Initial conditions for flows and depths (cross-sectional areas) throughout a river network are required for any time-marching (unsteady) solution of the one-dimensional (1-D) hydrodynamic Saint-Venant equations. For a river network modeled with several Strahler orders of tributaries, comprehensive and consistent synoptic data are typically lacking and synthetic starting conditions are needed. Because of underlying nonlinearity, poorly defined or inconsistent initial conditions can lead to convergence problems and long spin-up times in an unsteady solver. Two new approaches are defined and demonstrated herein for computing flows and cross-sectional areas (or depths). These methods can produce an initial condition data set that is consistent with modeled landscape runoff and river geometry boundary conditions at the initial time. These new methods are (1) the pseudo time-marching method (PTM) that iterates toward a steady-state initial condition using an unsteady Saint-Venant solver and (2) the steady-solution method (SSM) that makes use of graph theory for initial flow rates and solution of a steady-state 1-D momentum equation for the channel cross-sectional areas. The PTM is shown to be adequate for short river reaches but is significantly slower and has occasional non-convergent behavior for large river networks. The SSM approach is shown to provide a rapid solution of consistent initial conditions for both small and large networks, albeit with the requirement that additional code must be written rather than applying an existing unsteady Saint-Venant solver.

Setting initial conditions for unsteady simulations of the Saint-Venant
equations (SVEs) across large river networks can be challenging. Every element
of the river network must be given initial values of flow and depth, and
these values should be consistent with the inflow boundary conditions
(e.g., from a land surface model) at the starting time to prevent
instabilities. This issue has not been previously addressed in the
literature, arguably because (i) adequate initial conditions are fairly
trivial for small SVE reaches, (ii) hydrological models with large river
networks often do not use the full SVE

Saint-Venant equation modeling arguably dates from Preissmann's seminal work

Our experience with Saint-Venant river network modeling is that simple approaches to initial conditions often cause localized numerical instabilities, slow convergence of the time-marching numerical solution, and long model spin-up times. Herein, we investigate the initial condition problem for a Saint-Venant river network model for a given set of inflows from a hydrological model.

Model spin-up time is completed when the effects of initial conditions cannot be observed in the model results. Thus, by definition, the initial conditions cannot affect the unsteady solution beyond spin-up time. It follows that initial conditions are irrelevant to the quality of the time-marching simulation and are only important in how they affect the spin-up duration. We can imagine a “perfect” set of initial conditions with zero spin-up time, which would require initial flows and depths consistent with (i) the actual unsteady behavior prior to the model start time and (ii) the model boundary conditions; the latter includes both the bathymetric model for the river channels and the coupled hydrological model providing runoff and baseflows. Such perfect initial conditions are practically unattainable due to the sparsity of synoptic flow/depth data as well as unavoidable uncertainty and errors in both bathymetric and hydrological models. Another way to think of this is that perfect initial conditions also require perfect boundary conditions (perfect bathymetry and hydrology), or else some spin-up time is required to wash out inconsistencies. In general, the spin-up time will be affected by how far the initial conditions are from the theoretical perfect conditions.

The key point is that the exact observed river initial conditions (if such
were available throughout a network) will not eliminate or necessarily
reduce spin-up time if the observed data are inconsistent with the model
boundary conditions. Similarly, interpolations of sparse synoptic data will
not be a priori consistent with the boundary conditions and thus cannot
eliminate spin-up time. Inconsistency is a critical concept: the mismatches
between the initial conditions and the boundary conditions can lead to
unrealistic destabilizing impulses in time marching the SVE solution. Such
impulses can require extensive spin-up time to damp their effects. An extreme
example is a high runoff rate into an almost dry stream that can cause a
Gibbs phenomenon at a wave front and negative values for the computed
cross-sectional area

We argue that the primary goal of initial conditions is providing consistency with the boundary conditions to allow smooth, convergent spin-up of an unsteady solver. This task can be accomplished with synthetic initial conditions that are independent of observations. The only practical discriminators between using observed and synthetic initial conditions are (i) the effort required to prepare the initial condition data and (ii) the length of spin-up time.

As demonstrated herein, consistent synthetic initial conditions can be
readily generated for even large complex networks – a task that is daunting
for interpolation/extrapolation of sparse synoptic observations. Indeed,
developing consistent synthetic initial conditions only requires the channel
geometry and hydrological model that are used for the SVE time marching,
supplemented by a steady-state SVE solver (Sect.

Unfortunately, we cannot definitively prove our second discriminator; the wide range of possible hydrological models, channel geometry models, and synoptic observation systems makes it impossible to prove that initial conditions based upon observations will always require longer spin-up times. However, no one has proposed an approach for interpolation/extrapolation of observed synoptic data that provides simultaneous consistency with a hydrological model, a river geometry model, and the SVE. It follows that we are on firm ground in stating that any existing approach using observed data as SVE initial conditions will result in inconsistencies. Finally, as our work demonstrates that inconsistent initial conditions cause long spin-up times in SVE models of sufficiently large river networks, it can be argued that observed data should be deprecated as initial conditions until a consistent approach is demonstrated and the spin-up can be shown to be shorter than provided by synthetic initial conditions.

Approaches for specifying initial conditions for the SVE can be grouped into
three main categories: (i) a “synoptic start” applying an
interpolated/extrapolated set of sparse observational data, (ii) a “cold
start” with initial flow rates and flow depths prescribed either as zero

Based on our discussion above, the first approach (synoptic start) is unlikely to be efficient for SVE initial conditions in a large river network due to inconsistencies between observations and model boundary conditions as well as inconsistencies caused by interpolating/extrapolating sparse observations throughout a network. There are no proven approaches to analyzing consistency and melding observations to hydrological model runoff, so the river network model spin-up will be subject to random inconsistencies and instabilities that can delay or prevent convergence.

The second approach, a cold start, provides innumerable possible ways to
create initial conditions. For example, mean annual flows and depths
(e.g., from the NHDPlus data in the US) can provide a smooth and consistent
set of flows and elevations throughout a network. Although such cold start
initial conditions can be internally consistent, they may be far from the
flows/depths implied by the initial hydrological forcing. For example, a
river network model that is started with mean annual values would be
substantially in error if the initial hydrological inflows were from the
monsoon season. As a result of inconsistencies between the selected cold
start values and the hydrological inflows, a cold start can require extensive
spin-up time to dilute or wash out the error. Indeed, the spin-up time
dominated the computational time for the large SVE networks that we previously
modeled in

Herein, we investigate the third approach, steady-state initial conditions, as
a preferred method for initializing a large river network model. With this
idea, a set of consistent initial conditions is one that satisfies both the

Herein, we present an efficient approach to establishing a set of steady-state conditions that provides a consistent and smooth starting point for time marching an unsteady Saint-Venant simulation. A full model initialization problem has two parts: (i) determining a set of flows and water surface elevations that are consistent steady solutions of the SVE for starting an unsteady solver and (ii) determining the spin-up time needed to ensure errors in the initial conditions are washed out of the unsteady solution. The second problem is highly dependent on the network characteristics and the particular flow and boundary conditions during spin-up, so for brevity, this work deals quantitatively with solving the first problem and then illustrates the effects on the second problem.

The Saint-Venant equations for temporal (

The most obvious approach for finding steady-state initial conditions is to
time march an unsteady solver until a steady state is achieved. That is, we
apply the unsteady solver with time-invariant boundary conditions of

The PTM is outlined as Algorithm

Pseudo time-marching method

Solve SVE at time point

Compute error:

The PTM approach (above) results in a steady solution of the unsteady
Saint-Venant equations that satisfies both momentum and continuity for
time-invariant

A key point, implied by Eq. (

To efficiently compute the conservative initial

Propagation of flow rate

A simple DFS traversal

DFS traversal for

After the steady

These nonlinear equations are similar to the unsteady discrete equations,
except that

Steady-solution method

Call QTraversal()

Call bisection
routine BiSection(

Solve steady version of dynamic eqn
in Eq. (

The performance of PTM and SSM are examined with a series of test cases
ranging from simple uniform cross sections over short river reaches to
15 000 km of a real river network. To demonstrate the robustness and
performance of the SSM, we conduct tests from three perspectives: (i) effects
of different cross-section geometries; (ii) scalability with an increasing
number of computational nodes; and (iii) real-world river networks. Two
different computers are used: the cross-section and scalability tests are run
on a computer with 2.00 GHz Intel Xeon D-1540 CPUs and 64 GB of RAM, while
the large network tests are run on a computer with 2.52 GHz Intel i7-870
CPUs and 8 GB of RAM. In both cases, Ubuntu Linux is the operating system
and GNU C

Test cases for cross-section geometry effects were conducted for synthetic
geometry of simple river reaches without tributaries. Cases included
rectangular, parabolic, trapezoidal, and non-uniform cross sections, with a
range of channel lengths, widths, and computational nodes, as provided in
Table

Cross-section geometry test cases.

To demonstrate the scalability as the number of computational nodes
increases, we use the geometry and flow conditions of Case 4 in
Table

To examine the robustness of PTM and SSM for more realistic conditions over
both small and large scales, we use a section of Waller Creek (Texas, USA) as
well as the entire watershed of the San Antonio and Guadalupe river basins
(Texas, USA). The former is a small urban watershed for which dense
cross-section survey data are available, whereas the latter is a large river
basin that has been previously modeled with the RAPID Muskingum routing model

The Waller Creek study includes two stream reaches and the catchment area
illustrated in Fig.

San Antonio and Guadalupe river network from an NHDPlus V2 flowline.

To test the initial condition approach for a large river network, we use the
San Antonio and Guadalupe river basins (Fig.

As is often the case in large river networks, comprehensive cross-section
geometry data were not available for the San Antonio and Guadalupe rivers.
Indeed,

Because the geometry affects both PTM and SSM solutions, we tested four
different estimation approaches for synthesizing geometry (Cases A, B,
C, and D). Case A uses synthetic trapezoidal cross sections using the
approach applied in

The overall algorithm efficiency is evaluated by the number of Newton
iterations required for convergence to steady state. The number of Newton
iterations reflects the difficulty in converging the nonlinear solution and
is proportional to the simulation runtime. As this metric is independent of
computer architecture, it provides a universal measure of algorithm
performance. For SSM, we use the number of iterations to converge the area

Table

Newton iterations required to achieve convergence for benchmark geometry test cases. The converged results are identical for both methods.

By comparing the geometric data from Table

Computing initial conditions using models with varying numbers of
computational nodes for Case 4 in Table

Speed-up multiplier of SSM compared to PTM for Case 4 as a function of the number of computational nodes.

The results of initial condition convergence for two Waller Creek simulations
are shown in Table

Total Newton's iterations required to achieve convergence of the Waller Creek test case.

The results of the full river network computations are provided in
Table

The convergence behavior of the PTM for configuration B is shown in
Fig.

Total Newton's iterations required to achieve convergence for four configurations of the San Antonio and Guadalupe river network.

Convergence of the L2 norm between consecutive pseudo time-marching
solutions for the PTM with configuration

As alluded to in the introduction, obtaining an effective model initial
condition is only one step in the initialization of an unsteady model. A
second step is understanding at what time the model results are independent
of any errors or inconsistencies in the initial conditions – i.e., the
spin-up time. Some model spin-up time is generally unavoidable as we never
have exactly the correct spatially distributed initial conditions that are
exactly consistent with spatially distributed boundary conditions. In effect,
eliminating spin-up time requires a set of initial conditions that are not
only consistent with the boundary conditions at

As an illustration of the scale of the spin-up problem compared to the
initial condition problem, we have run the SPRNT unsteady SVE model

Spin-up for the San Antonio and Guadalupe river network with the SPRNT
unsteady SVE model initialized using the SSM approach. The positive and
negative 20 % perturbations are for the

In general, the PTM performed poorly except on very simple systems. As the
river network complexity increases, the PTM changes from being somewhat slower
than SSM to being non-convergent. Indeed, the PTM has only one advantage over
the SSM in providing initial conditions to an unsteady SVE solver:
specifically, no new code is needed as PTM uses the same unsteady SVE code.
However, using PTM for large systems requires a frustrating trial and error
approach to tuning the system to obtain convergence. In contrast, the SSM
provides a rapid solution to the initial condition problem because

Herein, we only tested two methods for initial conditions, both based on
finding the steady-state

Although it is possible that a synoptic start could perform better than PTM
or a cold start, it seems likely that any approach to
interpolating/extrapolating sparse observational data across a larger river
network will necessarily result in inconsistencies between the initial

Note that the poor performance of the PTM cannot be attributed to
inefficiencies in the unsteady solver. As discussed in
Sect.

For simplicity, the SSM algorithms presented herein are for common river
networks with only a single reach downstream of each junction. Such tree
networks are a subset of the more general DAG river networks. Extension to
more complex DAG geometry requires the definition of splitting rules to
uniquely define partitioning of

Fundamental to the SSM approach is the presumption that there are no places
of net flow accumulation or loss throughout the network. All reservoirs and
hydraulic structures are treated as pass through so that all the upstream
flow is propagated through the downstream reaches. However, if there are
known locations of accumulation or loss, the river network could be divided
into subnetworks with separate SSM solutions. For example, upstream of a
reservoir an SSM solution can be used to obtain the inflow to the reservoir,
which can be used with operating information from the reservoir to provide
the correct outflow

It is demonstrated that inconsistencies between initial conditions and
boundary conditions for a large river network solver of the Saint-Venant
equations can lead to long spin-up times or solution divergence. We note that
synthetic initial conditions are preferred over observed synoptic initial
conditions due to the ability of the former to provide smooth and consistent
spin-up. Two methods to compute synthetic initial conditions for flow (

Cross-section geometry test cases can be found in the open-source repository mentioned in
Liu (2014).
Test cases and files for scalability, Waller Creek, San Antonio and Guadalupe River network are
uploaded to a public repository under Texas ScholarWorks (

PTM requires starting conditions (or a first guess) of

The hydraulic radius at normal depth requires the area at normal depth and
the wetted perimeter at normal depth,

Thus, an initial guess for

Note that failure to converge for Algorithm

Bi-section method

Evaluate

The authors declare that they have no conflict of interest.

This material is based upon work supported by the US National Science Foundation under grant no. CCF-1331610. Edited by: Insa Neuweiler Reviewed by: Rodrigo Cauduro Dias de Paiva and one anonymous referee