Passive Microwave Radiometry at Different Frequency Bands for River Discharge Retrievals

The present era of climate change and expanding population requires major improvements in sustained observation of global river discharge. Floods and droughts are affecting food supplies, and suspected long‐term trends require appropriate data for evaluation. Orbital remote sensing can address this observational need. Here we use satellite Ka‐ (36.5 GHz) and L‐band (1–2 GHz) passive microwave radiometry (PMR) to monitor river discharge changes and determine what size rivers can be measured and the frequencies and polarization configurations that yield the most robust results. Selected satellite gauging reaches (SGRs) can be measured at near‐daily intervals from 1998 to present (Ka‐band) and 2010 to present (L‐band). The SGRs are 10–36 km in length; the dynamic proportion of water surface area within each varies with river discharge. Due to contrasting dielectric properties, water and land emit different intensities of microwave radiation; thus emission from a mixed water/land pixel decreases as the proportion of water within the pixel increases. Depending on the river and floodplain morphology, water flow area can be a robust indicator of discharge and the microwave sensors can retrieve daily discharge to ±20%. Instead of spatial resolution, it is the sensor measurement precision, geolocation accuracy, and channel and floodplain morphology that most strongly affect accuracy. Calibration of flow area signals to discharge can be performed using nearby ground stations (even if now discontinued) or by comparison to hydrologic modeling.


The Traditional Ground-Based Approach
River discharge is a particularly challenging measurement. Both ground-based methods and the satellite-based approach described here do not measure Q directly, but instead retrieve an indicator of Q. Then, a calibration from the measured indicator to Q is performed. River stage (s, height of water above a fixed datum) is the commonly used discharge indicator on the ground. In fact, since at least Pharaonic time in Egypt, ∼1000 BCE, s has been used to monitor river discharge (Elfatih et al., 1999). Our flow area approach is fundamentally the same as the stage-based method, but simply uses a different flow indicator. Thus, according to the flow continuity Equation (1): where Q is discharge (m 3 /s), w is flow width (m), d is flow depth (m), and u is integrated flow velocity (m/sec).
River stage (s) is the height of the flow surface above a fixed datum and, if the channel cross section is stable, it varies directly with d. It is used to frequently monitor Q over time. s is calibrated to Q by periodic (∼10 per year is typical in the U.S.) field traverses across the river with a current velocity meter, thereby providing u, d, and w measurements at high, medium, and low flows. The flow column is sampled at intervals across the river at different depth intervals (e.g., 20% and 80% of maximum depth) to retrieve u and compute total Q. After these repeated in-contact Q measurements, observed s is translated via an empirical "rating curve" to Q (Figure 1). Such field measurements should continue each year because, along most rivers, the channel cross section may change. Recently, acoustic doppler profiling is also being widely used to obtain the intermittent Q observations (Gonçalves et al., 2023;Simpson, 2001;Turnipseed & Sauer, 2010), again to calibrate s to Q.
The accuracy of the results is a complex topic and includes the accuracy of the individual in situ Q measurements and the construction and updating over time of the rating curves. Due to channel morphology changes, either gradually over years of time (Pinter, 2006), or abruptly after major floods, the "rating" relations also may change.
As well, differing water resources applications vary in their measurement needs. For some uses, reasonably accurate Q records that extend over a considerable period are more valuable than extremely accurate data covering only a short period (Grover & Hoyt, 1916). In this regard, the United States Geological Survey (USGS) operationally uses stage measurements to compute streamflow at over 8,000 stream gauging stations. USGS policy is to produce s measurements precise to the nearest 0.02 ft or 0.2% of stage, whichever is greater. The in-situ Q measurements for rating curve production may often be accurate to ±5% but can range between ±2% and ±20% depending on field conditions. Also, local judgment is required as to which measurements are accepted as accurate, and then incorporated into the station rating (Turnipseed & Sauer, 2010). Figure 1 illustrates a common situation: as overbank flood flows are attained, it is more difficult to measure the complete flow cross section and associated velocities. Retrieval of Q becomes less accurate and the s-versus-Q relation can be less precise. Typical rating curves include an uncertainty in the Q measurements of 18%-25% at peak discharge (Pappenberger et al., 2006); total rating curve errors increase when the river discharge increases and vary from 1.8% to 38.4% with a mean value of 21.2% (Di Baldassarre & Montanari, 2009).
In the U.S, and using s, instantaneous Q for a channel and floodplain may be recorded very frequently (hourly to daily). Similar stage-based methods are used internationally (Herschy, 2008) but temporal sampling and quality-control methods vary widely. For example, the frequency and method of in situ Q measurements are frequently left unpublished and a fixed rating may be used over decades instead of being adjusted by new Q measurements. Maintenance of gauging station rating curves is labor-intensive: as noted, measuring Q in a natural river channel can be challenging. In any case, Q may be published as daily, weekly, monthly and yearly statistics. If s is retrieved more frequently, for example, hourly, then a daily mean Q is calculated. Runoff (r, commonly expressed in mm) is the same information provided as water volume/contributing area. Thus: total m 3 discharge/contributing area (in m 2 ), both converted to mm units, with 86,400 s/day. Runoff volumes can be expressed as daily to yearly total values, and thereby directly compared to watershed precipitation and evapotranspiration (usually also in mm).

The Flow Area Approach
According to the flow continuity equation, flow area, a, can also be used to track discharge for a defined satellite gauging reach (SGR) (Equation 2).

= adu∕l
where a is SGR water flow area and l is water flow length.
As is the case for s, a varies approximately monotonically with Q (Bjerklie et al., 2005;Smith, 1997). Also, a applies to a defined river reach instead of a single flow cross section (Figure 2), and a SGR includes both the river Figure 1. The stage-discharge (s-versus-Q) rating curve enables Q monitoring via stage recording. In practice, the data may be plotted with log-log axes and inflection points may be identified in the exponential relations at different parts of the flow regime. Example from the Wabash river in the US; see Data Availability Statement for data source.
channel and its surrounding floodplain. Due to the coarse spatial resolution of most orbital sensors in this domain, a PMR SGR is operationally defined either as: (a) information retrieved from (swath) satellite information within specified radii of a single target coordinate, or (b) information retrieved from a single pixel within a gridded product. Examples of both SGRs are shown in Figure 2 along the Ayeyarwady River in Myanmar.
Q is assumed constant at both upstream and downstream ends of the SGR parcel ( Figure 2): the PMR method retrieves an integrated average Q over the river reach. In detail, Q within 10 km or longer river reaches may increase downstream from groundwater input through the hyporheic zone ("gaining reaches") or decrease by infiltration into groundwater ("losing reaches") (Mertes, 1997), or may be affected by influxes from side tributaries (Nickles et al., 2020). However, for most rivers, it is possible to define many SGR locations where the flow-in = flow-out assumption is valid within the precision limits of the measurement. Also, floodplain negative relief features may provide additional water surface area responsive to river flow status, if the water bodies are connected (Lewin & Ashworth, 2014).
Along river channels with steep-sided cross sections, and for in-channel flow, s can be a more sensitive monitor of Q than a. However, over 10 km or more of river reach, there may occur many locations where flow widens as Q increases. Depending on the river morphology in an SGR, a is a potentially robust indicator and may offer a significant advantage compared to s. Thus, for in situ gauging, the preferred locations are where at-a-station changes in water surface level apply to the entire river flux passing the gauging station and where the s-versus-Q relation is relatively stable through years of time.
In contrast, for flow area approaches, many rivers are multithread, anastomosing, or meandering, and the channels themselves change location over time ( Figure 3). In such cases, observation of a within a fixed SGR may more accurately record Q over time: increases in Q are accommodated by total flow area expansion, local channel deepening may be matched by aggradation (shallowing) at different locations, and channel migration is also accommodated within the SGR.

Choice of Satellite Sensors
Orbital sensors at various spatial resolutions and repeat intervals can be used to observe changes in a (Bjerklie et al., 2003(Bjerklie et al., , 2005Smith, 1997;Smith et al., 1996). To realistically address both global (Milliman et al., 2008) and regional (Hirpa et al., 2013) measurement needs; however, the following are required: (a) sensing with high  . SGR #26, Ayeyarwady River, Myanmar. Illustrated are the SMAP satellite 36 and 45 km L-band target parcels for the "M" (the SGR) and the "C" (calibration site; see text); both shown as white circles. The black square is the 10 km M pixel used for Ka-band SGR measurements. No Ka-band C pixels are illustrated (see text). Dark blue is permanent water, light blue is annual flood extent, and dark gray is maximum flood as imaged by the MODIS multispectral sensor at 250 m spatial resolution (Kettner et al., 2021). temporal resolution (near-daily revisits), (b) extensive geographic coverage, (c) minimal interference from cloud cover or poor solar illumination, and (d) the longest possible time series.
PMR fits these requirements and, at Ka-band, has been shown to be an efficient technique to monitor rivers using a (Brakenridge et al., 2012;Brakenridge et al., 2007;Brakenridge et al., 2021;Z. Kugler et al., 2010;Z. Kugler & De Groeve, 2007;Z. Kugler et al., 2019;Revilla-Romero et al., 2014;Van Dijk et al., 2016). Passive microwave sensors measure emitted microwave radiation of the Earth's surface in units of brightness temperature (Tb[K]). Global or near global data sets are available at various frequency bands since the 1980s. The satellite platforms which carried these sensors include the U.S. or international DMSP SSM/I and SSMIS, WindSat, TRMM, AQUA/AMSR-E, GCOM-w/AMSR2, GPM, SMOS, and SMAP. At the microwave bands selected, there is minimal interference by cloud cover or rain droplets, and data are obtained both day and night.
An apparent constraint to use of these data is their relatively low spatial resolution. Commonly, the data are gridded into equal area map projections with spatial resolutions of 8-40 km/pixel. For comparison, many small single-thread rivers are only 0.4 km wide or narrower during in-channel flow (although flood flows may reach widths of 4 km nearly every year). Thus, a during in-channel flow may occupy only 4% of a 10 × 10 km SGR area for much of the year (1.5-3× more if the river is meandering), but a may reach 40% or more during annual high flow. Due to their microwave dielectric difference, surface water and land emit strongly contrasting intensities of thermal microwave radiation. As a result, the bulk emission at selected frequencies, expressed as brightness temperature, from a mixed water/land pixel decreases as the proportion of water increases (Brakenridge et al., 2007). Because the objective is to measure such water/land proportion changes, precise and sustained geolocation accuracy and the measurement precision obtained by the sensor are more important than the spatial resolution.
Nevertheless, sensors with a larger ground "footprint" must provide higher measurement precision to record the same percent surface water area change. Also, the SGR must be large enough to accommodate large floods. For example, Figure 2 illustrates a 10 km SGR that is nearly fully inundated during flooding; it is barely large enough. In this regard, much of the engineering invested in these satellite platforms and multi-frequency radiometers has focused on geolocation accuracy, high sensor dynamic range, multi-resolution capability at different frequencies, and maintaining well-calibrated results over long time periods. The PMR approach makes maximum use of these capabilities.

The Spatial Ratio Approach for Flow Area Retrieval
Environmental factors other than water area affect bulk microwave brightness temperature emission, and these may impact the a signal unless somehow removed. This need is addressed by employing a spatial ratio approach as follows (Equations 3-5): where T b is brightness temperature measured by a passive microwave radiometer and is related to the physical temperature T and the emissivity e of the surface.
A lower brightness temperature is observed over a sensor resolution element containing river water bodies, T b (m), as compared to nearby footprints on land without river surface water, T b (c). Under a constant physical temperature, T b (m) decreases over the river parcel when the water area expands. Nevertheless, microwave radiation is influenced by many factors including physical temperature (T), polarization (P), radiometer inclination angle (θ), effective permittivity ( Thus, surface water change is conveyed by the emissivity as controlled primarily by the effective permittivity over the SGR (Brakenridge et al., 2007), whereas other factors such as soil moisture, surface roughness, vegetation cover, and atmospheric conditions affect the overall brightness temperature of both measurement and calibration targets. Regarding vegetation cover, a dense forest in a riparian environment can affect T b at high frequency such as Ka-band, whereas the lower frequency data at L-band are known to less affected. Also, the L-band signal is even less affected by atmospheric conditions compared to that from the Ka-band. Regarding radiometer observation angle (θ), it is constant for all radiometers with a conical scanning configuration (e.g., Ka-band radiometers and L-band SMAP). The SMOS satellite θ is variable, and data must be selected within a small range of θ to avoid complications in the PMR algorithm (Kugler et al., 2019).
According to Equation (3), T needs to be canceled out to obtain e that conveys information relevant to Q retrieval. This is achieved by dividing the measurement value, T b (m) obtained over the river, by the calibration observation, T b (c), unaffected by water surface. The calibration location can be chosen so that T is similar at both locations due to the long correlation length of regional temperature variability, at least in low relief terrains. This is a critical choice, however: such large land areas could themselves include changing surface water areas which could affect the ratio values.
In summary, a water flow area signal can be defined (Equation 5): As a expands within the SGR, emission falls, and the M/C ratio decreases also. Figure 4 demonstrates this approach for the SGR shown in Figures 2 and 3 using both L-band SMAP data and Ka-band GFDS-RW data. Also, sample MODIS images for January-August 2017 (low water to annual maximum flood) are provided for comparison.
In Figure 4, the Ka-band ratio is calculated by the Global Flood Detection System, GFDS (De Groeve et al., 2015) and it is compared to SMAP L-band data processed by us. The GFDS system obtains the ratio by, each day, selecting the "driest", highest radiance calibration pixel in a 9 × 9 pixel array surrounding the measurement SGR. In contrast, selection of the L-band C target was manually accomplished, and is guided by the need to: (a) locate C without any substantial overlap with M, (b) locate C over land that may contain small streams, but no major river or wetlands that could substantially affect its use as "background" microwave emission (Figures 2 and 5). Qualitatively, the two Ka-band and L-band time series in Figure 4 are strongly correlated but exhibit several differences. Thus: (a) the SMAP M/C signal range (when using H polarizations for both M and C) is less than the GFDS-RW M/C range, as expected given the larger footprint of the SMAP SGR ( Figure 2). (b) There are more frequent and intense short-lived fluctuations in the GFDS-RW signal as compared to the SMAP information, although both time series jointly record many peaks and troughs. See for example, the summer of 2018. Finally, (c) both data sets capture two flood events during the summer of 2017, but the GFDS-RW data show the later peak (8/18-8/22) to be more intense than the earlier one. The MODIS information in Figure 4 indicates that a was indeed greater in the second flood wave.

Comparison of Ka-Band and L-Band Microwave Data
Using this flow area signal, Ka-and L-band microwave data can be evaluated for use in the PMR method and as validated by ground gauging stations. The SMAP L-band data are obtained from ascending/descending and H and V polarization brightness temperature information (Table 1) by selecting observations nearest to the SGR target (M) or the calibration target C (see below): with all distances from the target <4.5 km. The SMAP spatial resolution and footprint is approximately 36 km, but a "blurriness band" extends from the core area to 45 km diameter due to one-way antenna gain. Nevertheless, the brightness temperature measurement is dominated by the    There is no fixed calibration target in the processing system used: instead, the calibration pixel C is chosen as the 95th percentile of the pixels in a grid of 9 × 9 pixels centered on the measurement pixel M. Calculation of the 95th percentile excludes outliers due to measurement error but still obtains a dry land comparison. From 1998 to present, the 36-37 GHz Ka-band sensors used by GFDS are: TRMM/ GMI, AMSR-E, AMSR-2, and GPM/GMI (see Table 1). The M/C signal is calculated by the GFDS processor and published online. Figure 4 illustrates this signal as provided by the GFDS "merge" product (De Groeve et al., 2015).
At GFDS, all values are obtained from level 1 swath satellite data, and then projected into the global grid. When multiple samples for one pixel are available in 1 day, an average of all samples for the day is computed, for both M and C, and the published ratios are calculated from these averages. Satellite observations are processed in real time as soon as they are available: lag times vary for different satellites from 3 hr (AMSR-2) to up to 24 hr (GPM/ GMI). The system thereby provides water surface metrics with daily or near-daily frequency across the world. As suggested by its name, the GFDS was designed primarily for automated flood detection ( The two PMR M/C time series in Figure 4 are independent. They are based on different sensors, orbits, microwave frequencies, polarizations, averaging procedures, and different methods of obtaining the C signal. Their general agreement highlights the overall consistency of the PMR method using the spatial ratio approach. However, comparisons to in situ discharge time series are necessary to identify which signal is the most accurate river discharge indicator.

Ka-Band Signal Comparison to Ground-Measured Discharge
We first investigate these data for a relatively small meandering river in the central U.S. and by comparison to data from a co-located USGS ground gauging station. Figure 5 shows the location of SGR 91 along the Wabash River near New Harmony, Indiana, USA and the L-band and Ka-band SGRs. The cumulative water area variation tracked by MODIS optical sensing is also illustrated. Figure 6 shows detailed views of a portion of the river reach within SGR 91 at high and low flow states. The flow area variation is evident. The Ka-band signal extends from January 1998 to the present; the SMAP L-band information only from April 2015-May 2021. Left shows the river during low flow, right shows a much higher flow. Forest-covered floodplain surfaces bound the river and are flooded on the right; also, a meander bend is re-occupied by the river and becomes a major flow route.

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We compute 4-day forward running means to provide a consistent time series updating each day without any gaps. The averaging slightly lowers temporal resolution compared to that obtained by the ground gauging station; it also slightly offsets the timing of flood peaks. We also retrieved the published USGS Q information for 2011-2021. They are based on river stage values (Figure 1) as per standard USGS methods. For comparison with the microwave information, 4-day forward means were also calculated from the gauging station information.
Figure 7 provides samples of the two independent time series of (4-day averaged) daily Q. The vertical axes for the daily Ka-band M/C ratio and Q are both arithmetic. Thus, it appears that the two series maintain an approximate linear relation to each other through 11 years of time. Figure 8, top, tests this hypothesis by a scatter plot and a linear least-squares regression. Any offset of timing in flood peaks and troughs will increase the scatter. The resulting equation (illustrated) serves as an approximate a/Q rating which can allow daily Q to be computed from the signal data. Note that for M/C values decreasing from 0.96 to 0.86, station-measured Q increases from 2,000  low flow (top plot in Figure 7). However, as flow increases, M/C remains higher than 0.97 instead of immediately tracking the station-monitored rise toward a peak at 4,000 m 3 /s (bottom plot in Figure 7). These errors are per expectation: the flow surface area changes during low flow are at the limit of the methodology to detect for this river reach, but flow is accurately determined to be within the low flow regime. As well, possibly the type of rainfall producing the rise from low flow conditions may determine whether M/C always accurately tracks rising Q. Thus, heavy and sustained local rains may also produce somewhat more surface water locally in the C parcel ("pluvial" flooding) and dampen the ratio response.
Through use of the M/C microwave signal, and the rating equation produced from Figure 8 (top) scatter plot, monthly runoff can also be computed for the SGR and compared to that observed by the co-located ground station (Figure 8, middle). Monthly runoff observed by the station and by PMR data is even more strongly correlated than was the case for "daily" information. Figure 8, bottom, compares these results in another scatter plot.
In this case, a second order polynomial regression applied to the satellite data predicts monthly runoff with a Nash-Sutcliffe efficiency (NSE) statistic of 0.81 ("Very Good") (Moriasi et al., 2007;Yilmaz & Onoz, 2020); and as compared to the ground station gauge.
The time series comparisons also indicate the specific intervals where anomalies occur and errors are introduced. Thus, for 2011-2013 (Figure 7), q is tracked remarkably well, save for false predicted Q peaks in December 2012 and 2013: these low M/C anomalies contribute to the scatter in Figure 8, top. Comparison to MODIS optical images indicates these peaks to be related to winter storm events which caused significant snowfall that soon afterward melted. For these intervals, both the SGR pixel (Tb(M) and the comparison values (Tb(c)) decline sharply in tandem, but the SGR decline was greater, and the M/C ratio dropped, even though the gaging station shows no increased Q. Similar false peaks occur in February 2014 and 2021 (not shown), for a total of four occurrences in 10 years. In each case, visible snow cover was established, but was transient and melted after a few days. These changes may have temporarily affected the M/C signal. Thus, if differential snow cover or snow melt consistently disturbs the results, frequent repeat optical sensors such as MODIS and VIIRS could flag those intervals for the affected PRM gauge observations and thereby account for this source of error.
In summary, determination of Q and runoff from the PMR approach with GFDS-RW Ka-band data can use a simple linear rating equation from a co-located ground station to calibrate the signal to Q units. A comparison of the two independent time series (remote sensing vs. station Q) in Figure 7 shows extended periods of strong correlation and good characterization of flood hydrographs. Figure 8, top, also shows this correlation of remote sensing (flow area) signal to Q, but with significant scatter. Useful daily information is retrieved but within broad error limits. The high/medium/low flow status of the river is generally very well constrained. Thus, a M/C ratio of 0.9 almost always indicates Q of between 3,500 and 5,500 m 3 /s (only several outliers out of 4,015 days). A ratio of 0.96 indicates Q is between 900 and 3,000 m 3 /s, again with only several outliers. Between 0.96 and 0.99, the upper limit of Q is well-constrained (<3,000 m 3 /s at 0.96 and <1,800 m 3 /s at 0.99), but the flow may be near this constrained maximum value or as low as only a few hundred m 3 /s. As a result, and for operational applications, publishing upper and lower limits from such satellite information could be a useful approach.

L-Band Signal Comparison to Ground-Measured Discharge
SMAP L-band information is now evaluated for monitoring changes in Q at the same SGR. See Figure 5 for the SMAP footprint. The larger SMAP SGR suggests that the M/C signal could be less sensitive to discharge: a 36 km target pixel covers a 10 times larger area than the GFDS pixel but the total river a at bankfull flow increases only ∼3.6 times. For the polarimetric SMAP L-band information, we can also test the sensitivity of different We expect H polarization brightness to be more affected by surface water proportion than V, and thus hypothesize that M(H)/C(V) may provide the most stable indicator of Q. In situ Q information is used to test this hypothesis and also to compare to the Ka-band results (Figure 9).
A 900-day interval from 1/1/2017 to 12/31/2019 was chosen to avoid gaps in the time series for both L-band and Ka-band. Four day forward-moving averaging was applied to all time series. As expected for this relatively small river, the Ka-band data provides the highest correlation to ground-measured Q (linear equation; R 2 = 0.75, Figure 9, top). However: the SMAP M(H)/C(V) signal also tracks Q with a R 2 of 0.68 (Figure 9, middle). In this case, the scatter plot indicates a non-linear monotonic relation. A third-order best-fit polynomial regression provides the rating equation used to calculate Q. In contrast, a linear best-fit provides R 2 of 0.58 (not shown); if applied, it would underestimate flood peak Q values. Finally, the bottom plots in Figure 9 show the results for M(H)/C(H): a linear regression produces R 2 of 0.56 (not shown) and a second-order polynomial regression a R 2 of 0.61 (bottom left, used to calculate Q, bottom right). Also, M(V)/C(V) correlates to Q with R 2 of 0.68 using a second-order polynomial fit (not shown).
The signal ranges also vary. SMAP M(H)/C(H) extends over only ∼0.08, whereas M(H)/C(V) ratios extend over ∼0.17. The Ka-band signal range is ∼0.10 ( Figure 9). The best choice of SMAP polarizations to use for the M/C ratio appears to be M(H)/C(V), which agrees with the known greater response of H polarization to surface water changes (Du et al., 2018). Most importantly, and despite the relatively small size of this river, it is clear that the SMAP L-band information can be used to monitor discharge at this SGR. Thus, provided adequate means can be found to calibrate the remote sensing signal to Q, the PMR method could very likely be applied using SMAP information to at least several thousand SGRs globally.

Rating Curves Using Ground Station Data
Like s, a variations are a useful indicator of Q changes. The above examples indicate how closely PMR-observed a does indeed track Q. However, a ground station was used to accomplish the needed calibration to Q units. Note that, at many locations where river monitoring may be desired, ground station data may be absent. Also, there is a range of possible calibrations to Q: from "not performed at all", to "calibrated via hydrologic models", to "calibrated by use of co-located ground station information." The last would be the most accurate approach: if nearby gauging stations monitor the river and if there is trust in their Q records.
If the periods of record of remote sensing and gauging station data do not overlap, the remote sensing information can still be calibrated to Q by comparison of flow duration curves. This is an approach commonly used to calibrate hydrologic models with non-overlapping ground-based data (Westerberg et al., 2011). Where the discharge information has been terminated, the PMR information, if strongly correlated, can effectively bring those stations back to life and extend their Q time series to the present and into the future (as PMR satellite data acquisitions continue). As is the case for model-based calibration, however (see below), there may be included bias: the older Q time series used to calculate the flow duration curves may be systematically offset compared to current conditions.
Even without calibration to Q, PMR flow series are still very useful hydrological information. Thus, there are many stage-only ground gauging stations in most nations, and these provide long time series valuable for analysis of seasonal to interdecadal changes (Elfatih et al., 1999), as well as information used in operational water management. PMR data can also be directly compared to stage information and, going forward, thereby used as an indicator of stage.
Calibration to Q remains highly desirable where it can be accomplished. Even if only a very few accurate ground-based Q measurements (e.g., at high, medium, and low flow conditions) are obtained during the period of microwave observation, these can nevertheless be used to develop approximate rating equations and allow the remote sensing to be converted to Q and r. However, there is no basis to assume a linear PMR signal/Q relation in general (though such may occur at some locations). Rating equations are purely empirical in character, and respond to the exceptionally varied channel and floodplain morphologies that global rivers exhibit. As a result, it is essential to obtain Q data over the widest possible range of flow conditions, so that the shape of the complete rating curve can be constrained without assuming linearity or single monotonic exponential functions.

Rating Curves Using Model Results
Hydrological modeling can also provide the information needed for Q/a rating curves. Such models are driven by precipitation and other meteorological observations, by topography, the drainage network topology, and other ancillary data. They produce times series of modeled Q all along the modeled networks: including at SGR locations. For example, global Water Balance Model (WBM) runs have been accomplished for 10 km SGRs at daily (4-day averaged) time steps for an arbitrary 5 years sample period (2003)(2004)(2005)(2006)(2007). The output can be compared to the coeval time series from GFDS-RW SGRs for rating curve construction (Brakenridge et al., 2012).
WBM is one of a family of such models (Sood & Smakhtin, 2015). It is a water balance/transport model with a water budgeting scheme that takes into account climate forcings and estimates various water stocks (soil moisture and groundwater) and fluxes (evapotranspiration, surface runoff, groundwater recharge and baseflow). WBM has been applied successfully in small watersheds at 200 m spatial resolution up to a global scale at 6 min grid (11 km at the equator) cell sizes. See (Cohen et al., 2011) for details regarding flow routing, precipitation products used, topography, and other parameters and variables. We use previously obtained model output here to illustrate the potential for PMR calibration to Q. Figure 10 provides a scatter plot and time series plot comparing monthly maximum, mean and minimum daily Q model output (5 years, 180 data pairs) to the matching M/C values provided by the GFDS: for SGR 26 along the Ayeyarwady River in Myanmar. Linear regression (R 2 = 0.67) provides a first-approximation rating curve.
This model-based approach can transform the remote sensing signal values to Q units at locations where model output is strongly correlated to the M/C time series. However, it requires confidence in the WBM results for the SGRs. Thus, the model might accurately calculate the seasonal and inter-annual flow changes (see time series in Figure 10) but with a significant positive or negative bias: the model-to-remote sensing agreement would remain high, but the calculated Q values would all be in error by the bias. Therefore, it is still desirable to obtain ground-based data to validate PMR results wherever possible. Also, WBM is only one of a family of applicable hydrological models: others may be more suitable for specific basins and supply the information needed for the construction of rating curves (Hou et al., 2018).

Factors Affecting Accuracy of PMR Discharge
Whether calibrated by independent model or ground station information, PMR has the potential to provide: (a) near real time (contemporary) information about river discharge, Q, and watershed runoff, r, (b) daily time series extending back several decades (for Ka-band) and 7-13 years (for SMAP and SMOS L-band), and (c) immediate evaluation of contemporary flow status as compared to the time series. The challenge is to evolve the method to a trusted technology, applicable globally and for a variety of uses. We consider that, in general, if measurements from a PMR SGR can consistently (over years) measure daily Q to within ±20% error range, then they can rival or exceed the accuracy of many in situ gauging stations (see text above). Due to issues previously described, it is likely that at some SGR locations this criterion can be met, and at others it cannot be: SGR-specific characteristics are critical, as is the case also at traditional ground gauging stations.
In this regard, end users have varying needs. Some require highly accurate Q and r, each day; others need long time series, so that statistics such as flood recurrence intervals can be computed. Thus, constraining the size of the "50-year flood" (2% annual exceedance probability) requires an annual flood time series of at least 25 years (IACWD, 1982), which existing PMR time series can provide, as well as a less accurate estimate of the 100-year (1%) flood. Other uses of river flow data include: irrigated agriculture, urban water supplies, evaluations of drought severity, transboundary water issues, hydropower siting, lock and dam operations, water quality and sewage permitting, riverine ecology management, flood risk (Slater et al., 2021), and flood alerts. Here we focus on several sources of uncertainty and error within the PMR method that may affect most uses, and consider whether a ±20% daily Q accuracy limit can be routinely obtained at suitable SGRs.

Effects of Rating Curve Selection
In all cases, if translation to Q is accomplished, then the rating curve used may be an important source of error. We further examine data for SGR# 26 and 91 as two examples along quite different rivers.
For SGR# 26, a Global Runoff Data Center-cataloged (GRDC, 2022) ground station (Katha) is located 40 km downstream and its measurements can be used to develop a station-based rating for comparison to the model-based rating provided above. This daily ground station record extends from 1998 to 2010 and does not overlap with the L-band time series; only the Ka-band information is available for comparison. Simple linear regression for the 1999-2010 GFDS-RW M/C daily satellite information (with some data gaps in 1998) versus daily Q for this station yields: −231,636x + 227,101, R 2 = 0.59, n = 4,380 using linear regression. . This is a decrease of 25%. However, the contributing watershed of the (downstream) gauging station is somewhat larger: 77,942 km 2 compared to 74,436 km 2 for SGR 26, or 5% larger. These results suggest that there may be negative model bias at this location, but that it is less than 25%.
Note that the previous analysis (and see Figure 11) illustrates how the use of quadratic or higher-order polynomials compared to linear equations may improve the calculated Q and r results. In general, the M/C signal, though constraining low flow states very well in time, is not expected to robustly record flow variation during low flow. Once flow is nearly bank-full, however, linear equations appear more suitable. By comparison, rating curves in the U.S. for ground gauging stations have historically used log/log plots (stage vs. Q); the final rating may include several "linear" (actually, exponential) relations that change, at the inflection points often visible in such plots. This emphasizes the empirical nature of rating equations, which are applied to natural river and floodplain reach morphologies. Monotonic relations exhibiting the best possible fit are the desired objective. Other model structures could in principal provide a more flexible approach, including providing a low flow threshold, and as based on better understanding of the relation of flow area to Q along rivers of differing morphologies.
For SGR# 91, we again compare WBM model-based rating curves to those using the local ground station. This is a much smaller river, without the monsoon-related seasonal hydrographs along the Ayeyarwady River that are reproduced well by the modeling. We compare possible rating equations using Ka-band microwave data (see Figures 9 and 11). For the WBM-based rating, a steeper linear equation results than for the station-based curve: −73,612.11x + 73,332.21, R 2 = 0.38, compared to −46,630x + 46,969, R 2 = 0.75. In this case, there appears to be significant positive model bias. The model-based rating yields Q at M/C = 0.94 of 4,137 m 3 /s whereas the station-based rating provides only 3,137 m 3 /s (−24%).
Although for this example, the WBM model less accurately tracks discharge than does the microwave information, the model-based rating, if used alone, still facilitates Q changes along this river to be accurately monitored by PMR except for the highest flow states (Figure 12). The two rating curves produce the most divergent results at the highest Q, where the WBM-calibrated Q at M/C = 0.91 is 6,345 m 3 /s compared to 4,536 m 3 /s (station-calibrated). In contrast, at the low flow M/C value of 0.97, both curves predict Q within 10% of each other. For most days in each of the 3 years in Figure 12, the calculated PMR Qs are within ±20% of each other.

Effects of SGR Location and Size
On the ground, river gauging station locations are commonly selected on a site-specific basis to achieve best performance. It is desirable for most of the Q variation to be accommodated by the stage changes. In contrast, for a useful PMR SGR, flow area must respond robustly to Q changes. The location of the SGR can be critical: performance of the method could be degraded or improved depending on its position and on the river morphology within the SGR (Hou et al., 2020).
In this regard, for the ∼10 km gridded Ka-band data considered here, a poor SGR location could completely fill with water during large floods: the pixel will saturate, and further increases in Q will not produce further in-pixel a changes in emission. To avoid this situation, a larger SGR could be selected: for example, a 2 × 2 array of 10 km pixels could be defined as the SGR and the signal averaged (Revilla-Romero et al., 2014). In contrast, for the much larger L-band SGRs shown in Figures 2 and 5, their precise location is less likely to affect the results as long as consistent geolocation is maintained over time. For these, a longer river reach and greater floodplain is being monitored and without the chance of saturation.
One method to evaluate the effects of SGR river and floodplain morphology on the results is to compare two SGRs along the same river close to but separated from each other ( Figure 13). In this example, for the 1-km wide middle Mississippi River, the WBM model was used to calibrate both the upstream (SGR# 507) and slightly downstream (SGR# 505) Ka-band information to discharge. Figure 14 provides a sample of the complete (1998-2022) time series of Q from each; the close correlation indicates that they are both accurately recording Mississippi River flow variation. Figure 15 is a scatter plot of the complete results from both SGRs.
We now compare also the SMAP information to the GFDS-RW data at the two SGRs. Figure 16 shows the SMAP footprint at SGR# 505; note that a footprint locational error of several km will have less effect on the results from these larger SGRs than for the 10 km ones. Figure 17 demonstrates that both SMAP SGR C/M signals correlate to the gauging station information, with similar R 2 values for the regressions.
The two close-by SGRs along the Mississippi independently record discharge changes at high temporal sampling and at both Ka-band and L-band frequencies. For this river, the exact location of a 10 km Ka-band SGR may affect result accuracy: SGR# 505 exhibits somewhat stronger correlation to station discharge than SGR# 507. However, the two L-band SGRs produce comparable results. The variety of possible sensing footprint sizes is an important advantage of the PMR method overall. For small rivers, the higher spatial resolution information may be desirable to maximize signal range, but greater care must be taken at appropriate location. For coarser spatial resolution L-band PMR, useful Q is clearly also obtainable and SGR location is less critical: 36 km of river and floodplain reach will be monitored.

Conclusion
PMR uses freely-available Ka-and L-band data from orbital sensors to observe, at near-daily intervals, the proportion of water area within defined SGRs. The SGRs are 10-36 km in length and width and can be individually located for maximum sensitivity to river water area changes. The dynamic water area proportion within a SGR tracks local river discharge (m 3 /s) and, thereby, upstream watershed runoff (mm). Presently, some Ka-band information used for this purpose is first processed into daily global gridded gauging signal products by the European Commission's Global Flood Detection System (GFDS), which merges data from the AMSR-2 and GMI sensors (for 1998-2002, from TRMM/TMI and AMSR-E) to achieve maximum geographic and temporal Figure 13. Left, the 10 km SGR 507 on the Middle Mississippi River near New Madrid, Illinois. Right, the 10 km SGR# 505, ∼20 km downstream of SGR# 507. The images are from Google Earth and each illustrate the complete SGR areas being monitored. Any major flooding would saturate these 10 km pixels. Figure 14. Sample of the two GFDS-RW time series of river discharge obtained at the two nearby but discrete SGRs along the Mississippi river for a period of 5 years at the start of the records in 1998. coverage (Table 1). Also, L-band information from the polarimetric NASA SMAP radiometer or from the European Space Agency SMOS satellite and other sensors currently being planned can be used (Table 1). For both Ka-and L-band, a flow area signal is calculated from the M/C ratio: this is the microwave brightness temperature of the SGR (M), through which the river flows, compared to the brightness temperature of the surrounding terrain outside of the river valley (C). This paper evaluated such data from both Ka-and L-band sources, and also compared it to nearby ground gauging stations and to model results.
As is the case for river stage measured on the ground, SGR flow area varies approximately monotonically as river discharge rises. On the ground, gauging station locations are chosen where flow is confined between stable banks so that stage is a robust indicator of discharge. For flow area approaches, the reaches are chosen where there is room for flow expansions: over point bars and channel islands, low floodplains adjacent to the channel, and at small tributary mouths. Along large floodplains in particular, negative relief features such tributaries, channel remnants (cutoff meanders), tie channels, accretionary swales, and lakes respond during periods of high flow as the local groundwater table (the hyporheic zone) rises. Thus, an important component of river discharge can be water originating from the perirheic zone (Lewin & Ashworth, 2014;Mertes, 1997) and this is observable via PMR. For multithread (braided), anastomosing, or meandering rivers, observation of flow area may be a more practical approach for discharge measurement than stage-based methods at a single fixed flow cross section.
The microwave information was evaluated for contrasting examples: the ∼2 km-wide Ayeyarwady River in Myanmar, the 1-km wide Mississippi River, and the 0.4 km-wide Wabash River. Also considered was the construction of the rating curves needed to translate flow area changes to discharge variation. Global-scale hydrological modeling can provide the needed information for rating curves without any nearby gauging station information. Such models produce times series of modeled Q all along the modeled networks; we used WBM global model runs accomplished for SGR locations at daily (4-day averaged) time steps for 5 years. For these test SGRs, ground gauging station data were also used to construct and compare rating curves. Model bias can significantly affect the discharge results for these SGRs if only model-based ratings are used. Depending on time interval chosen, the simple regression R 2 with M/C as predictor and Q as dependent variable varied from 0.63 to 0.81; for SMAP, the highest values were obtained using M(H)/C(V) polarization information; this configuration appears to have the highest accuracy.
Both microwave bands yield useful daily discharge and monthly runoff results at the tested locations. Ka-band information allows more precise location of SGR measurement sites; L-band information provides for monitoring  of longer river reaches and larger rivers and the results are less sensitive to exact SGR location. An appropriate metric for determining how accurately the PMR method can measure Q and r is the correlation coefficient R 2 value (linear or polynomial regressions): for SGRs that are near ground gauging stations with overlapping periods of record. If the microwave signal is first transformed to Q using model-based rating equations, then the Nash-Sutcliffe statistic offers the advantage of also evaluating any bias introduced by model results. Other statistics, here not explored, could also be useful to evaluate calibration using flow duration methods and non-overlapping time series.

Data Availability Statement
The United States Geological Survey (USGS) stream gauging information for the Wabash River station shown in Figures 1, 7 and 14, and the Mississippi River station data shown in Figure 19 are for: USGS 03778500 Wabash River at New Harmony, IN and USGS 07024175 Mississippi River at New Madrid, MO, respectively. The data are available at (sample access): https://waterdata.usgs.gov/nwis/dv?cb_00060=on&format=rdb&site_ no=03378500&referred_module=sw&period=&begin_date=2021-05-26&end_date=2022-05-26. Use desired time span. The GFDS-RW Ka-band data in Figures 4 and 7-9 and 11 and 12 and 14 and 15 are from the Global Flood Detection System online at (GDACS). The GFDS data used are the "merge" product. Sample access for particular SGR sites for these data is: https://www.gdacs.org/floodmerge/data.aspx?from=20200101&to=20241231&-type=html&alertlevel=green&datatype=1dAY&areaid=91. This is for years 2020-2024; change dates for different time periods. The SMAP L-band information and data used in Figures 4, 9 and 17 are available from the NASA National Snow and Ice Data Center Distributed Active Archive Center (NSIDC DAAC) at https://nsidc. org/data/spl1ctb_e/versions/1. The WBM model results for individual SGR sites in Figures 12 and 13) are available as "91calib.xls" files (substitute appropriate site ID number) at: https://floodobservatory.colorado.edu/SatelliteGaugingSites/.