In this section, we investigate the reason(s) why the results of L2013 differ from W1315 (see Fig. 4a, b). One possibility is due to the value of \(\alpha\) used. Another possibility is due to their difference in the g-dot and uplift rates used which may be due to the difference in their post-processing used. It is also possible that the uncertainties of the results have previously been underestimated. In the following subsections, we will study each of these possibilities individually.
Value of \(\alpha \)
As we noted earlier, the value of \(1/\alpha \) used by W1315 is \(-\,0.150\, \upmu {\hbox {Gal}}/{\hbox {mm}}\) which is slightly larger than the upper limit of range used in L2013 (i.e. \(-\,0.165\pm 0.012\,\upmu {\hbox {Gal}}/{\hbox {mm}}\)). So, can the difference in the inferred amplitude of the hydrology signal be due to the difference in the value of \(1/\alpha \) used?
To answer this, we used the methodology of W1315, but with \(1/\alpha \) equal the middle and lower limit of the range in L2013 (i.e. \(-\,0.165\) and \(-\,0.177\,\upmu {\hbox {Gal}}/{\hbox {mm}}\), respectively). The results are shown in Fig. 4c, d, which show that the value of \(1/\alpha \) has little effect on the signal amplitude in the Nelson River basin. This is due to the fact that the GPS uplift rate in the Nelson River basin is quite small (see Fig. 3). However, near the centre of GIA in Hudson Bay, where the uplift rate is much higher (the red area in Fig. 5d), the effect of the value of \(1/\alpha \) is more significant (see Fig. 5). For example, in the west and east side of Hudson Bay, the amplitude of the blue troughs is \(-\,10\) and \(-\,16\,{\hbox {mm}}/{\hbox {yr}}\), respectively in Fig. 5a, but \(-\,14\) and \(-\,20\,{\hbox {mm}}/{\hbox {yr}}\), respectively in Fig. 5c. Thus, the magnitude in the west and east side of Hudson Bay increases by 40 and 20%, respectively.
g-dot and uplift rates
If the value of \(1/\alpha \) is not the main cause of the difference in the inferred amplitude of the hydrology signal in the Nelson River basin, what else can be the cause? As we have seen earlier, due to the post-processing of the GRACE data, the g-dot values in W1315 are \(0.4\,\upmu {\hbox {Gal}}/{\hbox {yr}}\) smaller than that for L2013 (see Fig. 2a, b). This difference is approximately \(9.72\,{\hbox {mm}}/{\hbox {yr}}\) in equivalent water thickness (EWT) and can possibly explain the difference in the maximum value in Fig. 4.
To check this, we investigate the effects of the post-processed GRACE and GPS inputs on the resulting predicted EWT rates. This time, we follow the approach of L2013, but use different combinations of post-processed GRACE and GPS input data from W1315 or L2013. For visual comparison purpose, the EWT predictions along an east-west profile along \(52^{\circ }{\hbox {N}}\) are shown in Fig. 6. Here, the red solid curve is computed with the GRACE and GPS data both from L2013, and the blue solid curve uses GRACE and GPS data from W1315. On the other hand, the green dashed curve uses the data from Figs. 2a and 3b, while the orange dot-dash curve used the data from Figs. 2b and 3a.
Inspection of Fig. 6 shows that: (1) the peak value of the red solid curve agrees with the peak value in Fig. 4b. This is expected because they all use the method of L2013, and their inputs are almost the same. (2) The values of the blue solid curve (including the peak) agree reasonably well with that along the profile at 52N in Fig. 4a. Note that the blue solid curve and the profile in Fig. 4a are computed with different approaches, but the inputs are almost the same. This means that both methodologies give almost the same EWT prediction when the input files are the same. (3) If the GRACE input of L2013 is used with the GPS input file of W1315, then the orange dot-dash curve shows that the peak is slightly displaced but the peak amplitude is slightly smaller than the red curve. On the other hand, when the GRACE input of W1315 is used with the GPS input file of L2013, then the green dashed curve is closer to the blue solid curve. This means that the GRACE input has a larger effect on the EWT amplitude than the GPS input. The reason is that at the location of the EWT rate (east of Saskatoon), the difference in the GPS data is relatively small, so its effect is smaller.
As for the difference in the pattern south of the peak (see Fig. 4a), the cause is probably due to the difference in the density and location of the GPS network used (see Table 1) which results in more negative contours in the south in Fig. 3b.
In short, the main cause of the difference in the peak amplitude of the inverted EWT rate between W1315 (Fig. 4a) and L2013 (Fig. 4b) is mostly due to the post-processing of the input GRACE data, rather than the methodology to determine the hydrology signal or the value of \(1/\alpha \).
Post-processing step
So, the next question is: which steps in the processing of GRACE data are responsible for the difference? From Table 1, we see that the main difference between the processing of GRACE data of W1315 and L2013 is that the latter tried to restore the signal lost in de-striping filtering, but the former did not. Furthermore, an isotropic Gaussian filter is applied to the GRACE data in W1315, while a non-isotropic Gaussian filter is applied in L2013. This could cause more amplitude reduction in L2013 than in W1315.
Thus, the next question is: how does the isotropic Gaussian filter in W1315 or the non-isotropic Gaussian filter of L2013 affect the amplitude of the GRACE g-dot data in the Nelson River basin? The answer of the first part may be found in “Appendix 1”, where it is shown that the magnitude of the peak g-dot values can decrease significantly after Gaussian filtering if the filter radius is large. However, these peak g-dot values lie outside the Nelson River basin. Figure 7 in “Appendix 1” also shows that for the synthetic GIA-induced g-dot data in the Nelson River basin, their amplitudes are not significantly affected, although the contours become smoother when higher frequency signal is reduced. Figure 8 in “Appendix 1” confirms that this is also true for the GRACE g-dot data that contains both GIA and hydrology contributions. Therefore, the isotropic Gaussian filter in W1315 should not be the main cause of the difference in amplitude of the GRACE g-dot data or the EWT rate around the Nelson River basin.
What about the non-isotropic Gaussian filter? A comparison of Fig. 4a, b shows that the pattern of the EWT results for W1315 resembles the topography map of the Nelson River basin more than that of L2013. Since basin topography should affect the pattern of the hydrology signal, the distortion of the hydrology signal may be caused by the non-isotropic Gaussian filter of L2013. This hypothesis should be verified in future studies.
Unfortunately, the synthetic and real GRACE g-dot data in “Appendix 1” cannot be used to study the effect of the de-striping filter on amplitude reduction nor the faithfulness of L2013’s statistical test in restoring the signal lost in de-striping because the whole procedure of L2013 requires the GRACE time series as input data but not the spatial distribution of g-dot data (private communication of J. Huang).
So far, we have traced the main source of difference between the amplitude of the hydrology signals inferred by W1315 and L2013 to be due to the difference in their post-processing of GRACE data. However, a thorough understanding of the effects of post-processing to the amplitude of the GRACE data is beyond the scope of this paper and should be addressed in future research.
Uncertainties
Other important questions that arise in view of that are now: how large are the uncertainties of the L2013 and W1315 results? Are their uncertainties large enough that the difference in the inverted hydrology signal can be explained? How large are the uncertainties of the inferred total water storage? “Appendix 2” shows that the uncertainty of the hydrology component in EWT is about \(2.3\,{\hbox {mm}}/{\hbox {yr}}\) for L2013 and \(2.5\,{\hbox {mm}}/{\hbox {yr}}\) for W1315. Since the peak EWT value is about \(34\,{\hbox {mm}}/{\hbox {yr}}\) in Fig. 4b and \(24\,{\hbox {mm}}/{\hbox {yr}}\) in Fig. 4a, that means the differences between the peak EWT value in Fig. 4a, b are larger than their uncertainties and thus enough to be resolved. Since we only consider the measurement uncertainties here, but do not consider the uncertainties from the separation approach and irregular GPS distribution, maybe we underestimate the uncertainties. The uncertainties details are illustrated in “Appendix 2”.