We separate the total ice-shelf thickness into its constituent thicknesses of solid ice and firn air by following the method of Holland et al. (2011) with a few modifications. Since the floatation of an ice shelf and the propagation of a radar wave through an ice shelf both depend upon the relative proportions of ice and air, we formulate two corresponding equations from which two unknown quantities, ice and air thickness, are derived. The presence of a third unknown, liquid meltwater, is neglected on the basis that most surveys were undertaken early in the austral spring and there is no evidence of a perennial aquifer in LCIS (see Sect. 4.3).

If the ice is freely floating then the hydrostatic ice and ocean forces must balance at the ice base, so the total mass of the shelf ice and firn air equals that of the atmosphere and ocean displaced ρiI+ρaA=ρAS+ρo(I+A-S). Here I is the total solid ice thickness, A is the total firn-air thickness, S is the ice-shelf freeboard (surface elevation above sea level), and ρi = 918 kg m-3, ρa = 2 kg m-3, ρA = 1.3 kg m-3, and ρo = 1028 kg m-3 are densities of solid ice, englacial air (partly pressurised), atmospheric air, and ocean respectively, which are all assumed constant. Adopting a similar approach and separating the radar delay of ice from that of air using the simple, empirical complex refractive index method (e.g. Arcone, 2002), the TWTT of a radar wave through the ice shelf is T=2cniI+naA, where T is the TWTT, c = 3 × 108 m s-1 is the speed of light in vacuo, and ni = 1.78 and na = 1.0 are refractive indices of pure ice and air. Combining Eqs. (1) and (2) and eliminating variables as appropriate, we obtain expressions for the constituent ice and air thicknesses (and hence total thickness, I + A) as functions of known quantities and the measured TWTT and surface elevation: A=cρo-ρi2niT+ρA-ρoS/ρa-ρo+naρo-ρini I=cρo-ρa2naT+ρA-ρoS/ρi-ρo+niρo-ρana. Taking the temporal derivative of these expressions, we obtain the trends in ice and air thickness as a function of the trends in elevation and TWTT: ∂A∂t=cρo-ρi2ni∂T∂t+ρA-ρo∂S∂t/ρa-ρo+naρo-ρini ∂I∂t=cρo-ρa2na∂T∂t+ρA-ρo∂S∂t/ρi-ρo+niρo-ρana. Hence, we calculate ice and air trends directly from elevation and TWTT trends; we do not derive the ice and air thickness for each survey and then calculate their trends. This explicitly excludes potentially large errors inherent in steady corrections to the input data, particularly from the geoid and mean dynamic ocean topography. Evaluating the known quantities in Eqs. (5)–(6), we find that ∂A∂t=1.06∂S∂t-0.114c2ni∂T∂t∂I∂t=-0.598∂S∂t+1.06c2ni∂T∂t, where the TWTT is expressed as a solid ice equivalent for clarity.

Note that the derivation of Eqs. (5)–(6) from Eqs. (3)–(4) neglects temporal derivatives of all densities, of which the most variable is the ocean density. Repeating the derivation and retaining ocean density terms provides an expression in which 0.3 m yr-1 ice loss would require a ∼ 2 kg m-3 yr-1 reduction in ocean density, and 0.03 m yr-1 air loss would require a ∼ 0.1 kg m-3 yr-1 increase in ocean density. Such changes persisting over 15 years are clearly implausible, and we conclude that ocean density changes have negligible effect.

We apply the above method to eight radar surveys between February 1998 and December 2012 along a line traversing the centre of LCIS (Fig. 1b, red line). The surveys were carried out by ground-based field parties and a variety of aircraft flying at different heights and speeds, and many different radar instruments and methods for measuring elevation were used (Table 1). The processed elevation and TWTT data are shown in Fig. 2a and b. The most densely spaced TWTT data were gathered during the 2004 NASA-CECS airborne survey, so this is chosen as a baseline data set. For each elevation and TWTT measurement in the other surveys, we find the difference from the nearest corresponding measurement in the 2004 survey, discarding all observations that do not have a 2004 analogue within 1000 m. These elevation and TWTT differences are shown in Fig. 2c and d. There is a great deal of scatter in the differences, which could result from several factors, including the advection of ice topography across the survey line at ∼ 400 m yr-1 (Rignot et al., 2011). The differences are therefore binned spatially to extract the overall signals by averaging random noise, and linear trends in surface elevation and TWTT are calculated for the bins. Equations (5) and (6) are then used to determine the trends in ice and firn-air thickness from trends in surface elevation and TWTT. We apply this methodology in two ways: first considering the overall trends for the entire survey line and then dividing the survey into five bins, surrounding each of the five satellite crossover points (Fig. 1).

The 2012 British Antarctic Survey (BAS) ground-based survey was a mission of opportunity within a wider seismic season (Brisbourne et al., 2014) and deviated from the rest of the surveys, heading due south (Fig. 1b, yellow line). However, it did repeat a flight line from the 1998 BAS airborne survey, so to include the data we first calculate the mean difference between the 2012 and 1998 surveys along the meridional line, and then the mean difference between the 2004 and 1998 surveys along the primary line, then use these to obtain the 2012–2004 difference. The results are only included in the northernmost bin when we consider along-survey variability; they are not included in the whole-survey results.

Different techniques are available for picking radar return echoes from echograms, and so to ensure that our inter-survey trends are as robust as possible the ice surface and base echoes from all surveys were re-picked in a consistent manner. Automatic first-break picks on time-windowed and scaled traces were manually edited to remove or correct mispicks. For airborne surveys, TWTT was calculated as the difference between ice surface and basal returns from the ice-penetrating radar, thus minimising inter-survey biases by removing any error associated with the absolute accuracy of the radar. Basal return TWTTs from the ground-based survey data were corrected for the radar antenna separation. In the NASA IceBridge 2009 and 2010 and BAS 2011 airborne surveys, the altitude of the aircraft in specific sections caused the surface multiple return to appear at a TWTT similar to that of the basal return, significantly contaminating the picks. Therefore, the radargrams were overlain with an estimate of the surface multiple return calculated from the aircraft altitude and also an estimate of the basal return derived from the aircraft altitude, surface elevation, and hydrostatic assumption. Wherever the TWTT of these two signals was indistinguishable in the radargram, no basal return pick was recorded. Significant marine ice bands were omitted from all surveys, because basal returns become indistinct and the meteoric–marine transition may be visible instead.

The TWTT data are recorded with a wide variety of instruments and subject to different processing techniques to optimise the signal prior to picking. The TWTT precision in the echogram picked (i.e. the time between samples of return power: the reciprocal of the sampling rate) varies between surveys, with a mean of ∼ 4 m ice equivalent and a range of 0.13–8.8 m ice equivalent. The 15-year TWTT change is of comparable magnitude to this precision (see below). However, the first break of the return echo is actually known at higher precision because waveform fitting is used to interpolate between samples of the return echo power. Furthermore, each inter-survey TWTT difference, from which ice and air trends are calculated, is actually the mean of a population of thousands of individual point differences. These populations are well resolved by the TWTT precision, and so by using large numbers of data points we are able to detect mean inter-survey differences statistically at a precision much finer than that of the individual data.

TWTTs from the 2009 IceBridge survey were found to contain consistently shorter radar-wave delays than the 2009 McGrath ground-based survey despite being collected only 2 weeks earlier, with a mean equivalent ice thickness approximately 10 m lower and therefore a significant outlier relative to the other surveys. The data were investigated and re-picked, but the problem seems to result from transmit/receive switches not meeting their switching-time specification in the survey (https://data.cresis.ku.edu/#RDS). Therefore, the 2009 IceBridge TWTT data are neglected throughout this study, other than in a test recalculation to demonstrate their effect. The laser altimeter elevation data from this survey are used in all calculations.

Surveyed ice elevation data have several corrections applied to make them directly comparable. Corrections for the steady geoid and mean dynamic ocean topography are not required because the method employs only temporal differences in elevation, as shown by Eqs. (5) and (6). All data are de-tided using the CATS2008a_opt model (L. Padman, personal communication, 2014) and have a local sea-level rise of 4 mm yr-1 removed (Rye et al., 2014).

Most of the instruments used to derive elevation were well calibrated in the field (e.g. http://nsidc.org/data/docs/daac/icebridge/ilatm2/index.html), but the two BAS airborne surveys in 1998 and 2011 were not calibrated to the centimetre-scale accuracy required here. The 1998 survey passed over the open ocean in many locations, so these elevations were corrected for tides, EIGEN-6C geoid (http://icgem.gfz-potsdam.de/ICGEM/ICGEM.html), and DTU12 mean dynamic topography (http://www.space.dtu.dk/english/Research/Scientific_data and models/downloaddata), and then the mean difference from 0 (sea surface) of 1.01 m was removed from the entire data set. Repeating this procedure for the 2011 survey produced a 1.33 m offset but from only a small area of open-ocean data. Fortunately, it was possible to correct the 2011 elevations to match the well-calibrated 2010 NASA IceBridge laser altimeter survey that took place 10 weeks earlier. However, this was complicated by the issue of radar firn penetration. Radar altimetry penetrates the surface and reflects from within the firn layer, whereas laser altimetry reflects from the surface. North of 67.85∘ S there is no broad-scale spatial variation in the offset between data sets (Fig. 3), implying either uniform or no radar penetration. We regard the mean offset in this area, 1.59 m, as the calibration error and subtract it from the 2011 data everywhere.

The elevation estimates derived from the two BAS radar altimeter surveys need a firn-penetration correction to make them comparable to those derived from the laser altimeters and GPS. After the above calibration, the 2011 radar altimeter survey records a progressively lower surface than the 2010 laser altimeter survey to the south of 67.85∘ S, which we ascribe to firn penetration. This is qualitatively consistent with the known southward increase in firn-air content (Holland et al., 2011). Therefore, we correct both the 1998 and 2011 radar altimeter surveys by adding a linear fit south of 67.85∘ S to the difference between the IceBridge 2010 and BAS 2011 surveys (Fig. 3). Out of necessity, the correction includes implicit assumptions that there is no firn penetration north of this during either radar survey and that penetration to the south is identical in February 1998 and January 2011.

Satellite radar altimeter data are used to corroborate the surveyed elevation data and provide a context for the lowering. The satellite elevation time series combine radar altimeter data from the ERS-1, ERS-2, and Envisat satellites using an existing methodology (Fricker and Padman, 2012) but including new data to the end of 2011. These data consist of repeat measurements of ice-shelf surface elevation at satellite orbit crossing points, available approximately every 35 days during austral winters (April–November) during 1992–2011.

When analysing the data we found a strong correlation between changes in elevation and changes in surface backscatter for the period 1992–1993 (the first 2 years of ERS-1). This anomalous behaviour in the altimeter backscatter, which alters the shape of the waveform from which the elevation is deduced, occurs throughout Antarctica. This leads us to believe that these data may not be reliable, so we only use data from 1994 onwards in this study. Shepherd et al. (2010) and Paolo et al. (2015) also neglected data prior to 1994 in their analyses. This is important because studies of LCIS that include these early data (Shepherd et al., 2003; Fricker and Padman, 2012) derive very rapid lowering in the 1990s that is not found if the early data are neglected. To illustrate the lowering of LCIS we first consider the period 1994–2011 (Fig. 1a), though our main analysis focuses upon the 1998–2011 period covered by the radar surveys (Fig. 1b). During the latter period the LCIS lowering has the same general pattern, but the trends at the five crossovers covered by the survey line are slightly different. Importantly, the survey line does not sample the northern section of LCIS in which the fastest lowering occurs.

To compare elevation trends derived from the survey data to those derived from satellite radar altimeter data (Fig. 4 and Table 2) a single satellite elevation trend was derived that represents all five independent satellite crossovers. First, the mean elevation for the austral winter of 1998 was calculated for each independent crossover and subtracted from each crossover's time series. The resulting temporal elevation anomaly data were then treated as individual data points in a single merged time series, and from that a linear trend was calculated to compare to the surveyed trends. Linear trends were also calculated at each crossover, as presented in Figs. 1, 5, 6, and 7.

We consider the derived ice and air losses in the context of the ice and air mass balances of LCIS. The mass balance of the ice fraction of the ice shelf (i.e. excluding firn air) yields an equation governing the depth-integrated ice thickness ∂I∂t+I∇⚫u+u⚫∇I=aI-mI, where u is the two-dimensional horizontal ice velocity vector, aI is net surface ice accumulation, and mI is basal melting. The mass balance of the air fraction of the ice shelf yields a similar equation for depth-integrated air thickness ∂A∂t+A∇⚫u+u⚫∇A=aA-mA-d, where aA is the air trapped in the firn by accumulation, mA is the loss of air by surface melting, percolation, and refreezing, and d is the loss of air by dry compaction. The terms on the left-hand side of both equations are the unsteady term, divergence, and advection.

When analysing the results we map the terms in the ice mass balance Eq. (9) following a previous study (McGrath et al., 2014) that combined data from several sources. Divergence, advection, and mass input terms can be mapped from satellite-based observations of ice velocity (Rignot et al., 2011) and ice-shelf elevation (Griggs and Bamber, 2009), firn-air thickness derived from airborne radar measurements (Holland et al., 2011), and model estimates of net surface accumulation (Lenaerts et al., 2012). Though we also possess a spatial map of ice surface elevation change, an unknown fraction of this is caused by firn-air changes and so we cannot derive ice thickness change outside the temporal and spatial range of our survey data. Neglecting the unsteady term, we can derive a map of steady-state melting from the other terms. Prior to these calculations the ice thickness and velocity fields are smoothed over a 20 km footprint (masked outside the ice shelf) to remove small-scale noise that is amplified in the spatial derivatives. The firn-air mass balance Eq. (10) contains so many unknown quantities that we do not attempt to derive its terms.

Figure 4 shows the elevation and TWTT for each survey as mean differences from 2004 over the entire survey line, and Table 2 gives the “primary” derived trends for these “reference” data and also a variety of alternatives. Since the data points and their error bars refer to differences from 2004, the 2004 data are 0 for both elevation and TWTT, with 0 error. The surveyed elevation differences show a lowering trend (-0.066 ± 0.017 m yr-1) that is very similar to that obtained from the satellite altimeter data (-0.062 m yr-1); the trends are not expected to be identical due to method uncertainties and spatial and temporal differences in sampling. Crucially, there is also a decreasing trend in surveyed TWTT (-0.296 ± 0.17 m yr-1 ice equivalent), though there is considerably more inter-survey scatter in this quantity and uncertainty in the resulting trend (see Sect. 4.3). Combining these observed trends using Eqs. (5) and (6) reveals that the surface lowering is caused by a combination of air loss (-0.0367 ± 0.026 m yr-1) and ice loss (-0.274 ± 0.18 m yr-1). Ice loss is an order of magnitude larger than air loss, but surface lowering is approximately 10 times more sensitive to air loss than ice loss, so ice and air loss contribute approximately equally to the surface lowering. There is considerable scatter in the data and several sources of uncertainty in the methodology, but our conclusion that ice and air loss both contribute to the lowering is robust when several different combinations of data are used in the calculations (see Sect. 4).

We now consider spatial variability by binning the survey data around each satellite crossover (Fig. 5a). The derived ice loss is reasonably uniform along the line, while the derived air loss is noticeably higher towards the southern end of the survey line. However, the surveyed elevation trends at the southern end of the line show considerably more lowering than the satellite elevation trends. Inspection of the data underlying the time series in each bin (Fig. 6) reveals that the surveyed elevations are reasonable apart from the 1998 data in the southernmost bin (centred on 68.3 ∘S), which exceed the range of the figure. We consider the satellite altimeter data to be a more reliable measure of lowering because the 1998 surveyed elevation data are subject to calibration and firn-penetration corrections that are uncertain in this area (see Sect. 2.4). The TWTT data are not subject to these uncertain corrections, so we retain these and recalculate the ice and air trends with the surveyed elevation trends replaced by the satellite elevation trends (Fig. 7a). This has virtually no effect on the derived ice loss but removes the air loss completely from the southernmost bin, so that the air loss is concentrated on the centre of the survey line.

The air and ice losses shown in Figs. 5a and 7a are scaled so that their resultant surface lowering can be read on the left-hand axis. Figure 7a suggests that air loss contributes the majority of the lowering in the centre of the survey line, while ice loss also contributes to this lowering and is responsible for the lowering at both ends. It is unsurprising that the ice and air loss have different spatial patterns given their different (oceanic, ice-dynamic, and atmospheric) forcings.

Figure 8 shows the maps of each term in the LCIS ice mass balance Eq. (9). Thinning along flow lines causes a sink of ice through divergence (Fig. 8a), advection is generally a source of ice where the ice shelf flows from thick to thin (Fig. 8b), and modelled surface accumulation is almost uniform (Fig. 8c). Their sum, the steady melting map (Fig. 8d), contains obvious artefacts but also many features that match our existing knowledge of ocean melting beneath LCIS. For example, the results are in agreement with a simple ocean-layer model (Holland et al., 2009) that predicts strong melting along the grounding line and freezing in the thinner ice immediately offshore of islands and peninsulas on the western coast (also visible as negative values in the advection term). A more sophisticated three-dimensional ocean model (Mueller et al., 2012), forced only by tides, predicts large values of tidally driven melting next to Bawden Ice Rise and Kenyon Peninsula, which also seem apparent in Fig. 8d, though other areas of high melting near the ice front and south of Kenyon Peninsula are not consistent with the model.

Combining the estimated mean terms in the ice mass budget (Fig. 8) with the ice loss derived along the survey line (Figs. 5a and 7a) allows us to consider the full unsteady ice budget (Figs. 5b and 7b). The basic ice balance is between accumulation and divergence, with advection becoming important at the southern end of the line. If the ice shelf were in steady state the derived oceanic melt rate would be an order of magnitude smaller than accumulation and divergence (0.06 m yr-1). In fact, our derived ice loss profiles suggest a mean oceanic melt rate over the survey line of 0.26 m yr-1, peaking at 0.5 m yr-1 in the southernmost bin. These estimates are consistent with modelled patterns of melting (Holland et al., 2009; Mueller et al., 2012) and observations in a higher-melting region nearby (K. W. Nicholls, personal communication, 2014; Nicholls et al., 2012). Crucially, without basal melting the components of the mass budget are approximately balanced, so the majority of the melting is causing net ice loss. This emphasises that for ice shelves melted by cold ocean waters, relatively small absolute changes in melting can have a significant influence on the ice-shelf mass balance. In comparison, warm-water ice shelves such as Pine Island Glacier can have much larger melting perturbations (e.g. 5 m yr-1; Wingham et al., 2009), causing correspondingly large thinning rates, but these perturbations are a much smaller fraction of the mean melt rate (e.g. 100 m yr-1; Dutrieux et al., 2013).

The terms in the analogous firn-air budget are extremely uncertain. To put the derived air loss of 0.04 m yr-1 into context, we simply note that there was 10–15 m of air in the surveyed section during the 1997/1998 survey (Holland et al., 2011), and if fresh snow is deposited at a density of 350–450 kg m-3 (Kuipers Munneke et al., 2012b) then the accumulation of 0.5 m yr-1 ice implies the addition of 0.5–1 m yr-1 firn air each year before compaction is taken into account. Therefore, our best estimate is that the net air loss is only 5–10 % of the annual air input.

It is possible to visually assess the reliability of ice and air trends from appropriately plotted trends in elevation and TWTT. If the TWTT trend is expressed as a solid-ice surface-elevation equivalent, i.e. ∂Ts∂t=c2niρo-ρiρo-ρA∂T∂t, then comparing ∂Ts/∂t to the elevation trend ∂S/∂t allows us to determine the value of ∂A/∂t from Eq. (5). Any elevation trend that is more negative than ∂Ts/∂t implies a loss of air, with the air loss equal to 1.06 times the difference between ∂S/∂t and ∂Ts∂t. For this purpose, the two y axes of Figs. 4, 5a, 6, and 7a are scaled such that the left-hand axis shows both ice surface elevation (∂S/∂t) and TWTT expressed as solid-ice surface equivalent (∂Ts/∂t). Consideration of the numerator of Eq. (6) shows that ∂Ts/∂t merely has to be more negative than -0.107 × ∂S/∂t to imply a loss of ice; any ∂Ts/∂t that is negative enough to be distinguished in the figures implies some ice loss. In plain terms, Fig. 4 is scaled such that if the red line (scaled TWTT trend) is parallel to the green line (elevation trend) then the lowering is due solely to ice loss, and if the red line is flat then all of the lowering is due to air loss.

These criteria allow a simple visual assessment of the signal present in the available data. Our assessment of Fig. 4 is that the scaled TWTT is decreasing but this result is not robust; that is, it is dependent upon all data sets and removing certain surveys would remove the calculated trend. This reduces confidence in the conclusion that ice loss has occurred. However, we do not believe that the scaled TWTT data could support a trend that is more negative than the elevation trend, and therefore we are confident in our conclusion that air loss has occurred.

A formal analysis revisits these conclusions below, but this requires many assumptions about the nature of the errors and so is not necessarily superior. There are many sources of error in our surveys, which we divide into two classes. The first class of errors produces random intra-survey scatter, which affects the extent to which the data from each survey estimate the mean signal within that survey. The second class of errors creates a systematic signal across a whole survey, directly affecting inter-survey differences. The latter are of greatest concern because they have the largest effect on trends.

Predominantly intra-survey errors include the following:

instrument and processing error (including radar picking error, assumed intra-survey because all surveys were re-picked consistently)

We can easily quantify these random errors by considering, for each survey, the statistics of each population of differences of data points from their 2004 analogues (Table 3). Standard deviations are relatively large, 1–2 m for elevation and ∼ 10 m ice equivalent for TWTT, as expected from previous analyses of the error in individual point measurements (Holland et al., 2009). However, when all data are considered the standard errors are small due to the large sample sizes. Assuming that the differences are independent and normally distributed, 95 % confidence interval bounds for the survey mean are given by multiplying the standard error by 1.96, as shown by the error bars in Fig. 4. We estimate overall 95 % confidence interval bounds as ±0.04 m for elevation and ±0.5 m ice equivalent for TWTT. Thus, from a random error perspective, we are confident that all surveys differ significantly from 2004 apart from the elevation differences in the 2011 and McGrath 2009 surveys and both elevation and TWTT data sets in 2012. Simple examination of the error bars in Fig. 4 shows that variation within these random error bounds will have negligible effect on the computed trends.

Predominantly inter-survey errors include the following:

differences among survey instruments, calibration, and processing (radar altimeter penetration, ice-penetrating radar power and frequency, speed and altitude of acquisition platform)

An initial concern is that the NASA IceBridge and NASA-CECS surveys (high altitude, high speed, consistent radar systems, laser altimeter) differ from the BAS airborne surveys (lower altitude, slower, different radar, radar altimeter) and both differ from the ground-based surveys (low-frequency radar, GPS elevation). However, the three types of survey are interleaved in time, so such differences do not necessarily cause systematic trends. The issue is assessed by re-calculating the trends using different combinations of data (Table 2). Considering only the two BAS surveys produces broadly the same results. However, considering only NASA IceBridge and NASA-CECS surveys produces a much weaker surface lowering and no decrease in TWTT, so that the ice loss disappears. Systematically removing the surveys from the calculation reveals that it is neglecting the BAS 1998 survey that removes these trends. We know of no reason to neglect this survey, but this suggests that we treat TWTT and ice trends with additional caution.

The presence of meltwater in the firn would require us to adapt the methodology because it affects both the hydrostatic floatation and radar-wave delay of the ice shelf, as described by Holland et al. (2011), leading to different ice and air thicknesses being derived from the same TWTT and elevation. This potentially confounding issue is neglected because most surveys were sampled in November, before the onset of melt (Barrand et al., 2013), and instrumented boreholes have revealed no evidence of a perennial aquifer (K. W. Nicholls and B. Hubbard, personal communications, 2015). However, the two BAS surveys were sampled in summer and could be contaminated by the presence of meltwater. Repeating the derivation of Eqs. (3) and (4) but including the effects of meltwater produces new equations from which 0.57 m more air and 5.6 m less ice would be derived for every 1 m of meltwater present (Holland et al., 2011). A maximum LCIS meltwater content of 0.4 m (Holland et al., 2011) therefore implies a maximum underestimate of 0.23 m air and overestimate of 2.24 m ice. The summer of 1997/1998 was a high melting year (Tedesco, 2009), and if meltwater was present during the 1998 survey the derived air content should be higher and ice content lower, enhancing the air loss trend and reducing the ice loss trend. A linear regression to 0.23 m air error and -2.24 m ice error in 1998 and no meltwater-derived error in the other surveys yields maximum trend errors of -0.0137 m yr-1 air and +0.134 m yr-1 ice. Melt estimates for 2010/2011 are not available, but any 2011 meltwater would have the opposite effect on the inter-survey trends to 1998 meltwater and thus mitigate this issue.

For the airborne surveys, surface penetration could affect both radar altimeters and the surface pick of ice-penetrating radars. We have used a penetration correction in radar altimeter data (see above), and their agreement with the satellite elevation trend implies that deviation from this correction is not important. Our strategy of finding the ice TWTT by picking the surface and basal returns and differencing the result means that surface penetration could affect the TWTT. We examine this by comparing the radar surface picks with altimeter data. This test is imperfect because it introduces error from the aircraft altitude and surface elevation data and requires absolute accuracy in the radar data that is not needed of the TWTT differences used. The test cannot even be performed for the NASA IceBridge and NASA-CECS surveys because the absolute timing of the radar pulse transmission is not known to the required accuracy. The mean difference between altimeter-derived surface elevations and radar-derived surface elevations is 2.14 m for the BAS 1998 survey and 2.38 m for the BAS 2011 survey. The altimeter-derived elevation is higher than the radar-derived elevation in both cases, so the difference may be caused by surface penetration. This very limited data set suggests that radar firn penetration is of the order of 2 m, with an interannual variability of order 0.2 m.

These differences between radar surface picks and altimeter data are also the only independent information we have to quantify overall inter-survey error in TWTT differences. They are again imperfect in this role because they include errors in aircraft altitude and surface elevation data that does not appear in the TWTT differences used in Eqs. (5) and (6). Also, if the error in basal and surface picks is identical (e.g. from an absolute calibration error) then there is no error in their difference. However, if the surface and basal errors are uncorrelated and of the same magnitude then the TWTT difference error is the surface pick error multiplied by 2. We believe that an inter-survey error of 2 m ice equivalent for TWTT is a reasonable compromise, and this value is in good agreement with the deviation of the TWTT points from the trend line in Fig. 4.

The effects of unsteady dynamic ocean topography, error in the tidal correction, and inverse barometer effect should each contribute an inter-survey error of the order of 0.1 m to the surface elevation differences (L. Padman, personal communication, 2014; Padman et al., 2003; King and Padman, 2005). If these errors are uncorrelated, this would create a total error of about 0.2 m, and this estimate is consistent with both the deviation of the surveys from the linear trend and the difference in elevation between the two 2009 surveys (Fig. 4). In any case, the surface lowering from the satellite crossovers provides an independent test of the surveyed elevation trend, and the two trends are only slightly different (Table 2) as might be expected from the difference in spatial and temporal sampling.

Given these overall inter-survey error estimates (0.2 m elevation and 2 m TWTT ice equivalent), we used a Monte Carlo approach to estimate the resultant uncertainty in the elevation and TWTT trends. The trends were recalculated 500 000 times with all data points subject to a perturbation drawn from a normal distribution with 95 % confidence interval bounds equal to the error estimates. This yields a population of trends with 95 % confidence interval bounds of ±0.017 m yr-1 for elevation trends and ±0.17 m yr-1 ice equivalent for TWTT trends. Evaluating the terms as in Eqs. (7) and (8) and combining the errors in quadrature yields εAt=0.013εTt2+1.13εSt2,εIt=1.13εTt2+0.36εSt2, where εAt, εIt, and εSt are errors in ∂A/∂t, ∂I/∂t, and ∂S/∂t respectively. The symbol εTt represents the error in c/2ni∂T/∂t, TWTT converted to solid ice thickness. These formulae yield uncertainties of ±0.026 m yr-1 for ∂A/∂t and ±0.18 m yr-1 for ∂I/∂t.

In summary, formal error estimates suggest that both the ice and air loss derived in our reference calculation are robust. However, visual assessment of Fig. 4 suggests that the data support air loss more strongly than ice loss. Recalculating the trends with different combinations of the data (Table 2) shows that almost all possible calculations have significant air loss; the only way to obtain insignificant air loss is to include 2009 IceBridge TWTT data known to be erroneous. However, removing either the BAS 1998 or McGrath 2009 surveys is sufficient to render the ice loss insignificant. Any meltwater that was present during the BAS 1998 survey would further strengthen the air loss and weaken the ice loss. Our best estimate is that the lowering is a result of both air loss and ice loss, but there remains a possibility that air loss is solely responsible.

The preceding calculations apply to the whole-survey comparisons shown in Fig. 4. The latitude bins shown in Figs. 5–7 contain fewer data, so the intra-survey standard error should increase. Standard errors scale with the reciprocal square root of the number of data points, so the 95 % confidence interval bounds approximately double (±0.08 m for elevation and ±1 m ice equivalent for TWTT) when the data sample size is reduced by a factor of 5. Inter-survey systematic error should in principle remain similar, but on the shorter length scale of an individual bin, several intra-survey errors become inter-survey in character (differences in radar picking, survey path, and advection of ice features, which can be a significant fraction of a bin length in the along-survey direction). Scrutinising the time series in Fig. 6 suggests a reasonable confidence in the binned trends. In most cases a downward trend of the TWTT is apparent, suggesting some ice loss has occurred, and the scaled TWTT data would not support a downwards trend steeper than the satellite elevation, suggesting air loss has occurred. The steepest elevation trends and shallowest TWTT trends are in the centre of the survey line, implying greatest air loss.

The existence of mean rates of change in ice and air over our 15-year period implies an imbalance in the other terms of Eqs. (9) and (10) during this time. We consider the ability of each of these terms to cause the imbalance and therefore the ice and air losses. Whether the budget was ever balanced in the past, with the observed imbalance then implying that changes have occurred, is a separate question that we cannot answer.

We start with sources and sinks. Above-balance basal melting will cause ice loss but not air loss and can easily account for our ice loss signal. Any melting greater than a few centimetres per year can cause an imbalance (Fig. 7), and observations and models easily support the rates of ∼ 0.26 m yr-1 needed to explain the ice loss (Holland et al., 2009; Mueller et al., 2012; Nicholls et al., 2012). Above-balance surface melting and refreezing or dry compaction will cause only air loss, and it is again easy for these processes to account for the air loss signal observed here. Below-balance surface accumulation will cause air and ice loss at a ratio of 2 : 1–1 : 1 if snow is initially deposited at a density of 350–450 kg m-3 (Kuipers Munneke et al., 2012b) and compensating compaction changes are ignored. Below-balance accumulation of approximately half of the modelled value (Fig. 7b) would be required to solely explain our ice loss; the fact that our ice loss is an order of magnitude larger than the air loss suggests that below-balance accumulation alone cannot account for both. A small below-balance accumulation could, however, explain the air loss. Since the total input of air into the firn is 0.5–1 m yr-1, relatively small anomalies in surface melting, dry compaction, or accumulation are required to yield the observed 0.04 m yr-1 air loss.

We now turn to dynamic mechanisms. Above-balance ice flow advection will affect air and ice thicknesses in proportion to their relative gradients along flow. According to the results of Holland et al. (2011), increased advection would enhance the flow of thicker ice with less firn air across the survey line. The air thickness increases along flow by approximately 1 m for every 10 m decrease in along-flow ice thickness. Above-balance advection would therefore cause air loss but accompanied by ice gain approximately 10 times faster, which entirely contradicts our observed signals. Above-balance ice flow divergence will cause air and ice losses in proportion to their relative thicknesses, approximately 1 : 30 for characteristic ice and air thicknesses of 10 and 300 m. The largest velocity change in the literature is an acceleration of 80 m yr-1 between 2000 and 2006 surveys of northern LCIS (Haug et al., 2010; Khazendar et al., 2011). If this acceleration caused unbalanced divergence over a length scale of 100 km, it would cause ice loss of ∼ 0.24 m yr-1 and air loss of ∼ 0.008 m yr-1. Above-balance divergence could explain the ice loss, but not the air loss, if maintained at this level and not accompanied by above-balance advection.

In summary, the ice loss we observe could be explained by above-balance basal melting and/or ice divergence, and the air loss could be explained by below-balance accumulation and/or above-balance surface melting and/or compaction. Our results therefore suggest that at least two different forcings caused the lowering of LCIS during our survey period. Elsewhere around Antarctica, rapid ice-shelf thinning is thought to be driven by unbalanced ocean melting (e.g. Shepherd et al., 2004; Holland et al., 2010; Padman et al., 2012; Khazendar et al., 2013), and our robust evidence of a firn-air loss from LCIS in response to surface processes is the first direct evidence of an exception to this. The existence of at least two different mechanisms underlying the change is also consistent with our observation that the ice and air loss signals have different spatial variation along the survey line.

The surveys do not encompass all of the known ice-shelf lowering (Fig. 1), and it is likely that the balance of ice and air losses, and their driving mechanisms, varies in different regions and periods. In particular, our surveys do not capture the rapid lowering in northern LCIS. Ice divergence may play a part in this, since the known acceleration of LCIS is northward-intensified (Haug et al., 2010; Khazendar et al., 2011), but there are also good reasons to expect changes in surface melting to be largest in the north (Holland et al., 2011; Trusel et al., 2013; Luckman et al., 2014). The pattern of changes in basal melting is unknown.

Our results have important implications for the future stability of LCIS and thus the AP Ice Sheet. Previous ice-shelf collapses are thought to have been accomplished by surface meltwater-driven crevassing (van der Veen, 1998; Scambos et al., 2003; van den Broeke, 2005; Banwell et al., 2013) and ice-front retreat past a “compressive arch” in strain rates (Doake et al., 1998; Kulessa et al., 2014). We conceive several interconnected mechanisms by which LCIS stability could be compromised: (1) ice-front retreats past a compressive arch; (2) increased surface melting causes firn depletion and meltwater-driven crevassing; (3) decreased ocean freezing or increased melting depletes marine ice, permitting the propagation of crevasses; (4) collapse of the remnant LBIS opens a new ice front at the northern margin of LCIS; (5) ungrounding from Bawden Ice Rise removes an ice-front pinning point; (6) ice thinning and acceleration enhances the propagation of crevasses and weakens shear zones.