Observing Rivers With Varying Spatial Scales

Abstract The National Aeronautics and Space Administration/Centre national d’études spatiales Surface Water and Ocean Topography (SWOT) mission will estimate global river discharge using remote sensing. Synoptic remote sensing data extend in situ point measurements but, at any given point, are 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 quantities 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 coefficient and leads to an increased “effective” friction coefficient. An exact expression is derived for the increase in the effective friction coefficient, 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 that the approximated variability index is most faithful when the river parameters obey lognormal statistics. The effective friction coefficient, which can vary from a few percent to more than 50% of the point friction coefficient, is proportional to the riverbed elevation variance and inversely proportional to the depth. This has significant implications for estimating discharge from SWOT data.


Introduction
In this supplement, we present mathematical details for the results presented in the paper.
In this section, we give detailed derivations of the various forms of the Saint-Venant equations used in the text. The conventional writing of the momentum equation is in terms of the discharge (see, e.g., (Chaudhry, 2008), equation 12-15): where Q is the discharge; A is the wetted cross section; U is the flow velocity; H is the flow depth; S 0 = −∂ x Z 0 is the bed slope, and Z 0 is the bed elevation; S f is the friction slope; q is the lateral inflow and u x is the downstream component of the lateral inflow velocity. In general, one can ignore the lateral inflow momentum (i.e., u x ≈ 0), even if the lateral inflow to the mass conservation equation is not negligible, and we will do so here.
Following Ponce and Simons (1977), and subsequent authors, one obtains the kinematic wave approximation if only the bed slope, term I, is retained. Retaining the bed slope and pressure (terms I and II) results in the diffusive wave approximation. Term III represents the loss of kinetic energy downstream, and adding it to terms I and II results in the the steady dynamic wave. Finally, keeping all the terms results in the dynamic wave.
Neither Z 0 nor H can be measured easily using remote sensing, but the water surface elevation, h = Z 0 + H, can be measured using either radar interferometry (e.g., SWOT) or altimetry (lidar or radar). Combining terms I and II, the momentum equation can be written in a form more amenable to remote sensing observations: The two terms in the final parenthesis can be rewritten as follows where we used the continuity equation, eq. (2) in the paper, in the middle step. Adding these two terms, the term proportional to ∂ t A cancels out, and the resulting equations is given by equation (4) in the paper.
We define p(x, t), the reach-average of a hydraulic variable p(x), as the result of the convolving p(x) with smoothing kernel, f (x): To conserve constant values and have an associated reach scale, L R , we assume averaging (the factor of 1/2 in the last equation is chosen so that L R is the reach length for uniform weighting). The hydraulic variable p can then be decomposed into reach-averaged the variability occurring at scales smaller than L R . By construction, one will have that δp(x, t) = 0; i.e., the small scale variations are zero-mean over the reach length. From the properties of the convolution, reach-averaging and differentiation commute Therefore, if p is differentiable, p is also differentiable, and the reach-averaged Saint-Venant equations are well defined.
In prior studies, (Garambois & Monnier, 2015) and (Durand et al., 2014;Yoon et al., 2016;Durand et al., 2016) assumed uniform weighting for a reach defined between x u and x d , the upstream and downstream coordinates, respectively, and the weighting function, f U , was given by In this case, the reach-averaged steady gradually varying flow term can be integrated Although uniform weighting agrees with the conventional meaning of reach averaging and handles control points well, smoothing kernels with better spectral properties may be preferable for modeling purposes, since the spectral leakage of the uniform window X -4 RODRIGUEZ ET AL.: MULTI-SCALE RIVER OBSERVATIONS is high. The discussion in the paper applies to general smoothing kernels, including the uniform kernel.
In this section, we provide the detailed derivation of the steps from equation (15) to equation (24) in the paper.
As in the paper, we start with where p i is the ith river hydraulic parameter, and α i is the corresponding exponent, and we take the friction parameter, ρ, to be the i = N p parameter.
Taking the logarithm of both sides and using log( i z i ) = i ln z i results in where we have used ln(p α i i ) = α i ln(p i ). Reach averaging both sides and using the results in S2, yields Because of the nonlinearity of the logarithm, ln p i = ln p i in general, unless p i is constant.
In fact, using Jensen's inequality (Jensen, 1906), one has that ln p i ≤ ln p i . We characterize this difference through a positive indefinite variability index κ i defined by The left-hand side of equation (16)can then be expanded as X -5 where κ Q , the discharge variability index, is defined analogously to κ i . The right-hand side of equation (16) becomes Equating left and right-hand sides, moving the κ Q term to the right hand side, one gets where we have used the fact that, by convention, ρ is the last parameter and α ρ = −1.
Gathering the last three logarithms Using this equation, one can write the reach-averaged discharge equation as and we define the total variability index as κ T . Taking the exponential of equation (16), which is equation (24) in the paper, as was desired.
Starting with the hyperplane equation for the parameters derived in Section (4.3) RODRIGUEZ ET AL.: MULTI-SCALE RIVER OBSERVATIONS one can obtain a set of N p equations for the variabilities and co-variabilities of the log hydraulic parameters by multiplying by η j and reach-averaging There will be N p variances, and N p (N p − 1)/2 covariabilities, so it is possible to use equation (26) to express the co-variabilities in terms of the variances for 1 < N p ≤ 3. The case of N α = 1 is not consistent with the assumption of constant discharge. For N α > 3, the variances do not uniquely determine the co-variabilities, and there will be multiple solutions that conserve the discharge.
Since it is usually the variance of the parameters (rather than the log parameters) that is known, we use the weak fluctuation limit for equation (26) as an approximation In the case when N p = 2, the solution is 1 2 = − α 1 2 1 + α 2 2 2 α 1 + α 2 The case of N p = 3, leads to the set of equations which can be easily inverted.
We model a constant-width periodic riffle and pool bathymetry, Z 0 (x), using equation (30) in the text, which has bed slope, ∂ x Z 0 , given by equation (31). The bathymetry, illustrated in Figures (7) and (8) in the text, represents a riffle and pool sequence, of pe-