1. Introduction
Rivers play an important role in the global hydrological cycle, in connecting the atmospheric, terrestrial, and marine processes of the water cycle, and in transporting about two-fifths of the global total rainfall over land back to the ocean [
1,
2,
3]. In this process, rivers provide water resources for agricultural irrigation, human life, ecosystems, shipping, hydropower, etc., which are closely related to the survival and development of human beings, animals, and plants, but may also bring water supply conflicts and major flood disasters to upstream and downstream countries or regions. In these contexts, hydrological data, especially river-discharge information, as important data for monitoring the state of rivers, is crucial to the study of the terrestrial branch of the global water cycle, the changing patterns of freshwater availability that affect large populations, the improvement of hydrological models, and flood-events monitoring.
At present, the river discharge is generally obtained by measuring the volume of water passing through a certain point in unit time at the hydrometric station (also known as a hydrological station, which monitors the water level, velocity and flow direction, etc.). Since the discharge is the product of the flow area and mean velocity, the field measurement of river discharge requires the direct measurement of the vertical-velocity profile using a current meter on the cross-section perpendicular to the flow direction of the river [
4,
5]. Although the field measurement of the discharge may be very accurate, this collection method limits the temporal and spatial resolution of the discharge dataset, that is, data collection is sparse, time-consuming, and discontinuous. Therefore, obtaining the discharge data with a higher spatial and temporal resolution depends on expressing the discharge as a function of the measurements that are easier to obtain. For example, establishing a functional relationship between the water level and discharge (i.e., rating curve) to monitor the discharge through the water level. Such monitoring is usually achieved by using a river-water-level gauge, which draws a “rating curve” using the river water level (higher than an arbitrary datum) and the discharge simultaneously measured at regular intervals [
6]. After the rating curve is drawn, approximately continuous monitoring of the discharge can be achieved through the water level obtained by a pressure sensor. Although hydrometric stations are important for river-discharge monitoring, their high construction and management costs, sparse and uneven spatial distribution, as well as the barriers to data sharing among countries have led to our limited understanding of global river discharge. Given that global warming may accelerate the water cycle, the lack of discharge information has become an increasingly acute problem [
1,
7]. Whether it is to develop a process-based understanding of runoff at large spatial scales (i.e., how water flows into and through rivers), or to calibrate and constrain hydrologic models to predict the impact of future discharge changes on the terrestrial hydrologic cycle, it is necessary to improve the temporal resolution of river discharge in a larger global spatial coverage [
1].
In the absence of in situ measurements from hydrometric stations, remote-sensing technology (i.e., optical remote sensing, microwave remote sensing, and satellite altimetry) has become a new way to supplement discharge information [
8,
9,
10,
11,
12,
13,
14,
15]. However, the remote-sensing method cannot directly measure the river-discharge information, as it only provides the river widths or river-surface heights. Researchers have tried to obtain useful discharge estimation by combining the gauged measurements with remote-sensing data [
6,
15,
16,
17,
18]. In addition, the combination of remote-sensing data and hydrologic models (i.e., data assimilation) is also a common way to estimate river discharge [
9,
19,
20,
21,
22,
23]. However, the aforementioned methods cannot calculate river discharge independently without in situ measurements, prior information, or auxiliary data (e.g., hypothesis), let alone the number of hydrometric stations in the world continues to decline [
8,
24,
25]. Therefore, it is urgent to study a method of calculating river discharge only using remote-sensing data. In 2014, Gleason and Smith found that the AHG between the cross-sections of the same river had a correlation, which is called the AMHG [
26]. They also proposed a new method for estimating river discharge that does not require any prior knowledge, or gauged or auxiliary data, i.e., they established a power-law-function relationship between the width and discharge solely using river-width data. Subsequently, several scholars verified this AMHG-discharge-retrieval method in different watersheds around the world [
1,
27,
28,
29,
30,
31], proving that this method can be used to estimate river discharge in the absence of gauged measurements, but also finding that the accuracy of the river discharge obtained by this method is relatively low.
Many researchers have also pointed out in their respective articles that the upcoming SWOT mission developed by the National Aeronautics and Space Administration (NASA) and the Centre National D’Etudes Spatiales (CNES) can simultaneously obtain river width, water-surface slope, and water-surface height, and associating these observations with existing discharge estimation methods can realize the calculation of the river discharge [
20,
21,
32,
33,
34,
35]. More importantly, solely using the SWOT measurements can complete the calculation of the river discharge [
1,
36], which can further reduce the dependence on the discharge calculation of in situ measurements. It can be predicted that the river-discharge information obtained by SWOT will bring great progress to global hydrological research. River widths obtained from SWOT can be used to calculate the river discharge based on AMHG. Some scholars have studied the accuracy of river-discharge estimated by SWOT measurements and the AMHG-discharge-retrieval method compared with other methods [
1,
26,
27,
29,
37]. The results showed that the discharge accuracy calculated by the AMHG and SWOT data is deficient, which may result from rivers not all having significant AMHG relationships [
38]. However, the AMHG-discharge-retrieval method has the advantage of simple and fast calculation compared with other methods [
1,
36,
39].
Helsel and Hirsch [
40] recommended using the Duan [
41] deviation adjustment factor Δ to adjust the estimated value of the AHG coefficient (i.e., the coefficient
a in Equation (1)) to improve the systematic bias of the discharge estimation. However, gauged discharge measurements are required to calculate the deviation-adjustment factor, and thus this method cannot be applied to areas without in-situ-measured data. Bonnema et al. [
36] corrected the parameters of the AMHG using the discharge and river width derived by the HEC-RAS model, making the accuracy of estimated discharge by AMHG is comparable with Metropolis Manning (MetroMan) and Mean Flow and Geography (MFG). However, the calculation of the discharge and river width by the HEC-RAS model needs to use SWOT measurements. Hagemann et al. [
38] presented a novel Bayesian AMHG–Manning (BAM) algorithm for discharge estimation from only remote-sensing data, implementing a Bayesian formulation of streamflow uncertainty using a combination of Manning’s equation and the AMHG. A dataset of simulated widths, slopes, and heights from 19 rivers was used to evaluate the algorithms using a set of performance metrics. Results across the 19 rivers indicated an improvement in the performance of the BAM method over previously tested methods and highlights a path forward in solving discharge estimation solely using satellite remote sensing. However, the BAM method requires a prior distribution of discharge and hydraulic equation parameters. Mengen et al. [
31] presented a novel decile thresholding method for estimating river discharge solely using the Sentinel-1 time-series data within an automated workflow. The results show that their novel approach is a significant improvement relative to the optimized AMHG method proposed by Gleason and Wang [
28], although the aforementioned improvements to the AMHG-discharge-retrieval method indeed weaken the systematic deviation between the estimated and gauged discharge. Among them, the improvement achieved by Bonnema et al. [
36] does not require the in-situ-measured data, but requires SWOT measurements and the HEC-RAS model, which is complicated. In addition, other improvements are realized by the correction of the AHG parameters, and still require in-situ measurements or some prior information. Moreover, there still exist large systematic deviations compared to the discharge estimated by Gleason and Smith [
26], Hagemann et al. [
38], and Mengen et al. [
31] with the gauged discharge, and the accuracy difference of estimated discharge in the dry season and wet season has not been improved yet. Therefore, it is necessary to study new AMHG-discharge-retrieval methods to further improve the accuracy of the estimated discharge.
In the research of Gleason and Smith [
26], Gleason et al. [
27], as well as Gleason and Wang [
28], in order to control the AHG coefficient
a and exponent
b (as shown in Equation (1)) given by the genetic algorithm (GA) within a reasonable range, they used the tolerance range of discharge. However, when the tolerance range is set too large, although the AHG coefficient
a and exponent
b of the cross-sections conform to the AMHG relationship of the river reach, they will deviate from the truth situation of the river reach, ultimately resulting in a deviation in the estimated discharge.
Therefore, this paper will carry out an accurate discharge calculation based on the SWOT-river-width data and the AMHG-discharge-retrieval method, proposing a CAMHG method to control the AHG parameters derived from the GA using an approximate prior discharge of the river reach, ensuring the AHG parameters are more in line with the truth situation of the river reach. It is verified in the reach of the Yangtze River, where the Hankou, Shashi, and Luoshan hydrometric stations are located, to prove the feasibility of the CAMHG-discharge-retrieval method proposed in this paper.
5. Conclusions
In this paper, the simulated SWOT-river-width data and the AMHG-discharge-retrieval approach are used to jointly calculate the discharge. It is found that the discharge estimated by the original AMHG-discharge-retrieval method has a large deviation, the RMSEs of the river discharge are greater than 8852.526 m3/s, and the RRMSEs of the Hankou, Shashi, and Luoshan reaches are 100.1%, 1137.1%, and 48.6%, respectively. In addition, there also exists a difference in the accuracy of the discharge estimation in the dry season and wet season, i.e., the RMSE of the estimated discharge in the dry season is three times higher than that of the wet season. Based on the results of the analysis of the original AMHG-discharge-retrieval method, a CAMHG-discharge-retrieval approach constrained by a prior discharge is proposed in this paper, which can obtain AHG parameters that are more consistent with the truth situation of the river reach, and then improve the accuracy of the discharge estimation.
Our results show that the CAMHG-discharge-retrieval approach can obtain a more accurate discharge estimation than that of the original AMHG method, and the RMSEs of the discharge estimated by the CAMHG-discharge-retrieval approach are reduced to more than half of the original one, except for the Luoshan reach. Compared with previous research, the CAMHG-discharge-retrieval approach not only significantly improves the accuracy of the estimated discharge, but also reduces the difference between the accuracy of the discharge estimation in the dry season and the wet season, e.g., for the Hankou, the RMSE is 17967.503 m3/s for the wet season and 6183.772 m3/s for the dry season using the original AMHG-discharge-retrieval method; however, the RMSE is 6047.949 m3/s for the wet season and 3857.448 m3/s for the dry season using the CAMHG-discharge-retrieval approach.
In addition, Gleason and Wang mentioned that it is not recommended to use Equations (8) and (9) to estimate the AMHG relationship of the river reach. However, this paper found that the AMHG relationship calculated by Equations (8) and (9) was reasonable at three river reaches in the middle and lower reaches of the Yangtze River basin, and after using a priori discharge to constrain the AMHG-discharge-calculation process, a more accurate discharge estimation solely using remote-sensing data was achieved. Therefore, the CAMHG-discharge-retrieval approach proposed in this paper can further improve the applicability of the AMHG.
However, the applicability of the CAMHG-discharge-retrieval approach based on a constraint of prior discharge needs to be further evaluated globally. The detailed evaluation of the CAMHG-discharge-retrieval approach in different watersheds and geographical conditions is our future research goal.
In a word, based on the analysis of the shortcomings of the original AMHG-discharge-retrieval method, this paper proposes a remarkable CAMHG-discharge-retrieval approach constrained by prior discharge, which improves the application potential of the AMHG and can serve as the SWOT discharge calculation in the near future.