hagen, O. Liestøl, F. Palsson, and B. Erlingsson (1996), The thermal regime of sub‐polar glaciers [68] 4. From the reconstructed time series, a mass loss of mapped by multi‐frequency radio‐echo sounding, J. Glaciol., 42,23–32. −17.8 ± 3.7 m w.e. is estimated over the period 1958–2001. Boé, J., L. Terray, F. Habets, and E. Martin (2007), Statistical and dynam- The annual net mass balances are consistently negative over ical downscaling of the Seine basin climate for hydro‐meteorological studies, Int. J. Climatol., 27, 1643–1655, doi:10.1002/joc.1602. the 44 year period, but there are no statistically significant Braithwaite, R. J., and Y. 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