These authors contributed equally to this work.

Lateral migration of meandering rivers poses erosional risks to human settlements, roads, and infrastructure in alluvial floodplains. While there is a large body of scientific literature on the dominant mechanisms driving river migration, it is still not possible to accurately predict river meander evolution over multiple years. This is in part because we do not fully understand the relative contribution of each mechanism and because deterministic mathematical models are not equipped to account for stochasticity in the system. Besides, uncertainty due to model structure deficits and unknown parameter values remains. For a more reliable assessment of risks, we therefore need probabilistic forecasts. Here, we present a workflow to generate geomorphic risk maps for river migration using probabilistic modeling. We start with a simple geometric model for river migration, where nominal migration rates increase with local and upstream curvature. We then account for model structure deficits using smooth random functions. Probabilistic forecasts for river channel position over time are generated by Monte Carlo runs using a distribution of model parameter values inferred from satellite data. We provide a recipe for parameter inference within the Bayesian framework. We demonstrate that such risk maps are relatively more informative in avoiding false negatives, which can be both detrimental and costly, in the context of assessing erosional hazards due to river migration. Our results show that with longer prediction time horizons, the spatial uncertainty of erosional hazard within the entire channel belt increases – with more geographical area falling within 25 %

Meandering rivers migrate in their alluvial plain due to differential erosion and deposition along the outer and inner banks. Among other processes, this migration primarily happens because the shear stresses exerted on the cut bank are relatively high and the shear stresses on the point bar are relatively low, leading to erosion on one side and deposition on another

Both local and nonlocal mathematical theories have been postulated to understand the dynamics of river migration

Along with working on the predictive accuracy of earth and environmental science models, over the past few decades, there has been a growing interest in and acknowledgement of the cascading chain of uncertainties, especially when these models are used to evaluate risks and engineer solutions

There has been previous work on introducing probabilistic analysis to geomorphic modeling by introducing Monte Carlo simulations over parameter values as well as incorporating distributions of rainstorm intensity, discharge, sediment motion, and sediment supply

In the next sections, we will go over methodological details describing how to make geometric models stochastic and generate risk maps using inferred parameter distributions. We present the results from our simulation experiments on synthetic data and on a real case study. After the results, we discuss the advantages of this method over purely deterministic prediction setups. Finally, we discuss some of the unique challenges in generating risk maps for migrating rivers – growing uncertainty with increasing prediction horizon, identification of adequate geometric models that capture the first-order dynamics, and the nonlinearity introduced by cutoffs. We end the paper with some generalizable conclusions from this analysis.

Here, we introduce the recipe to extend a deterministic geometric model of river migration into a stochastic model. We first motivate river migration as a problem concerning an evolving curve. We introduce a simple geometric model that can guide channel evolution. We then go through the various additive stochastic terms that can be used to account for model structure deficits. We propose a novel application of smooth random functions for the river migration dynamics, which is different in characteristics than other space–time processes like discharge time series or flow velocity fields. Once the stochastic model for river migration is defined, we prepare the inverse problem by selecting the observations and defining the likelihood function. We finally introduce a novel counting algorithm as a counting problem for a 1D manifold in a 2D space and explain how to generate risk maps from the initial and final channel geometry – which is much more efficient than tracking the geometry during the entirety of evolution time.

In geomorphic models, like most predictive models in earth and environmental sciences, we plug the input variables in the model and get the output. However, the nature of the dependent variable is different – unlike a time series or a geospatial field, it is a dynamic 1D manifold, when channel evolution is viewed in its totality. In the context of fluvial geomorphology, we generally assume the channels to be smooth planar curves (a one-dimensional manifold in a real Euclidean plane

This illustrative figure depicts river migration as a problem of evolving planar curves.

In deterministic modeling paradigms, for a given input channel geometry and parameter values, a unique output geometry is expected. Observed deviations over and above the deterministic model

While geometric geomorphic models lose accuracy due to simplifying assumptions, models that describe the system in greater detail - by incorporating hydraulics and soil mechanics – do not necessarily improve the predictive capability. This is because many other factors play a role. An attempt at adding more subprocesses to the system description also requires that the representation of those subprocesses is correct – i.e., the model is able to represent the true dynamics of the system. Besides, more data are needed to feed these models. So there is a tendency to accumulate errors when we go to more detailed models that resolve various spatial processes. Therefore, geomorphic models tend to have uncertainties associated with them irrespective of the detail with which they try to describe the hydrologic and geomorphic systems.

The specific sources of uncertainty in geomorphic models are as follows: (a) model structure uncertainty, which arises from the fact that the equations used to describe the evolution of a system like (a meandering channel) are approximate. Model structure deficits can lead to a combination of systematic (overestimation or underestimation of the migration rates) and random deviations from the real system response. (b) The measurements of the input, elevation, and system response itself suffer from systematic and random deviations. This is called observational uncertainty. These deviations may be stationary in time or show some dependence on external drivers. This uncertainty becomes especially more pronounced when we want to carry out inferences about the system invariants – like its parameters or model structure – using these data. (c) The other source of uncertainty is unknown parameter values. Many combinations of parameter values, the ones we have not measured, can produce a close fit to the observed migration of the channel geometry to a comparable degree, leading to parameter uncertainty

Given all these cascading chains of uncertainty, there is prudence and value in quantifying them for risk assessments

Model structure deficits can be thought of as an aggregation of small deviations that come from several subprocesses not being incorporated in the model equations. We can therefore assume that errors due to model structure will follow some parametric distribution in the limit. The simplest case that is often employed is the normal distribution at each location. However, if we want more structure to the additive term, we will have to upgrade from random variables to random functions, which are called stochastic processes.

Mathematical models of earth and environmental systems can be considered the sum of a deterministic term and a random variable

This figure demonstrates an additive scheme that allows us to model river migration as a stochastic process.

Modeling geomorphic evolution as a stochastic process comes with a unique challenge in the case of migrating rivers – at the scales we are interested in, channels appear to be smooth (Fig.

In the context of channel migration, a likelihood function

Illustration for the generation of a risk map using parameter distribution inferred from satellite observations (see Algorithm 1).

The goal of this study is to devise a framework that helps understand and quantify the risk of each pixel of land getting eroded within a time frame by the evolving channel. Once we have the inferred model parameter distribution from the observed channel migration, we can use it to perform Monte Carlo simulation and generate multiple channel evolutions, the spread of which incorporates both parametric and model structure uncertainty. To create a pixeled risk map over the alluvium, we need to count the number of evolved channels that cross a pixel. The ratio of the number of channels crossing a pixel

Once these constraints are assumed, Fig.

Generating risk map for river migration.

This figure illustrates the algorithmic counting scheme that allows us to establish whether a channel eroded past a geographic location (

In this work, we have not taken into account the width of the channel and only take into account the centerline of the channel. This is because incorporation of the width poses additional computational challenges. For example, the optimal methods to sequentially assign each on the centerline to specific points on the bank have not been identified yet. Also, as our risk estimation algorithm is based on comparing the static configurations of initial and final channel geometries, it is not straightforward to identify which one of the two banks is creating the risk. However, the resolution of these challenges is not key to the assessment of our risk map generation framework, and therefore we leave it as a question to be researched further.

Unlike evaluating the performance of a deterministic forecast, where observed system realization is compared to the prediction using a distance metric (e.g., root mean squared error or correlation coefficient), evaluating probabilistic forecasts is more challenging as the prediction is available as probabilities. Here we describe a metric that can allow us to compute the performance of the probabilistic risk map. We also define the same metric over a subset of pixels to show the performance in avoiding false negatives. We start by defining the root mean squared departure (RMSD), where the departure is the difference in the probability value assigned to a pixel and the actual observed erosion in that pixel.

To test our model, we focus on the Ucayali River in the western Amazon Basin. The Ucayali is a single-threaded, meandering river that runs from the Andes through the Peruvian Amazon and eventually joins with the Marañón River to become the Amazon River. The gauging station at Requena, Peru, reports a mean annual discharge of 12 100 m

A Landsat image depicting the region of Ucayali River, with demarcated bends, used in this analysis to assess the proposed geomorphic risk estimation framework (image courtesy: the US Geological Survey).

Our description of the stochastic geometric river migration model is put to the test in a systematic manner using a suite of simulation experiments. Through these model runs, we wish to understand the following aspects of the proposed mathematical description: (a) how well are we able to constrain the parameter values of the model using the satellite observations? (b) How well are the risk maps produced using our stochastic description, in combination with inferred distribution of parameters, able to provide improvements in reliability over deterministic descriptions, (c) Are the underlying deterministic geometric models, like Howard–Knutson, able to capture the first-order dynamics of the migrating river system adequately?

The validity of the framework is tested using synthetic data in order to check the convergence of the posterior to the “true” parameter values that generate the observed river migration. Such controlled simulation experiments, where the parameter values of the system are known a priori, help in ascertaining the ability of the inference algorithm and observational data to constrain the system parameters. In the more detailed simulation experiments, we use the channel centerlines from the case study. We infer three parameter values of our stochastic description – migration rate

The prior distribution of migration rate and the friction factor can be chosen based on expert opinion and specific characteristics of the case study. In our case, where the river is rapidly migrating, we chose a relatively high value of the migration rate constant. The specifications of the prior distribution that we used for our inference are as follows: uniformly distributed within the range of [100, 500] for the migration rate constant and [0, 0.03] for the friction factor. For the standard deviation of the additive error term, we use exponential distribution as its prior within the range [0.01, 10] and with the distribution parameter

We then move on to study the effect of risk map generation on a real meandering river system using satellite observations. In real systems, the underlying dynamics of river evolution are expected to deviate from the model that we deploy to forecast it, thus putting the utility of our probabilistic framework to the test. We assume our stochastic description is the observation-generating process; i.e., each simulation from the model, using a sampled parameter value from the posterior, is a forecast of one possible course of river channel evolution, which is finally perturbed by a smooth random noise. A large ensemble of such forward simulations therefore gives the aggregate statistics of our forecast. We conduct extensive simulations for inference and prediction using the data from Ucayali. We infer parameters using the river profile in the year 1985 as the initial channel geometry and the river profile in the year 1995 as the final channel geometry. We run numerical experiments in multiple batches to understand the effect of (a) systematically increasing the number of bends on which the model parameters are inferred and (b) increasing the time horizon for which prediction is made. We systematically infer the model parameters using data from one, two, three, and four bends. We then predict the risk map for a time horizon of 5, 10, 15, 20, 25, and 30 years, thus capturing the spatial dynamics of prediction uncertainty and confidence as the forecast lead time increases. In this analysis, we have not explicitly considered the width of the channel, and we simply evolve the centerline. As Ucayali is a fast-migrating river, there is enough channel migration in multiples of 5 years that the proposed framework can be evaluated without incorporating with width. However, for slowly migrating rivers or with small forecast time horizons, the orthogonal correction of channel width over the centerlines should be performed.

In this section, we present the results from running our inverse and forward simulations using our stochastic description (Eq. 2). The results shed light on the ability of the likelihood function (Eq. 10) to help infer the unknown model parameter values using data from the past evolution of the channel. Finally, the forward simulations are evaluated based on various performance metrics for their corresponding risk maps (Fig.

The results from the numerical simulations on synthetic data show that the likelihood function on the summary statistic is able to retrieve the underlying parameter values that generated the data (Eqs. 9 and 10). Figure

The figure shows the posterior probability distribution of two parameters – migration rate constant and friction factor. The spread of the posterior captures the confidence in learned parameter values. Top row – panels

Results from the numerical simulation containing real observations (Fig.

After learning the posterior parameter distribution, we run the forward model (Eq. 2) using parameter samples from that distribution and employ our new framework to generate risk maps (Algorithm 1). The risk maps encapsulate the spatially distributed information about the probability that a geographical location in the alluvium will be eroded away by the migrating river within a certain time frame (Fig.

This figure showcases risk maps generated for various forecast time horizons. Each row represents the simulation when the model is trained using one, two, three, and four bends. The panels jointly represent the effect of increasing temporal and spatial scales of the forecast for river channel migration. Our framework is able to furnish spatial and temporal spread of erosion probabilities that deterministic paradigms are unable to provide.

The effect of increasing forecast time horizon and spatial extent on the certainty and uncertainty forecasts (

We observe that the lengthening of the forecast time horizon increases uncertainty (Fig.

This shows the comparison of aggregate error (root mean square deviation) between probabilistic and deterministic forecasts.

The inference of parameter values should, in principle, be more precise as more observational data become available. However, in the case of river migration, there seem to be diminishing marginal gains in constraining parameter values as we include data from more bends. Figure 6 shows the posteriors for various simulation experiments. The posterior distribution does not become narrower when we use four bends to infer parameter values instead of one. This can be attributed to the fact that each bend might have a tendency to converge to a slightly different parameter value, and the overall posterior distribution ends up being wide when we force the model to have the same parameter values for each bend. However, when we only use a single bend for parameter inference, the learned distribution is narrow as the model can relatively easily fit to the shorter channel stretch. The result suggests that if the modelers have ample computational resources, they should use different migration rates and friction factors for each bend so that the most appropriate values can be learned in a spatially explicit manner. We also notice that trying to predict the river evolution for multiple bends using the same posterior distribution results in, to a certain degree, a relatively poorer prediction. While these results are qualified by their application to one case study, it appears that the prediction of risk due to erosion has better performance if it is done on a bend-by-bend basis.

Given the probabilistic nature of our framework, it enforces a forecast timescale on each case study for which the uncertainty in erosion will be maximum. Beyond that timescale, the river is expected to sweep its meander belt, and the forecast will start becoming more confident about the possibility of erosion, with the asymptotic behavior of all pixels in the meander belt getting an erosion probability of 1 for a large enough timescale. However, this will happen for long timescales of thousands of years. Conversely, for planning horizons of several decades, predicting the evolution of a river channel with high confidence is difficult (Fig.

The results show that uncertainty-aware forecasts are more reliable than deterministic forecasts, which do not explicitly report their uncertainties (Fig.

Figure

Nonetheless, one important feature of geomorphic uncertainty, which is expected qualitatively by forecasters but not possible to reproduce by deterministic paradigms, is that regions close to the river on the cut-bank side are highly likely to be eroded, and regions farther away from the channel are unlikely to be eroded; more intricate patterns of risk emerge in regions at a moderate distance away from the cut bank. This length scale is dictated by the prediction time horizon and shifts outwards as the prediction time horizon increases. Given our stochastic formulation, if two rivers have comparable parameter values and similar parametric and model structure uncertainties, the bigger rivers with bigger bends will have a bigger zone of confidence for erosional hazard. Figure

Parameter inversion can also be performed in the deterministic paradigm, where a cost function, which generally encodes the aggregate error in the model prediction, is minimized by systematically exploring the parameter space. However, in highly stochastic physical processes, like channel evolution, the observations deviate from the first-order dynamics due to the aggregate effect of the neglected subprocesses. These deviations have a random character to them, hence confounding the inversion of parameters with uncertainty. Bayesian inference then becomes an attractive choice as a paradigm to learn distributions about model parameter values from observations. Within the context of parameter inference, when the observations are very informative, we learn very narrow posterior distributions, and when the observations are not very informative, they do not constrain the posterior distributions to unjustifiably narrow regions of parameter space (Fig.

The motivation to switch to stochastic models in a purely predictive framework is obvious – driven by the need to quantify uncertainty and have more reliable forecasts for river migration. However, from a scientific and mechanistic perspective within the context of fluvial landscape evolution, such modeling choices also seem to be the most natural.

The counting algorithm (Fig.

From the simulation experiments (Fig.

In this current formulation, we are using an additive stochastic term

To address this limitation, further research is needed on more sophisticated formulations of extending deterministic river channel evolution models into their stochastic counterparts. Model (in)validation can be pursued in a systematic way

Another limitation of this work is that we have not included time horizons where cutoffs take place, as our counting algorithm (Fig.

In this paper, we propose a framework to extend deterministic numerical models for river migration into probabilistic models. Our framework allows for the incorporation of uncertainty in model structure and parameter values while also accounting for the effect of stochastic variations in the geomorphic systems. The framework is agnostic to the underlying numerical model used to capture the first-order dynamics of the migrating river. However, in this case study, we have conducted our analyses using a geometric model, with the formulation of Howard–Knutson, where the migration rate is dictated by a weighted sum of local and upstream curvature. From our analysis, we conclude the following.

Deterministic models are uncertain and are additionally unable to capture the stochastic variability in the geomorphic system.

Some of this uncertainty can be faithfully incorporated into the modeling exercise by using an additional additive stochastic term and using the inferred distribution of parameters instead of single values.

We propose a recipe that uses smooth random functions as an additive stochastic term and then employs MCMC simulation to infer a posterior parameter distribution, which is conditioned on the observed migration in the river system.

The recipe is shown to work in principle by using it on synthetic data (by inferring the parameter values that generated the data).

We show that the method is able to identify regions in the vicinity of the river that are likely and unlikely to erode in the next

We also propose a counting algorithm that enables us to ascertain whether a river has crossed a spatial location by just looking at its initial and final geometry. We mention the assumptions and constraints of this algorithm.

We see that the Howard–Knutson model may not always be able to capture the first-order dynamics of the migrating river. We therefore suggest choosing the underlying deterministic model based on the geomorphic complexity of the case study.

We also suggest further research into the extension of our additive stochastic term, enabling it, for example, to have variable statistical features in space and time.

The code and data used in this study are provided at

OW and MPL conceptualized the study. OW developed the methodology, wrote the initial code, and wrote the paper. BN, with guidance from OW, built on the initial code, conducted detailed simulation experiments, generated and analyzed the results, and plotted the figures. OW, KBJD, and MPL supervised BN. MPL supervised OW and KBJD. All authors contributed ideas and provided feedback during the research phase. KBJD and MPL also contributed insights on the geomorphic modeling of migrating rivers. All authors contributed to the review and editing of the paper during the writing phase.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

Omar Wani thanks the Swiss National Science Foundation for support with its Early Postdoc Mobility Fellowship under grant number P2EZP2 195654. Michael P. Lamb acknowledges funding from the Resnick Sustainability Institute at Caltech under NSF awards 2127442 and 2031532.

This paper was edited by Sagy Cohen and reviewed by Keith Beven and one anonymous referee.