Watersheds draining into the Gulf of Alaska (GoA) experience large seasonal and inter-annual variations of water in the form of rain, snow, and ice, but accurate constraints on these variations have been difficult to obtain. Over larger geographic regions, water variations can be inferred directly from the Gravity Recovery and Climate Experiment (GRACE) data. However, because GoA variations occur over such a small region, the inferred average value of water flux increases as the applied smoothing of the GRACE data decreases. We use this observed scaling together with scaling results obtained from forward models to infer a seasonal amplitude of 115 ± 20 km3 of water and an average contribution to sea level rise over the two years of data of 0.31 ± 0.09 mm/yr. These results suggest that accelerated melting that began in the late 1990s, as inferred from altimetry, continues unabated.
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 Seasonal and inter-annual (decadal and secular) hydrological variations in the narrow region that drains into the GoA are driven by fluxes in rain, snow, and glacier mass. Precipitation drives a large seasonal signal whereas glaciers provide a reservoir for inter-annual mass fluxes. The net variation has large effects both regionally and globally. Regionally, the freshwater input from both precipitation and meltwater affects coastal currents [Wang et al., 2001]. Runoff is difficult to measure, however, because the five largest river basins only contribute 25% to the total [Wang et al., 2004]. The rest is accounted for through smaller, unmeasured contributions along the coast.
 Meltwater contributions from large glacier complexes near the GoA are a significant component of the observed global sea level rise. The watershed of the GoA contains most of the complexes in the region, accounting for about 10% of the total global area covered by glaciers not directly associated with the ice sheets of Greenland and Antarctica [Dyurgerov, 2002]. Arendt et al.  monitored 28 glaciers in the GoA region using two sets of aerial altimetry profiles taken 5 to 7 years apart, from the mid-1990s to 2001. They concluded that the total discharge had increased, when compared to the historical average, to −96 ± 35 km3/yr of equivalent water, or 0.27 ± 0.10 mm/yr of sea level rise. A negative sign on the volume rate indicates a loss of mass from the region, while a positive sign on the sea level rise indicates that the globally-averaged sea level increased due to mass loss in the region.
2. Mass Balance From GRACE
 Large seasonal and inter-annual variations of water will cause changes in gravity over the GoA region. Gravity variations observed by the GRACE satellite mission have been used to constrain the total variations of water mass over much larger areas (e.g., the Amazon [Tapley et al., 2004; Wahr et al., 2004; Davis et al., 2004] and Mississippi [Wahr et al., 2004] river basins). Unfortunately, the error in the GRACE observations significantly increases as the wavelength of the signal decreases, and thus smoothing is commonly applied to the data to limit this error [Wahr et al., 1998]. Due to the topography of the GoA region and prevailing precipitation patterns, we do not expect immediately surrounding areas to have similar large variations in water mass. Thus, smoothing greatly decreases the amplitude of inferred water mass variations over a region as small as the watersheds draining into the GoA.
 Because of the increasing error with decreasing wavelength, GRACE data cannot be used to resolve the changes of individual glaciers. Thus, we sample an area which surrounds most of the glacier complexes included in the Arendt et al.  database (Figure 1), in order to determine the total change in surface mass. This choice of region differs from previous GRACE studies in that it is not defined by a particular drainage basin. This area does not correspond directly to the drainage basins of the GoA, in that it excludes the contributions from the Alaskan Peninsula and Kodiak Island and includes some regions in the Yukon and British Columbia that drain into the Arctic Ocean. Whereas the total area is quite large (7.01 × 105 km2) compared to the area of the included glaciers (0.87 × 105 km2), it is small compared to regions previously analyzed using GRACE data.
 To obtain regional averages from the Level 2 GRACE data released by the Center for Space Research, we apply a standard technique of adding the contributions from different wavelengths together over a Gaussian-filtered regional mask [Swenson and Wahr, 2002]. Prior to obtaining the regional estimate, we scale the errors for each individual GRACE coefficient to achieve a χ2 of unity to a best-fitting annual sinusoid and trend [Wahr et al., 2004] and then calculate the standard error σ due to satellite measurements using
where Ωregion is the angular area of the region, Δδℓmc and Δδℓms are the cosine and sine components of the satellite error for degree ℓ and order m, and Wℓmc and Wℓms are the smoothed spherical harmonic expansion of the regional coefficients [see Swenson and Wahr, 2002]. This approach differs from equation (28) of Swenson and Wahr , because it does not assume either that Δδℓmc and Δδℓms are equal or that these coefficients are independent of m. Because a few of the Level 2 GRACE monthly fields report coefficients only to degree and order 70, we truncate our calculations to that level. This level of truncation has a small effect on results that use Gaussian smoothing with a radius of 250 km or more. We do not include the C2,0 coefficient due to its large, uncertain variation. We also have not included any correction for geocenter motion. Both of these omissions are accounted for through a scaling procedure we explain below.
 As an illustration of the geographic extent of the surface mass variations present in the region, which may be caused by changes in surface water, snow cover, ice, or ground water, Figure 1 shows maps of the best-fitting annual amplitude and “trend” over the time period of the GRACE data. Note that in addition to the large annual signal over the region of interest in this study, there is also a coherent annual signal over the Pacific Northwest. However, the largest trend (by a factor of two) is present over the drainage basins of the GoA.
Figure 2 shows the time series of the surface mass variation (in mm of water equivalent) over the region shown in Figure 1. In order to be able to better judge the amplitude of the temporal variations, the constant value over this time span is removed. For comparison, 100 mm of surface mass over this region corresponds to 70 km3 of water equivalent or about 0.2 mm of globally-averaged sea level rise. A strong seasonal cycle and a trend are clearly present in the data.
Table 1 (500 km smoothing) shows the coefficients in equivalent water resulting from a least-squares fit to annual cosine (in-phase, maximum at January 1) and sine (out-of-phase, maximum at April 1), trend, and constant (not listed) terms, and the solid line in Figure 2 represents the best fit. The error bars reflect well the variation of the data about the fit in that the reduced chi-square value is close to unity (0.93). The seasonal signal is nearly completely out-of-phase, peaking near April 1. Removing either the first or the last data point (or both) does not significantly change the estimated coefficients.
Result of fit to time series of the regional averages shown in Figures 2 and 3.
9.4 ± 9.4
71.0 ± 8.0
−63.5 ± 12.6
−0.6 ± 4.2
42.9 ± 3.6
−41.5 ± 5.5
−2.3 ± 3.2
27.8 ± 2.8
−28.3 ± 4.2
−3.2 ± 2.7
16.7 ± 2.3
−19.3 ± 3.4
 The analysis in Figure 2 does not provide an accurate estimate of the total mass variation in the region. The Gaussian smoothing applied to the averaging kernel reduces the amplitude of the estimate because it introduces surrounding regions with smaller mass variations into the overall average. Thus, as the radius of the Gaussian smoothing is reduced, the amplitude of the resulting estimates should increase. To verify this, we have repeated the analysis with different levels of smoothing. Regional averages with fits are shown in Figure 3 with estimated coefficients given in Table 1. The error estimates given in Table 1 are the satellite errors only (i.e., those present in the GRACE data) and do not include contributions due to leakage from surrounding mass change or omission errors; this will be addressed below. As expected, the amplitude of the temporal variations increases as the level of applied smoothing decreases.
 To gain insight into the dependence of mass flux estimates on the adopted spatial smoothing, we generate two forward models of the geoid variation caused by mass changes over the region. The first model [Tamisiea et al., 2003] is based upon previous estimates of the geometry of glacial melt [Arendt et al., 2002]. Because the mass changes (e.g. precipitation) causing the annual variation are likely to be distributed across the region, the second model assumes a uniform mass change over the averaging region. If the observed geoid change is due to water variations in this region, then the inferred surface mass from both the models and the GRACE data should vary in the same manner when different levels of smoothing are applied. If this scaling relationship holds, we can use the scaling derived from the model data to determine the total mass estimate for the region.
 Because the amplitudes of forward models have large uncertainties, Figure 4 compares the GRACE data and the forward model results by normalizing each result by the corresponding value obtained assuming a 500-km-radius Gaussian filter. The scaled forward model results are identical for all levels of smoothing and differ only when no smoothing is applied. Figure 4 demonstrates that both the GRACE inferences of water variation over the region and the forward models exhibit an identical scaling relationship within the uncertainties.
 To obtain final estimates for the total annual amplitude and rate for this region, we should be able to scale the result at any level of smoothing by a scale factor determined from the forward models. However, estimates from the GRACE data are also contaminated by geoid variations due to sources outside the study area that leak into the regional averages. For the annual amplitude, the largest contamination occurs from hydrology. To account for this, we adjust the GRACE result (see Figure 4) with estimates derived from the LaDWorld-Danube hydrology data set [Milly and Shmakin, 2002; Shmakin et al., 2002]. When using this data, we start by removing a constant at each grid point for the time period overlapping the GRACE data. In order to conserve mass, we then calculate a gravitationally self-consistent ocean load that complements the hydrology, assuming water is exchanged directly between the continents and the oceans. The contamination is then calculated by removing the signal present in LaDWorld over the averaging region, creating a new geoid time series based upon the modified LaDWorld and ocean load, and repeating the same analysis used for the GRACE data on the resulting geoid variations. This correction has the effect of reducing the total inferred annual amplitude of mass change over the region by approximately 20 km3. We find that the contribution from an ECCO [Stammer et al., 2003] generated model of bottom pressure is small, and we include this contribution in our error estimates.
 Contamination of the trend is more difficult to establish because the magnitude and sign of many of the contributing effects, such as secular changes of the polar ice sheets and the contribution due to ongoing glacial isostatic adjustment, are uncertain. The contamination from distant sources should be nearly constant for all levels of scaling and thus will contribute a smaller percentage to observed value as the level of smoothing is reduced. The present-day geoid anomaly due to local ice variations during the late Pleistocene or early Holocene is expected to be small due to the low value of effective viscosity in the region [Larsen et al., 2003]. However, the collapse of the large ice complexes in Laurentia and elsewhere has an ongoing global effect on the geoid. Using a suite of different Earth models, we find that this signal will contaminate estimates of the trend by less than 3 km3/yr. Adjustment driven by glacial variations over the last millennia may impact our inferences; calculations based on the Larsen et al.  ice history would increase our estimate of mass change by <10% [see also Tamisiea et al., 2003].
 Accounting for these uncertainties in the error estimates, as well as the omission of the geocenter motion and C2,0, we infer an annual amplitude of 115 ± 20 km3 (0.33 ± 0.06 mm of sea level change) and a trend of −110 ± 30 km3/yr (0.31 ± 0.09 mm/yr of sea level rise) for the period 2002–2004. This scaling analysis mitigates the error introduced by omitting the geocenter variation and C2,0. The geocenter motion and value of C2,0 are very small in the forward models, with most of the power from the signal present in the higher degrees and orders. Applying this scaling technique to the LaDWorld data set as an example, we removed these two terms from the predicted geoid and found that the solution scaled from the 500-km smoothing result is nearly identical to the true value over the region. As longer time series become available, tighter constraints can be placed on the possible contamination factors, reducing the error estimates and providing further insight into the nature of the inter-annual variations.
 Gravity field studies provide a direct and integrated measure of mass variations in the hydrological system, and thus have a major advantage over other methods of monitoring the cryosphere-hydrosphere system. A simple analysis of GRACE data over the GoA region indicates large mass variations, but an estimate of the actual mass flux in the northwest Pacific requires a robust, quantitative correction for the effects of smoothing on the estimates. We have shown that inferences of annual and inter-annual variations can be obtained by combining the GRACE data with forward models of the geoid anomaly field.
 Our technique is not a replacement for other recent analyses [Swenson et al., 2003; Seo and Wilson, 2005] that determine an optimal balance, for a given region, between leakage error and satellite errors. Rather, it can be used to determine the total mass flux in regions where the results vary significantly with applied smoothing. This demonstration provides a generalized route to bounding the recent mass balance of other small ice sheet and mountain glacier systems.
 Ocean circulation in subpolar regions is more strongly impacted by coastal freshwater discharges than circulation in either the tropics or subtropics [Wang et al., 2004]. Our ability to obtain estimates of annual amplitude in surface mass can be used as an additional constraint in the modeling of the seasonality in coastal currents. Furthermore, our inferred inter-annual trend (0.31 ± 0.09 mm/yr equivalent global sea-level rise) suggests that the high rate of discharge inferred in the second half of the 1990s from aerial altimetry [Arendt et al., 2002] continues unabated.
 We would like to thank an anonymous referee for comments. This work was supported by NSF grant EAR-0125518, and NASA grants NNG04GL69G and NNG04GF09G.