The GRACE data used here are the GRCTellus global mascons (JPL RL05; mass concentration) solution data (Watkins et al., 2015; Wiese, 2015). This GRACE total water storage anomaly (TWSA) product is a 0.5∘ grid based on the spatial variability of the 3∘ measurements. The TWSA data for the Mississippi subbasins are aggregated over each subbasin using the area-weighted averaging method presented by Riegger and Tourian (2014). Due to satellite battery management and other issues, there are some missing months in the GRACE dataset. In total, 12 of the 151 monthly values are missing in our period of study. To fill missing months, linear interpolation between the previous and following months was used.

Monthly streamflow measurements (Qo) were obtained for select discharge gauge stations (US Geological Survey, 2015). The gauge stations were selected based on data availability, drainage area and location throughout the Mississippi River basin (i.e., along major tributaries). The 12 sites were distributed throughout the Mississippi River basin with 3 along the Ohio River (1–3), 3 along the upper Mississippi River (4–6), 5 along the Missouri River (7–11) and 1 near the outlet of the Mississippi River (12) (Fig. 1). Rodell and Famiglietti (1999) estimated that the minimum region size in which GRACE could resolve water mass variability would be about 200 000 km2, a smaller size than our smallest basin. The GRACE mascons (Watkins et al., 2015) are statistically independent and are at a 3∘ resolution (around 90 000 km2). Although multiple sites are from individual tributaries, they are distributed along the river such that the difference in drainage area between two sites is roughly 100 000 km2 or more.

All relevant gauge information, such as river name, drainage area and period of record, is contained in Table 1. It is essential to note that potential cold weather months (November through March) were excluded from this analysis for USGS streamflow to minimize the impacts of snow and ice influence on the total water storage. For example, if basin-wide storage increases due to snow accumulation, it is likely that there will be no correlated change in discharge at that time. Thus, the storage change measured by GRACE for those months is not directly linked to discharge until some later period. The sensitivity of the results of this study to the selection of April through October as the non-frozen period is likely to be minimal in this region.

There are other possible sources of storage variability that should be considered when using GRACE measurements, such as vegetation growth and groundwater pumping. Regarding vegetation biomass, Rodell et al. (2007) affirms that the seasonal and interannual biomass variations are typically smaller than the uncertainty in the GRACE TWSA measurements, and based on the global maps of vegetation biomass (Rodell et al., 2005), this holds true for the Mississippi River basin. Significant pumping occurs in the High Plains located in the basin; however, as it is a shallow-water-table aquifer (Scanlon et al., 2012; Brookfield et al., 2018; Nie et al., 2018), the storage changes would still be linked to baseflow generation. In other words, the portions of the basin which are experiencing water table decline due to human activities would still exhibit the same general storage–discharge relationship.

To identify potential relationships between monthly discharge (Q) and basin storage (S), GRACE TWSA data are used to represent storage variability and paired time series of Q–S are determined for each subbasin. Mean monthly observed discharge (m3 s-1) is converted to depth units (cm per month) by cumulating flow rates for each month and dividing by the drainage area upstream of each site (Table 1). Only non-winter months were selected to limit the impacts of snow processes on Qo–S relationships. Following work by Kim et al. (2009), we focus on the fact that most summer storage variability in the Mississippi River basin is not due to surface water storage, but instead to subsurface storage (including the vadose zone). Our assumptions are applied to the recession of the streamflow records, namely that baseflow drives the portion of streamflow that underlies monthly peaks, and that this baseflow amount can be regressed against storage to achieve the storage minimum with calculated uncertainty. Pairing Qo with S, we also assume that an average monthly discharge corresponds to the GRACE TWSA for the same month, which derives from a single measurement in the month concerned. However, the GRACE solution integrates temporal information from several ground tracks through the study region into the monthly gravity field, a single value carrying information for a whole month. Note that we focus on storage anomalies rather than absolute water storage to determine the discharge relationships because of the inability to quantify absolute storage based only on GRACE measurements.

To investigate baseflow (Qb) relationships, a forward-looking “low-flow filter” is developed and applied. The rationale for the filter is that both baseflow and event flow are represented in the discharge record at any time, but only the baseflow portion of streamflow serves to infer drainable storage. Hence, we assume that the storage-driven portion of discharge generally increases with increasing S, here represented by GRACE TWSA. To build the Qb–S relationship, the Qo–S paired series is sorted from the minimum to maximum value of S. Because Qo is assumed to increase with S, Qb for a given S is set to the forward-looking minimum Qo. Next, a Qb value is estimated for each S, based on minimum measured values of Qo: 1QbSi=minQoSii=1n, where n is the number of forward-looking values remaining in the paired series. In other words, the filter looks at the next nQo values paired to the next n larger S values, selecting the minimum Qo as baseflow. The value of n can be subjective depending on the series size. Here, we used 20 % of the number of pairs (18 months), after analyzing the model's sensitivity to n (Fig. S1 in the Supplement). The process defines the low-flow envelope in the Qo–S series, where the variations in discharge above the minimum value are due to short duration rainfall–runoff events not captured in the monthly GRACE TWSAs. Here, we term the low-flow series as baseflow (Qb) but acknowledge our definition of baseflow may differ from other studies.

Building on previous studies (e.g., Kirchner, 2009; Reager et al., 2014), which suggest that summer river discharge and drainable storage generally show an exponential relationship, we assume a relationship for total discharge and estimated baseflow in the form of Eq. (2): 2Q=αeβS, where Q is the non-winter discharge (Qo) or estimated baseflow (Qb), α and β are coefficients, and S is basin storage defined here as GRACE TWSA.

To transform TWSA into an absolute water storage value, referenced herein as drainable storage (Se) that directly influences discharge, a storage offset must best estimated. For example, Riegger and Tourian (2014) proposed a definition of time-dependent water absolute storage Se(t), using Eq. (3): 3Se(t)=TWSA(t)+So, where TWSA(t) is the monthly storage anomaly, and So is an unknown constant storage offset. So only shifts the Se(t) series without impacting its temporal variability. Figure 2 shows how the TWSAs provide the same fit (e.g., R2) and exponential coefficient (β) accounting for the change in discharge with changing storage. Only the leading coefficient (α) changes in response to the value of the storage offset (So) being added to each TWSA. The intent of Fig. 2 is to demonstrate that TWSA and S can be used interchangeably by replacing α to account for the resulting desired storage units. The storage offset cannot be measured directly but should correspond to the long-term mean water storage for the region of interest. Based on the assumption that baseflow is driven by storage (Se) and therefore a linear function of storage, the relationship between discharge and TWSA can provide insights for estimating the representative So value, which provides an opportunity to estimate drainable storage.

As discussed, we assume there is an exponential relationship between storage and discharge. However, because we only base our Q–S relationship on measurements, we use GRACE TWSA as a surrogate of storage. Figure 3 shows all non-winter (April–October) monthly observed discharges (Qo) and the relationships between discharge and storage for all 12 subbasins. In general, the figure shows that the Ohio and Upper Mississippi subbasins (1–6) exhibit similar behavior in terms of magnitude and variability of discharge, whereas the Missouri subbasins (7–11) have much less variability and smaller discharges for a given storage. Note that, the variability observed in the Missouri subbasins (7–11) series is due to high Q–S points resulting from flooding in April to July 2011 (Reager et al., 2014), where the four largest storages are from these months. Figure 3 also shows how the Qb–S relationships capture the minimum flow conditions for the observed storage–discharge series (i.e., minimum flow envelope). The variability above the Qb–S curve represents short-duration event discharges not captured by storage-driven discharge.

The resulting α, β and R2 values for the Qo–S and Qb–S relationships are shown in Fig. 3 and listed in Table S1 in the Supplement. In general, the relationships fit the Qb–S pairs with a median R2 of 0.89 ranging from 0.46 to 0.92. For overall discharge, which includes event variability, the median R2 drops to 0.63 ranging from 0.40 to 0.80. The α values range from 0.15 to 1.5 (cm per month) for baseflow and 0.22 and 2.7 (cm per month) for streamflow and differ between the major tributaries. In general, α tends to decrease as minimum observed discharge decreases. For example, values along the Missouri River are noticeably lower than those along the upper Mississippi and Ohio rivers. As expected, both αb and αo are highly correlated with mean annual low-flow (R is 0.99 for baseflow and 0.96 for streamflow).

Comparing the two relationships, αb is equal to roughly 65 % of αo ranging from 52 % to 75 %. Note that, the ratio αb/αo represents the mean baseflow fraction at each station when the TWSA is zero (i.e., Qb=αb and Qo=αo), which corresponds to the mean storage observed during the GRACE period. Although baseflow fractions are difficult to assess and vary based on estimation methods (Cheng et al., 2016; Eckhardt, 2008; Gonzales et al., 2009; Lott and Stewart, 2016; Zhang et al., 2017), the values reported here are consistent with those in the literature. Zhang and Schilling (2006) reported ratios ranging from 65 % to 75 % for sites along the Mississippi River. Arnold et al. (2000) reported a ratio of 65 % in the upper Mississippi River. Beighley et al. (2002) reported a median ratio of 55 % for the Susquehanna River, which boarders the Ohio on its eastern boundary.

The β values (i.e., exponential coefficient that scales discharge based on S) range from 0.02 to 0.1 for baseflow and 0.04 to 0.1 for streamflow and differ between the major tributaries. Based on a qualitative assessment, β appears to decrease as the amount of water regulation increases. For example, the Missouri River is known to be highly regulated and the associated β values are noticeably lower than those for the upper Mississippi and Ohio rivers. In a regulated system, basin storage can increase with little change in river discharge because water is being stored in lakes/reservoirs. In this case, the Missouri river has several very large reservoirs (e.g., Lake Oahe, Lake Sakakawea, Fort Peck Lake), which may explain the relative weaker relation between Q–S. This is one of this method's limitations, creating an uncertainty from the inability to include specific basin characteristics. For this reason, the relationships for heavily regulated rivers only reflect the reservoir storage availability observed during the study period. Of interest is the difference in βo and βb along the Missouri River, where βb is roughly 35 %–62 % of βo compared with the other rivers where βb is 84 %–110 % of βo. This difference, which is due to disproportionally lower βo values for the Missouri River, suggests that storage changes are mitigated more for baseflow than for event-flow conditions in regulated systems (Fig. 3). As expected, the β values are correlated with streamflow variability, defined here as the ratio of mean annual low-flow divided by mean annual flow for non-winter months (Qm-min/Qm), where R is -0.89 and -0.94 for baseflow and streamflow, respectively. The correlation of α to low-flows and β to streamflow variability supports the physical meaning of Q–S relationships (Kirchner, 2009; Reager et al., 2014).

A unique aspect of the Q–TWSA relationship described in Eq. (2) is that it can be used to estimate the storage offset (So) in Eq. (3), which enables the conversion of TWSA to drainable storage. For example, solving Eq. (2) for TWSA when streamflow is approximately zero, yields the maximum negative TWSA for the associated Q–TWSA relationship. If we set the storage offset to the maximum negative TWSA in Eq. (3), we can convert TWSA to drainable storages, where the basin storage is zero for the near zero flow condition. This is the fundamental concept supporting the assumed Q–S relationships. The challenge is defining near zero streamflow because an exponential relationship cannot be solved for S if Q is zero. Here, we assume near zero streamflow is approximately 0.01 % to 0.1 % of the minimum monthly non-winter observed discharge (see Qmin in Table 1). Although this is not exact, it is bounded by observed streamflow and provides discharges that capture the extreme hydrologic conditions associated with zero drainable storage. For example, 0.1 % Qmin corresponds to mean monthly discharges ranging from only 0.1 to 4.5 m3 s-1 between sites. Using the above approach and the Qo–TWSA relationships in Fig. 3, Fig. 4 shows the non-winter (April–October) drainable storage for each subbasin during the study period, where the colored regions represent the range in storage measured by GRACE for the two estimates of storage offset (So for 0.1 % Qmin and 0.01 % Qmin).

As the Mississippi River station (site 12) resulting storage offset ranges from 96 to 123 cm (i.e. 109±14 cm) and the observed basin-wide TWSA ranges from -9.7 to 14.6 cm, we estimate the absolute drainable storage as 2900±400 to 3600±400 km3. Considering that the Mississippi River site drains all 11 subbasins with sites 3, 6 and 11 representing the upper Mississippi, Ohio and Missouri river outlets, respectively (2.3 million km2). There is roughly 600 000 km2 of drainage area above site 12 not captured by three outlet gauges. Using the average storage per square kilometer from the three subbasins, we estimate storage for the remaining area. Cumulating the subbasin and ungauged storages, we estimate that the Mississippi River basin storage offset varies from 3100 to 4000 km3 for non-winter months (site 12* in Fig. 4), i.e. approximately one-tenth of the maximum storage in the largest US reservoir: Lake Mead. Although there should be no difference in the storage offset from the two approaches, a difference of roughly 10 % is found, which may result from the storage per unit area from the subbasins overestimating the storage in the ungauged area. Although the range of mean storage is 800 to 900 km3, it represents less than 30 % of the lowest storage estimates. Thus, we provide one of the first drainable storage estimates for the Mississippi River basin and its major tributaries. These values cannot be validated as there are no current measurements of such an amount. Most large-scale models (e.g, PCR-GLOBWB, van Beek and Bierkens, 2009) are not fully coupled with groundwater models and contain structural errors in their ability to represent the GRACE-observed storage variability (Houborg et al., 2012; Scanlon et al., 2018). Thus, the comparison would not be direct. The storage offsets listed in Table S2 can be used to covert GRACE TWSA time series to absolute drainable storage time series and determine corresponding α values.