Theoretical basis for at-many-stations hydraulic geometry (AMHG)
Colin J Gleason, Department of Geography, University of California, Los Angeles
Jida Wang, Department of Geography, Kansas State University
Supporting Information
1. Datasets
Of the 50 river reaches used in this study, 34 are identical to those used by Gleason et al.
[2014]. These reaches are all ~10km long, mass conserved, and abut or contain a gauge. In these
data, w was measured via Landsat satellite products and Q given by the gauge, and full
description of these reaches is given by Gleason et al. Q and w data for the additional 16 river
reaches in this study were generated using hydrodynamic models, with model inputs specific to
each reach. Outputs from these models were generously provided by colleagues who maintain
them for the reaches in this study. An example of the modelling framework for each river
follows. In each case, flows are imposed at the top of the reach are propagated downstream over
measured bathymetry until steady state conditions develop following finite element analysis
using conservation laws. This stable model run then yields w and Q at given cross sections, and
this procedure was invoked for all modelled data. In some reaches, tributaries and outflows are
simulated as intersecting the main channel at specific locations. Reaches of the Po, Sacramento,
Garrone, Ganges, Seine, Ohio, Severn, St. Lawrence, Tanana, Kanawha, Mississippi, and
Wabash Rivers are included in these data, and reaches contain from 40 to 500 cross sections,
cover 50 to 400km, and typically include one year of daily output.
2. Calculation of intersection percentage index pint
We show in this letter than AMHG arises because individual AHG rating curves intersect
at or near the same point (Qc_w, wc) within the range of observed data. Now, we represent the
degree of AHG convergence in a standardized intersection percentage index (pint), defined as the
number of rating intersections within the range of observed Q data as a percentage of the number
of total rating intersections (including those on the AHG rating curves beyond the observed
range). A high pint value suggests that most AHG curves intersect one another within the range of
̅̅̅̅̅̅̅̅̅̅
observed data, thus yielding a strong chance for a strong AMHG and 𝑄𝑐 = 𝑚𝑜𝑑𝑒(log⁡
(𝑄𝑥 ))
̅̅̅̅̅̅̅̅̅
and⁡𝑤𝑐 = 𝑚𝑜𝑑𝑒(log⁡
(𝑤𝑥 )). Figure S1 shows that pint less than ~15% produces an AMHG r2 less
than ~0.80, suggesting that the degree of rating curve convergence within the range observed
data has a control on the linearity of AMHG. The definition of pint further suggests that as long as
the topological relations among AHG curves are unaltered, the pint value will be unaffected.
This property of pint is enormously useful for remote sensing of discharge in a mass
conserved reach. In practice, we do not know Q but have observed w from remote sensing.
Although we are unable to establish accurate AHG rating curves due to the unknown Q, we
require that w and Q in a cross section must follow a linear relation in log-log space following
Leopold and Maddock’s classical conception of AHG. In this log-log space, a linear re-scaling of
log(Q) will not change the topology of AHG curves: these synthetically scaled rating curves are
pulled or compressed in the x-direction but their relative position to one another does not change
as they share the same conserved Q. Therefore, for any arbitrary minimum and maximum
log(Q), we can solve 𝛼 and 𝛽 in this equation set:
{
𝛼 ∙ log(𝑤)𝑚𝑖𝑛 + 𝛽 = log⁡(𝑄)𝑚𝑖𝑛
𝛼 ∙ log(𝑤)𝑚𝑎𝑥 + 𝛽 = log⁡(𝑄)𝑚𝑎𝑥
(S1)
Then log(Q) can be re-scaled by using any observed w in this cross section:
log⁡(𝑄)𝑠𝑐𝑎𝑙𝑒𝑑 = 𝛼 ∙ log(𝑤)𝑎𝑛𝑦 + 𝛽
(S2)
Note that this rescaling requires the power-law form of AHG to hold in order to preserve
topology, and requires mass to be conserved. By doing such rescaling, we have created a 2-D
topological relation similar to that of the true AHG rating cuves solely from observed width
values from which we can estimate pint (Figure S2). pint is calculated as before: by taking the
percentage of rating intersections within the now synthetic Q range. Therefore, by using solely
remotely sensed widths, we are able to gain an a priori estimate of pint and the strength of
AMHG and therefore the likely success or failure of discharge estimation, making pint a critical
index for future remote sensing of river discharge.
3. Proposed a priori AMHG slope estimation range.
Gleason et al.’s [2014] AMHG-enabled discharge estimation method simultaneously
tunes a reach’s AMHG and local AHG curves to estimate discharge. To do so, a prior estimate of
AMHG slope is provided by the user, and AMHG intercept is calculated from this value. Since
this letter has shown AMHG slope and intercept to be functions of Qc_w and wc, future discharge
estimation should use these quantities as parameters. wc is easily calculated from remotely sensed
imagery, but Qc_w cannot be calculated from images alone, as we have shown that the previously
proposed proxy to do this is coincidental. In this letter, we therefore propose an AMHG slope
value of -0.3 be used for all rivers, which corresponds to a Qc_w of 2,150m3s-1.
Gleason et al.’s [2014] performed a thorough sensitivity analysis of the discharge
estimation algorithm, and found a parameter tolerance of 0.1 for both AMHG slope and intercept
to yield the most accurate discharge estimation across their 34 rivers. When coupled to our
proposed slope of -0.3, this gives Qc_w values ranging from 316 to 100,000m3s-1. Figure S3
shows this range imposed on a histogram of observed AMHG slopes, and includes 35/39 rivers
with strong AMHG. The 11 rivers to the right of the proposed range have a Qc_w greater than
100,000m3s-1, which in a strong AMHG would represent the spatial mode of time mean
discharge through the reach. Since this is unlikely for almost any river on earth, these rivers to
the right of the proposed range are poor candidates for discharge estimation as it is highly likely
AMHG is not observed here. This can be confirmed a priori, as pint is less than 15% for these
rivers, suggesting that discharge estimation should not be attempted. Therefore, our proposed
range should cover most natural rivers observable by current satellite technology with moderate
resolution, and heuristic optimization of both AMHG and local AHG has been shown to be
effective at solving for Q even with this large range [Gleason et al., 2014].
Supporting Figures
Figure S1. Relationship between AMHG r2 and observed intersection percentage index (pint) for
34 mass-conserved river reaches. Blue shows pint calculated using known Q, and red shows
estimated pint from observed w only
Figure S2. Comparison of observed and estimated intersection percentage index (pint) values
shows that synthetic stretching of rating curves results in stable topological relationships between
rating curves, as evidenced by the congruence between observed and estimated pint.
Figure S3. This figure shows a histogram of each river’s observed AMHG slope. Gleason et al.’s
[2014] discharge estimation algorithm simultaneously tunes cross sectional AHG parameters and
AMHG using heuristic optimization, with a recommended tolerance of +/- 0.1 for each variable.
We here propose an initial AMHG slope value of -0.3, and the green polygon indicates this value
with the previously proposed tolerance. This polygon includes 35 of the 50 rivers given here, but
11/15 rivers not covered by the proposed range are not suitable for discharge estimation, as they
have AMHG slopes greater than -0.2, where Qc_w is obviously outside the range of observed data,
and pint is less than 15%.
References
Gleason, C. J., L. C. Smith, and J. Lee (2014), Retrieval of river discharge solely from satellite
imagery and at-many-stations hydraulic geometry: Sensitivity to river form and optimization
parameters, Water Resour. Res., 50(12), 9604–9619, doi:10.1002/2014WR016109.