Observing Rivers with Varying Spatial Scales

The NASA/CNES Surface Water and Ocean Topography (SWOT) mission will estimate global river discharge using remote sensing. Synoptic remote sensing data extends in situ point measurements, but, at any given point, is generally less accurate. We address two questions: 1)What are the scales at which river dynamics can be observed, given spatial sampling and measurement noise characteristics? 2) Is there an equation whose variables are the averaged hydraulic antities obtained by remote sensing, and which describes the dynamics of spatially averaged rivers? We use calibrated hydraulic models to examine the power spectra of the different terms in the momentum equation, and conclude that the measurement of river slope sets the scale at which rivers can be observed. We introduce the reach-averaged Saint-Venant equations, that involve only observable hydraulic variations, and which parametrize within-reach variability with a variability index that multiplies the friction coeffcient and leads to an increased "effective" friction coeffcient. An exact expression is derived for the increase in the effective friction coeffcient, and we propose an approximation that requires only estimates of the hydraulic parameter variances. We validate the results using a large set of hydraulic models and find the approximated variability index is most faithful when the river parameters obey lognormal statistics. The effective friction coeffcient, which can vary from a few percent to more than 50% of the point friction coeffcient, is proportional to the river bed elevation variance and inversely proportional to the depth. This has significant implications for estimating discharge from SWOT data.美国国家航空航天局（NASA）与法国国家空间研究中心（CNES）合作的地表水与海洋地形（Surface Water and Ocean Topography，SWOT）任务，将通过遥感技术估算全球河流径流量。大尺度遥感数据能够拓展原位（in situ）点位测量的覆盖范围，但在任意单点上的测量精度通常更低。我们针对两个核心问题展开研究：1) 结合空间采样与测量噪声特征，可观测河流动力学的尺度是多少？2) 是否存在这样一个方程，其变量为遥感获取的平均水力参数，且能够描述空间平均后的河流动力学过程？我们采用校准后的水力模型，对动量方程中各分项的功率谱进行分析，最终得出结论：河流坡度的测量决定了可观测河流的尺度。我们提出了河段平均圣维南方程组（Saint-Venant equations），该方程组仅包含可观测的水力变化量，并通过一个变异性指数对河段内的变异性进行参数化：该指数与摩擦系数相乘，进而得到提升后的“有效”摩擦系数。我们推导了有效摩擦系数增量的精确表达式，并提出了一种仅需水力参数方差估计值的近似计算方法。我们通过大量水力模型对上述结果进行验证，发现当河流参数服从对数正态分布时，近似得到的变异性指数贴合度最高。有效摩擦系数可介于单点摩擦系数的百分之几至50%以上，其与河床高程方差呈正比，与水深呈反比。这一结论对基于SWOT数据估算河流径流量具有重要的指导意义。