INTRODUCTION
Introduction
River channels are highly dynamic landforms that can shift across valley bottoms over various timescales, from events to millennia Increasingly, remotely-sensed imagery is utilized to define channel boundaries and assess changes in river planform, including lateral migration and channel width These measurements are crucial for developing effective management strategies for erosion in riparian corridors, contributing to sediment budgets, and distinguishing between single- and multi-threaded river planforms Additionally, they enhance our understanding of erosional hazards linked to migrating streams.
Understanding the effects of human alterations on river systems and managing riparian habitats is crucial for effective river corridor management Accurate quantification and reporting of uncertainty in river migration measurements are essential to avoid misguiding costly management strategies, such as bank stabilization and invasive restoration practices.
The enhanced availability and resolution of aerial photography, satellite imagery, UAV imagery, and LiDAR or digital elevation models (DEM/DTMs) have significantly improved our ability to analyze river channel migration These advancements allow for greater precision, broader spatial coverage, and increased temporal frequency in monitoring changes to Earth's surface.
Despite the availability of extensive remotely sensed data and advancements in software capabilities, there remains a lack of a consistent methodology for quantifying and addressing uncertainty in studies of fluvial change through remote sensing Recent research has made strides in understanding the uncertainty associated with measurements of channel width and lateral migration from remote imagery, highlighting the need for improved approaches in this field.
Despite advancements in measuring topographic changes, such as those outlined by Brasington et al (2003) and Wheaton et al (2010), a comprehensive framework for assessing river migration rates remains underdeveloped This gap highlights the need for rigorous and repeatable methods to quantify uncertainty in river migration, similar to those established for other topographic measurements.
This paper aims to establish a comprehensive framework for assessing uncertainty in river migration and width change estimates It summarizes relevant research and methods, tests various approaches for estimating channel migration and uncertainty, and explores the influence of spatial autocorrelations, riparian vegetation, and geomorphic conditions on this uncertainty While not exhaustive in addressing all possible analytical tools, the paper highlights key considerations and components of measurement uncertainty The findings extend beyond river channels, offering guidance applicable to measuring changes in various delineated boundaries, such as glacier dynamics, coastal erosion, wetland extent fluctuations, vegetation changes, cliff retreat, and political boundary disputes, ultimately aiding in generating and constraining uncertainty estimates for related models.
Error and uncertainty in Geographic Information Systems
Measurements of planform change over large spatial and temporal scales are often obtained from remotely-sensed images and analyzed within geographic information systems (GIS), which effectively manage spatial, temporal, and thematic components These components detail a measurement's location, timing, and descriptive attributes, revealing magnitudes and rates of change While temporal errors are typically absent due to known image acquisition dates, spatial uncertainty remains a concern, as it refers to the potential error in horizontal and vertical measurements Estimating total possible error helps establish a level of detection (LoD) threshold, which distinguishes statistically significant measurements from 'nondetects'—instances where measurement error surpasses the actual change observed By quantifying uncertainty in river planform change measurements, we can provide clear and valuable data to inform best management practices along riparian corridors.
When documenting channel platform and migration changes using aerial images, it is crucial to account for uncertainties in image georeferencing and orthorectification, as well as in channel delineations derived from manual or algorithmic methods (Libby et al., 2016) The choice of transformation method—such as linear, polynomial, kriging, or spline—significantly affects orthorectification uncertainty, with a second-order polynomial transformation being the preferred option for most applications due to its ability to minimize image distortion and georeferencing errors (Hughes et al., 2006) Additionally, the quantity, quality, and spatial distribution of georeferenced control points (GCPs) play a vital role in determining georeferencing uncertainty, as detailed in Table 2-1 (Lea and).
Delineation error in river channel mapping has been less extensively studied compared to other types of errors Gurnell et al (1994) quantified digitization error in a highly sinuous single-threaded channel, finding an average offset of ±2 meters over 18 river kilometers when using 1:10,000 scale maps However, their study did not assess the spatial variation of this error or the potential impacts of factors such as overhanging vegetation and shadows, which can also influence semi- and fully-automated channel delineations (Güneralp, Filippi, & Hales, 2013, 2014) This highlights the need for further research in mixed braided and anastomosing river morphologies to better understand delineation errors.
Werbylo et al (2017) discovered that digitizations from multiple users showed no significant average differences in channel width; however, variations at specific sections reached up to 37 meters The study concluded that high-resolution imagery leads to more consistent digitizations, with flow conditions being the primary factor influencing errors and inconsistencies in delineations Notably, higher flow levels resulted in more reliable delineations.
Accurate estimation of uncertainty is crucial for eliminating unreliable measurements and preserving reliable ones This is especially important for short-term assessments of minor changes, as overestimating uncertainty can render these measurements ineffective (Liro, 2015; Lea & Legleiter, 2016; Donovan & Belmont).
In situations where uncertainty cannot be accurately quantified, it is essential to apply upper and lower bounds of uncertainty to the results, as suggested by various studies (Kiiveri, 1997; Crosetto & Tarantola, 2001; Donovan et al., 2015; Passalacqua et al., 2015; Lauer et al., 2017) Additionally, identifying areas where measurements may primarily reflect random variability can help convey reliability to end users The use of probability and fuzzy positional boundaries has been recommended as a method to estimate probabilistic positional uncertainty in Geographic Information Systems (GIS) (Kiiveri, 1997; Wheaton et al., 2010) Ultimately, effectively assessing and communicating uncertainty enhances the quality of results and their subsequent applications.
Factors affecting uncertainty in remotely-sensed images and measurements of planform change
2.2 Techniques and developments in quantifying uncertainty
Georeferencing involves positioning scanned aerial photographs on a coordinate plane using georeferenced control points (GCPs), ideally sourced from orthorectified images for enhanced accuracy Despite this, significant errors can arise from inaccuracies in GCPs, associated imagery, and geometric distortions caused by various factors, which can compromise the reliability of image-based change detection analyses To accurately estimate georeferencing uncertainty, it is crucial to use distinguishable 'hard' points, such as buildings or permanent structures, rather than 'soft' points like vegetation or roads Research suggests employing a minimum of 5 to 8 hard GCPs to effectively reduce uncertainty, with little benefit gained from exceeding 8-10 GCPs Additionally, evenly distributing GCPs can minimize image warping, particularly when placed along floodplains Studies indicate that utilizing second-order polynomial interpolation during georectification significantly reduces errors and warping compared to higher-order transformations.
2016) A summary of factors affecting georeferencing uncertainty is found in Table 2-1
2.2.2 Approaches to river channel digitization and classification
After assessing the uncertainty in georeferencing aerial imagery, these images are utilized to digitize stream boundaries through either automated or manual techniques Typically, channel-margin delineations are established by the edges of riparian vegetation, which helps minimize variability caused by changing water levels (Winterbottom, 2000; D A Gaeuman, Schmidt, &).
Wilcock, 2003; Nelson, Erwin, & Schmidt, 2013; Rowland et al., 2016; Werbylo et al.,
In certain cases, such as estimating discharge or assessing changes in channel width for specific flow values, it is crucial to consider variability in relation to water stage (Bjerklie et al., 2005; Smith and 2017).
Delineating vegetated channel margins can lead to errors, particularly in braided and anastomosing systems, where width-related metrics are sensitive to changes in water stage An alternative method involves identifying the break in slope at the top of near-vertical channel banks; however, this technique requires high-resolution topographic data and is ineffective when banks are poorly defined Although the edge of riparian vegetation serves as a clear boundary, researchers must assess data quality and their specific research objectives when determining the appropriate channel margin.
The choice between manual and automated delineations in project planning is guided by the specific questions and goals of each project, as both methods can address a range of inquiries but excel in different contexts This article outlines the suitable scenarios and questions for each approach A comprehensive overview of methods and software for analyzing river planform properties and dynamics using remotely sensed imagery is provided by Rowland et al (2016), referencing various studies that contribute to this field, including works by Micheli & Kirchner (2002), Aalto et al (2008), and Fisher et al (2013), among others.
Manual delineations are widely recognized for their accuracy (Blundell & Opitz, 2006), but semi- and fully-automated methods offer significant time savings by reducing the labor-intensive process of manual delineation (Güneralp et al., 2013; Rowland et al., 2016; Schwenk et al., 2017) Researchers aiming to cover extensive spatial and temporal scales may choose automated methods, which can compromise accuracy for broader analyses and efficiency However, these studies are best suited for larger and more dynamic river systems (Peixoto et al., 2009; J A Constantine et al., 2014; Schwenk et al., 2017), as manual delineations can effectively reach lower-order streams better than automated techniques When performed by knowledgeable users, manual methods enhance the accuracy of delineations and classifications by allowing for the interpretation of complex or unusual features, thus adapting to diverse hydrologic and geomorphic conditions In contrast, automated classifications are bound by specific input criteria, increasing the likelihood of misclassification in scenarios that fall outside their designed parameters.
Despite the limitations mentioned above, automated approaches such as
SCREAM (Spatially Continuous Riverbank Erosion and Accretion Measurements;
SCREAM (Rowland et al., 2016) is increasingly utilized across various rivers with different morphologies and image resolutions, effectively identifying and rasterizing channel locations This tool provides outputs such as channel width, migration, sinuosity, bank aspect, and channel islands While automated channel classification algorithms like RivWidth (Pavelsky & Smith, 2008) and SCREAM yield similar bankfull width estimates for diverse river morphologies, SCREAM uniquely accounts for exposed channel bars and islands, leading to slightly higher discrepancies in width estimates for multi-threaded channels.
Vectorized streambank delineations are utilized to establish a single channel centerline, and by analyzing shifts in this centerline over time, researchers can estimate linear migration rates.
In river channel studies, polygons are often used to delineate areas for measuring changes in channel extent (D A Gaeuman et al., 2003; Rhoades et al., 2009; Donovan et al., 2015, 2016) A straightforward method for assessing migration involves calculating the difference in channel centerline positions, allowing for efficient linear migration measurements over specified lengths Both linear migration rates and areal changes can be normalized relative to channel width, facilitating comparisons among rivers of varying sizes (J M Hooke, 1980; Donovan et al., 2015; Spiekermann et al., 2017; Sylvester et al., 2019) However, estimating migration from centerlines may obscure individual bank movements, limiting the understanding of the mechanisms influencing meander-bend migration (Miller & Friedman, 2009; Schook et al., 2017) Therefore, measuring migration separately for cutbanks on the outer bends and point bars on the inner bends provides more insight into the local-scale mechanisms driving meander migration, despite requiring more computational resources.
Methods & Study area
3.1 Measuring migration and spatial autocorrelation
We develop guidelines to evaluate uncertainty for channel migration derived from manual channel delineations using a set of 13 aerial photographs spanning 76 years and
The study of 120 river-kilometers of the Root River in Minnesota, USA, utilized 441 images collected over nearly eight decades to analyze the relationship between uncertainty and various factors These factors include image resolution, acquisition date, local lighting, vegetation type and cover, as well as channel planform.
Streambank delineations and interpolated centerlines from each georeferenced image (Souffront, 2014, M.S Thesis) were used to calculate migration magnitude and rate at 10-meter increments using the Planform Statistics Toolbox (Lauer, 2007; Lauer &
The Planform Statistics Toolbox, as noted by Parker (2008), quantifies total migration by calculating the distance between nodes on the initial and terminal channel centerlines However, it does not account for meander bend cutoffs, which we manually identified and excluded from the analysis.
We initially measured migration at 10-meter increments along the entire river reach, but such close intervals may lead to autocorrelation due to the natural coherence of river movements and systematic offsets in digitization This autocorrelation results in non-independent measurements, underestimating standard errors and biasing statistical comparisons To address this, we computed Geary’s C to estimate the length scale of migration rate influences from spatial autocorrelation and local biases Unlike Moran’s I, which is suited for global autocorrelation, Geary’s C effectively detects local autocorrelation, making it more relevant for analyzing channel planform adjustments at the meander-bend scale Geary’s C values range from 0 to 2, with values near 1 indicating weak autocorrelation, values approaching 0 indicating positive autocorrelation, and values close to 2 suggesting negative autocorrelation For better interpretability, we transform Geary’s C values into standard correlation coefficients for plotting in correlograms, and scripts for calculating Geary’s C are available in the supplementary file, ‘GearyC.R’.
We analyze georeferencing uncertainty analyses using 13 sets of GCPs (n = 185 –
In a study spanning 120 km of the Root River, georeferencing error was quantified using an independent set of Ground Control Points (GCPs) derived from a high-resolution composite image from 2015 Instead of relying on the original GCPs, the research employed least-squares fitting algorithms for georectification transformations to minimize georeferencing offsets This approach ensured that the error assessment was comprehensive, as it included areas beyond the original input GCPs The selected 'hard' GCPs, which are immobile or unlikely to have shifted, were identifiable in both historical and 2015 imagery Additionally, the study evaluated the spatial correlation of GCP error across various distances to ascertain the presence and extent of spatial correlation in georeferencing error.
The article presents a detailed analysis of the Root River in southeastern Minnesota, highlighting its location within North America It focuses on a 120 km stretch of the river, utilizing 13 years of overlapping aerial photography A specific 11 km study reach is examined closely, with a single channel centerline divided into 10-meter points These points are color-coded to indicate the presence of shadows on the streambanks, while vegetation coverage is categorized similarly to assess its impact on the riverbanks.
We selected an 11-km stretch of the Root River characterized by diverse morphological features and varying levels of overhanging vegetation and shading to assess digitization uncertainty This analysis aimed to understand how fluvial and riparian conditions influence this uncertainty Additionally, the reach provided an opportunity to investigate whether the presence of point bars affected the consistency of manual riverbank delineation, similar to the outcomes observed with semi-automated algorithms.
In a study by Güneralp et al (2013, 2014), a single user delineated an 11-km vegetation-streambank boundary four times over 13 years, resulting in 52 delineations without relying on previous iterations The user was not restricted to a specific map scale, reflecting typical working conditions When vegetation obscured the bank, delineation was made through the tree crown or at the bank-vegetation interface Channel centerlines were interpolated from each bank delineation to measure migration rates, with a focus on ensuring consistency across delineations Centerline offsets were calculated at 10-meter increments to assess uncertainty from digitization inconsistencies, leading to 78 centerline comparisons and approximately 86,000 measurements Geary’s C values were computed from these comparisons to evaluate the length scales of autocorrelation introduced by digitization in migration measurements.
To determine whether image resolution influenced digitization uncertainty, first we visually compared the means and distributions of false migration of all 11-km of the
This study employs Kruskal-Wallis nonparametric analysis of variance (ANOVA) on 13 images to assess the impact of image resolution on false migration We investigate how a single user's digitization inconsistency is influenced by overhanging vegetation or shadows in channel reaches, as noted in previous research (Güneralp et al., 2013, 2014) Channel centerlines are classified at 10-meter increments, with values of 0, 1, or 2 indicating the extent of bank obscuration by shadows A similar classification is applied to vegetation, reflecting its presence or absence obscuring one or both banks across different years The Kruskal-Wallis test identifies significant differences among groups, and where differences are found, a Kolmogorov-Smirnov test is conducted to evaluate the stochastic increase in uncertainty for each class (Massey, 1951; Fay & Proschan, 2010).
3.4 Spatially-variable level of detection
We generated a spatially-variable level of detection (SV-LoD) raster for each year with images that included total uncertainty from georeferencing and digitization (Eqs 1,
The positional offsets in the x- and y-planes at each Ground Control Point (GCP) were analyzed by interpolating the sum of squares using second-order polynomials This approach resulted in a low mean Root Mean Square Error (RMSE) and effectively minimized image warping across all eight potential transformations (Hughes et al., 2006; Lea & Legleiter, 2016).
We compared the percent and magnitudes of migration measurements retained (n
We conducted a comprehensive evaluation of the quantity and quality of retained measurements by comparing the retention rates between SV and uniform LoD thresholds Our analysis confirmed that SV thresholds significantly enhanced the quantity of retained measurements Additionally, we assessed the quality by examining whether the distributions under SV thresholds exhibited a leftward shift, indicating a reduction in measurement values compared to uniform LoD thresholds This shift suggests that SV-LoDs are more effective at retaining low-magnitude measurements that are typically discarded by uniform LoDs To validate our findings, we visually inspected the distribution shifts and applied one-way Kolmogorov-Smirnov tests Finally, we employed linear regressions to analyze the percent retention in relation to factors such as pixel resolution, the initial year of imaging, temporal measurement intervals, and river-averaged migration, including its natural logarithm.
In geomorphology, common methods for addressing nondetects include removing them or substituting with values such as 0, 0.5, 0.7, or the limit of detection (LoD) Alternatively, some researchers retain nondetect measurements to establish upper and lower bounds based on uncertainty However, these approaches can introduce errors in estimating net change and uncertainty, as well as skew statistical parameters by adding arbitrary values To address these issues, we propose and test three alternative methods developed by statisticians from disciplines outside Earth science, including Maximum Likelihood Estimation (MLE) and imputation techniques.
Regression on Order Statistics (ROS) and Kaplan-Meier (KM) are distinct methods used for estimating summary statistics in the presence of nondetect data Their effectiveness varies based on factors such as data distribution, the proportion of nondetects, sample size, and detection limits (Helsel, 2005) In geomorphic change detection research, it is common to bracket results with the sum of uncertainties, a method that has both advantages and disadvantages (Anderson, 2018) For those interested, our scripts for executing Maximum Likelihood Estimation (MLE), ROS, and KM are provided in the supplementary R file.
Maximum Likelihood Estimation (MLE) aims to determine the most probable mean and standard deviation by fitting both observed and non-detect data to a distribution specified by an expert While MLE typically assumes a normal distribution, it is often applied to transformed lognormal data However, MLE tends to perform poorly with small datasets (n < 50 detectable observations) that exhibit significant skewness or contain outliers, especially when compared to Robust Ordination Statistics (ROS) or Kaplan-Meier (K-M) methods ROS is less reliant on distribution shape assumptions, utilizing probability plots of detectable data to estimate non-detects K-M is widely used in medical, industrial, and water chemistry statistics for analyzing censored data, as it does not assume a parametric distribution but requires a minimum of 8-10 measurements, with less than 50-70% being non-detects It can be biased if the highest or lowest values are non-detects and necessitates multiple levels of detection, making it suitable for a Site-Specific Limit of Detection (SV-LoD) rather than a uniform Limit of Detection (LoD) For a comprehensive understanding of K-M, refer to Hosmer et al.
(2008) Additional guidance and details on MLE, ROS, and K-M are provided online (Huston & Juarez-Colunga, 2009; ITRC, 2013)
We conducted a quantitative evaluation of various approaches by comparing their predicted mean (μ), median, distribution fit, and standard deviation (σ) against known values from modeled distributions (n = 400) with varying nondetect proportions (8-30%) The optimal estimates for mean, median, and variance from the MLE, KM, and ROS methods were identified based on the smallest differences from the modeled values To rank the distribution fits relative to the original modeled distribution, we employed a Pairwise Wilcoxon Rank Sum test with an adjusted p-value, following the Benjamini and Hochberg (1995) method to minimize false-positive rates and facilitate distribution comparison This adjustment enhances the sensitivity of the test to differences in distributions beyond just central tendencies.
We visually validate the quantitative findings by plotting the empirical cumulative density functions (ECDFs) from each approach, facilitating a discussion on the most suitable contexts for each method Additionally, we explored different measurement sample sizes, specifically using n=100.
Results & Discussion
4.1 Spatial autocorrelation for measurements of migration and uncertainty
Correlograms of autocorrelation values reveal a diminishing spatial autocorrelation of channel migration rates over scales of approximately 1-4 channel widths (50-200 meters), beyond which autocorrelation becomes weak or absent The trends observed in migration rate autocorrelation closely align with those of user digitization inconsistency Consequently, it remains unclear at which length scales autocorrelation indicates digitization inconsistency rather than coherent units of channel migration This finding implies that similar autocorrelation scales apply to both manual channel digitization and river migration across diverse geomorphic conditions.
Fig 2-4 Correlograms of spatial autocorrelation for measurements of channel migration
Lag distance refers to the measurement length for autocorrelation analysis Geary’s C values, generally ranging from 0 to 2, were adjusted to fit the standard correlation scale of -1 to 1 By analyzing migration across distances of at least 6 channel widths (approximately 400 meters) in the Root River, we ensured that autocorrelation did not interfere with the statistical outcomes and interpretations.
When testing the assumption that GCPs exhibit local spatial correlation (D
In a study by Gaeuman, Symanzik, et al (2005), it was observed that strong autocorrelations (values > 0.7 or < -0.7) were infrequent, occurring in only 6-28% of cases, while weak autocorrelations (between -0.3 and 0.3) were predominant, accounting for 62-81% of the data Additionally, 14-38% of the data showed moderate autocorrelations (values between 0.3-0.7 or -0.3 to -0.7) Notably, the few ground control points (GCPs) exhibiting strong autocorrelation did not correspond to any specific spatial scale, indicating that the assumption of similarity between nearby and distant GCPs is invalid This lack of local or global autocorrelation highlights the necessity for spatially variable Levels of Detail (LoDs), as neither neighboring nor distant GCPs demonstrated consistent similarity in magnitude or autocorrelation.
This study examines an 11-km stretch of the Root River to enhance existing research on factors affecting digitization offset The average digitization uncertainty was found to be 1.4 meters, which aligns closely with previous findings by Gurnell et al (1994), who reported an uncertainty of 2 meters The pixel resolution within this segment varied from 0.5 to 5.8 m², encompassing nearly the entire pixel range of 0.3 to 5.8 m² across all 441 images from the 120-km Root River mainstem Notably, despite the differences in pixel resolution, the distributions of digitization offset did not show significant variation.
A Kruskal-Wallis Rank Sum Test revealed no systematic relationship between digitization offset and pixel resolution, indicating that neither image date nor pixel resolution significantly affects digitization inconsistency for a single user Consistent results across various image resolutions (0.5-5.8 m²) suggest that the average and median digitization uncertainty for an experienced user ranges between 1.5 and 2 meters, eliminating the need for recalculation in future studies However, follow-up evaluations may be necessary for studies involving resolutions outside this range or differing geomorphic conditions The framework for assessing digitization uncertainty presented here can be adapted to other environments to investigate potential variations Similar methodologies were utilized by Werbylo et al for braided and anastomosing planforms.
(2017), who found that measurements of at-a-section channel width derived from multiple digitizers differ up to 20% of channel width, while river-averaged widths exhibited no significant differences
Digitization uncertainty was generally consistent across all years (i.e., high precision) in cases where the bank is masked by shadow and/or vegetation cover (Fig 2-
6) Higher degrees of inconsistency (> 5 meters) occurred along meander bends with
The analysis of 11.2 km of the Root River over 13 years reveals consistent magnitudes and distributions in digitization, despite variations in pixel resolution and types of vegetation cover Users exhibit inconsistency in defining vegetation boundaries, particularly in areas with multiple boundaries, leading to minor digitization offsets of less than 1 meter scattered uniformly across the river reach To accurately delineate the channel-vegetation boundary, users should identify the vegetation boundary that best represents the dominant discharge Ground-truthing is recommended for ambiguous areas, while high-resolution topography can help identify streambank-floodplain transitions through local curvature peaks This framework is essential for evaluating consistency in regions with significant variations in riparian vegetation and geomorphic conditions.
Figure 2-6 illustrates the inconsistencies in digitization categorized by the presence of shadows and vegetation on one or both adjacent streambanks, as well as cases where neither is present The numbers next to each boxplot represent the sample size for each category, highlighting the impact of environmental factors on digitization accuracy.
Georeferencing uncertainty showed significant variation across the study area and years, typically decreasing over time due to advancements in camera technology and self-calibrating sensors (Clarke & Fryer, 1998) This uncertainty generally does not impact river width measurements unless distortion occurs at scales smaller than the channel width, which is why we did not analyze its effect on width calculations Additionally, there was no significant correlation between the root mean square error (RMSE) of georeferencing and the mean pixel resolution for each year, as shown in Table 2-2 The RMSE often reflected the 75th percentile for most years, indicating that a few extreme outliers can skew the RMSE, making it higher than the median in a long-tailed distribution We will discuss the implications of relying on this inflated RMSE value in the following section.
Fig 2-7 Spatially variable georeferencing uncertainty, in meters, across the x- and y- coordinate planes (bottom and top, respectively) For total uncertainty, we used Eq 1 and
The final error ellipse for each pixel is calculated using two components, with color gradient scales varying across panels to reflect the differing ranges of error in the x- and y- planes.
The distribution of georeferencing uncertainty for each image year is illustrated, highlighting the variability based on a set of 185 to 302 georeferenced control points (GCPs) The root mean square error (RMSE), represented by red dots, indicates the mean of each distribution and serves as a common uniform uncertainty threshold However, applying the RMSE threshold results in significant data loss for measurements falling below these red dots.
4.4 Calculating and evaluating LoD thresholds
Final limits of detection (LoD) thresholds are determined by the combined uncertainties of georeferencing and digitization, while spatial variance (SV) LoDs account for individual pixel variations and root mean square error (RMSE) LoDs provide an average sum of squares However, extreme outliers in georeferencing errors can significantly skew the RMSE, inflating the mean error and misrepresenting the majority of values within a lognormal distribution Consequently, relying on an inflated RMSE value for LoD can lead to the exclusion of most low-magnitude migration measurements, which are often classified as 'nondetects.' This results in a skewed migration distribution where only a few high values dominate, ultimately distorting the mean migration rate, a critical metric in the analysis.
This article provides an overview of image characteristics, georeferencing control points, and associated errors for each year of imagery analyzed Using 2015 imagery as a reference layer, we calculated offsets and errors, with the understanding that this layer is considered the most spatially accurate representation of the study area Consequently, the 2015 imagery does not include error values.
Table 2-2 presents summary statistics for annual imagery and the related digitization uncertainty The application of second-order polynomial interpolation on SV error enhanced the retention of migration measurements across all years, supporting findings from the initial implementation of this approach (Lea & Legleiter, 2016).
In a comprehensive analysis of 67 comparisons, SV-LoD thresholds consistently captured a higher percentage of migration measurements (average 81%) compared to uniform LoD thresholds, which only retained an average of 52% The data shows that SV-LoDs are effective in retaining more measurements of smaller magnitudes while capturing fewer measurements of larger magnitudes, as illustrated in Figures 2-9.
For three different years, we evaluated migration measurements that were retained with the SV-LoD, but not the uniform LoD Visual observations of values retained by
Fig 2-9 A comparison of probability density functions among of all measurements
The analysis of migration measurements from 1981 to 2011 reveals that the SV-LoD (red) retained a higher proportion of low-magnitude measurements while capturing fewer large-magnitude ones, indicating an improvement in both the quantity and quality of the retained data Notably, the SV-LoD suggests that just over half of the recorded values represent real, verifiable changes, typically characterized by gradual systematic shifts in the river, supported by multiple images and visual evidence from LiDAR hillshade imagery.
Conclusions, recommendations, and future challenges
The significance of calculating and disclosing uncertainty in GIS-based measurements has gained recognition in earth-science literature In the early 1990s, Anders and Byrnes (1991) highlighted the necessity of addressing the primary sources of this uncertainty.
The empirical cumulative density functions (ECDFs) for the modeled migration data are presented, along with three methods used to handle nondetect measurements These ECDFs represent the majority of model iterations (n = 400) and are designed to estimate summary statistics such as mean, median, and variance Visualizing these distributions aids in the interpretation of the results found in Table 2-2.
A flow chart (Fig 2-13) outlines the process for handling nondetect measurements, addressing the uncertainty in boundary delineations obtained from aerial images Over the following decade, research expanded to estimate uncertainty and levels of detection (LoDs) through traditional error propagation methods (Edwards & Lowell, 1996; Kiiveri, 1997; Crosetto & Tarantola, 2001) Additional studies quantified the impact of specific variables on uncertainty (D A Gaeuman et al., 2003; Nelson et al.).
In recent years, researchers have developed various methods for addressing nondetect measurements below the limits of detection (LoD) (Shumway et al., 2002; Martín-Fernández et al., 2003; Helsel, 2006) However, a gap persists between these advancements and their implementation in earth-science research (Lea and Legleiter, 2016; Donovan & Belmont, 2019) This article compiles and evaluates relevant methodologies and applied research focused on calculating planform changes from remotely-sensed imagery The following sections offer a detailed framework that includes general guidance and specific considerations for assessing uncertainty in planform change measurements.
Our study utilizes 441 images collected over eight decades, representing diverse riparian conditions and geomorphic environments, to establish a framework for managing uncertainty in river analysis While our findings primarily focus on the Root River in Minnesota, which features a single-threaded meandering pattern, the insights gained are relevant to various river scales and forms We acknowledge the need for specific considerations related to our analysis, as detailed in the accompanying background material The methodologies and recommendations for assessing uncertainty can also be applied to other remote sensing contexts, such as monitoring glacier dynamics, coastal erosion, wetland changes, and vegetation shifts We encourage readers to tailor our findings to their unique scenarios and datasets, and we provide supplementary R scripts to aid in replicating our analyses Additionally, while our approach emphasizes linear change measurement, it is important to note that more complex river systems may require different methods for assessing erosion and deposition.
Inconsistencies in streambank delineations were found to be similar regardless of image quality, shadows, or vegetation cover, indicating that an experienced user can maintain consistent precision across various environments, as demonstrated along the 120 km stretch of the Root River Although user-defined delineation inconsistencies are a primary source of uncertainty, image quality becomes a significant factor when pixel size surpasses the necessary resolution for detecting the riparian-fluvial boundary (0.5 – 3.5 m²) Additionally, the type of vegetation present can influence the accuracy of delineations based on pixel resolution, while larger pixel sizes may complicate delineations in narrow channels or small tributaries, further affecting the accuracy of delineation offsets.
To ensure accurate river delineations, especially in areas where image quality is compromised, it is essential to utilize field measurements or high-resolution topography Meander bends with diverse vegetation types present the greatest uncertainty due to difficulties in identifying the appropriate vegetated boundary Future research should address the integration of spatially variable delineation uncertainty Single-user digitization is recommended as it significantly minimizes uncertainty, achieving a reduction of approximately 0.5 meters compared to multiple users, who can introduce a central tendency error of ±2 meters and a range of up to ±37 meters Establishing a standard for delineation will mitigate errors and biases in the long-term monitoring of river channels and riparian conditions that depend on multiple manual delineations Whenever feasible, it is advisable to delineate the vegetated boundary that most closely represents bankfull width to maintain consistency, particularly in reaches with complex features like braided or gravel-bed rivers.
Our analysis of georeferencing uncertainty aligns with prior research advocating for second-order polynomials to enhance the integration of retained measurements while minimizing image distortion We observed notable variations in georeferencing uncertainty in images from before the 1990s, attributed to lower image quality and fewer dependable control points The lack of local and global autocorrelation for Ground Control Points (GCPs) underscores the necessity for spatially variable Levels of Detail (SV-LoD), as GCP errors were uncorrelated regardless of their proximity Additionally, spatial autocorrelation for delineation bias and migration rates was evident across 1-6 channel widths (50- to 400-meters) The degree of autocorrelation linked to coherent migration patterns is likely influenced by river size and may not be applicable to other systems Future research should investigate autocorrelation further, as this could enhance models of river meander migration.
Our analyses reveal that spatially-variable Levels of Detection (SV-LoDs) enhance the quality of retained measurements compared to traditional RMSE-based Levels of Detection SV-LoDs effectively identify and discard erroneous large-magnitude migration values, which often result from georeferencing errors or image warping, while RMSE-based methods may retain these inaccuracies due to their magnitude Additionally, SV-LoDs capture small yet valid channel adjustments that typically fall below the RMSE threshold We advocate for the adoption of SV-LoDs to improve the accuracy of uncertainty quantification and the overall quality and quantity of retained measurements Utilizing the Planform Statistics Toolbox alongside Lea and Legleiter’s Matlab script allows for efficient and accurate assessment of linear river migration and spatially variable uncertainty For complex river systems, such as braided channels or those with numerous vegetated islands, area-based methods may provide a more effective means of measuring changes and estimating spatial variability.
When applying a limit of detection (LoD) threshold to distinguish between significant and nondetect measurements, various strategies can be utilized to address nondetects, depending on the research objectives and expert insights In the analysis of both linear and areal channel changes, instances of zero or minimal change are typically categorized as nondetects, even though this classification does not necessarily indicate that no change has occurred.
We advise using expert judgment to assess whether measurements of zero or nearly zero change are significant, as most river channels show minimal changes between photos This practice can enhance the accuracy of data distribution by minimizing the risk of mistakenly filtering out genuine geomorphic changes (Anderson, 2018) It is important to note that this recommendation is specifically for evaluating nondetect measurements in known stagnant areas and does not apply to all measurements We also explored the effectiveness of three new methodologies, including Kaplan-Meier, in this context.
Regression on Order Statistics (ROS) and Maximum Likelihood Estimation (MLE) are effective methods for estimating statistical parameters such as mean, median, standard deviation, and distribution fit for modeled distributions with known proportions of nondetects MLE and Kaplan-Meier (K-M) estimators perform well in approximating the mean of raw data, particularly with small sample sizes (n = 100), while MLE is superior for larger samples (n > 1000) ROS excels in estimating variance across all sample sizes and improves median estimates as sample size increases K-M consistently demonstrates robustness in overall distribution fitting The choice of method for handling nondetects should be tailored to individual cases, guided by method descriptions, external resources, and analytical results.
This article presents a thorough overview of research on uncertainty in river planform change studies Over the past decades, advancements in technology and critical thinking have enhanced our understanding of uncertainty in remotely-sensed data, and this field is expected to evolve further Future research should focus on refining methodologies to enhance the accuracy of fluvial change measurements However, there remains a lack of consensus on quantifying uncertainty, attributed to the complex nature of calculations, diverse evaluation tools, and sometimes the absence of uncertainty estimates To address these challenges, we advocate for the development of simpler, more generalizable, and open-source tools for assessing planform change and its associated uncertainty, facilitating a unified approach for comparison and measurement across studies.
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Study area and Data
We conducted an empirical evaluation of timescale dependence using 12 sets of aerial photographs that cover 120 km of the Root River in Minnesota This meandering, single-threaded river, characterized by its sand and gravel beds, flows into the Mississippi River The images analyzed span a significant timeframe of 76 years, with data collected from the years 1937, 1947, 1953, 1976, 1981, 1991, 2003, 2006, 2008, 2010, 2011, and 2013.
We chose the Root River for its three unique geomorphic settings, which allow us to investigate the variations in measurement-scale dependencies and channel migration patterns associated with each environment These settings are remnants of the Late Pleistocene and Holocene glaciation and base level changes, showcasing different levels of valley confinement, slope, and sinuosity (Souffront, 2014; Belmont et al., 2016a) Although this study does not aim to analyze the impacts of land use and flow changes on migration rates, it provides valuable insights into the geomorphic characteristics of the river.
Fig 3-1 The Root River watershed and three distinct geomorphic zones as defined by
According to Souffront (2014), the studied area features distinct slopes and varying degrees of valley confinement across different zones The total length of the delineated river reaches 120 kilometers, with Zone 3 measuring 42 kilometers, Zone 2 at 38 kilometers, and Zone 1 spanning 32 kilometers.
Table 3-1 Root River zones and morphological characteristics geomorphic setting provides useful context to interpret our results
The 120 km study reach is partially within the so-called ‘Driftless Area’ of the upper Midwestern United States, which has been unglaciated for the past 500 kyr
The Root River watershed, influenced by glacial meltwater and outwash from the Last Glacial Maximum, features deep valleys formed by the Mississippi River's incision before this period The landscape includes alluvial valleys bordered by rolling uplands, predominantly forested in steeper regions, while gently sloping areas are utilized for corn and soybean farming Row crops dominate approximately 75% of the watershed, particularly in the previously glaciated western section Additionally, shallow karst formations underlie much of the area, contributing to the region's unique geological characteristics.
15 m of alluvial deposits overtop carbonate bedrock Mainstem valleys and larger tributaries run across mantled karst with alluvial deposits exceeding 30 m
Improved agricultural management practices in the 1940s significantly reduced upland erosion from agricultural fields; however, the legacy of historical erosion continues to be a major source of sediment, evident in the large alluvial terraces and floodplain deposits along the modern Root River Additionally, historical milldams and small hydroelectric power dams can be found along certain tributaries, along with levees situated on the mainstem outlet in the downstream reaches of Zone 3.
Over the past 40 years, the Root River hydrologic regime has seen substantial increases in both low and high flows, with increases of 80% and 60% respectively These changes are primarily attributed to improved artificial drainage of agricultural lands and a rise in precipitation levels.
Research on sediment loading changes over time remains limited, despite the land use history showing similarities to the extensively studied Coon Creek across the Mississippi River (Trimble, 1999).
2009) The Root River watershed exhibits some of the steepest relationships between discharge (Q) and total suspended solids (TSS) throughout Minnesota (Vaughan et al.,
In 2017, research highlighted significant near-channel sediment sources that are particularly susceptible to erosion during high flow events (Stout et al., 2014; Belmont et al., 2016a) This study integrates three unique geomorphic environments, emphasizing the varying impacts of erosion in these areas.
(120 km) and temporally (76 years) robust set of historical air photos provides an exemplary opportunity to explore timescale dependence of migration measurements along an alluvial river experiencing increased flow.
Methods
3.1 Measuring and evaluating temporal change
Approximately 2,880 km of streambanks were digitized from 12 sets of scanned georeferenced images (Souffront, 2014; M.S Thesis) and used to interpolate channel centerlines for each year (1937, 1947, 1953, 1976, 1981, 1991, 2003, 2006, 2008, 2010,
In a study conducted between 2011 and 2013, curvature-driven cut-bank migration magnitude was quantified at 10-meter increments along channel centerlines for 66 image combinations using the Planform Statistics Toolbox The analysis did not differentiate between cut-bank migration, down-valley translation, or bend expansion/contraction, as preliminary comparisons showed less than 1% variation in results Total migration was calculated as the distance between nodes on the initial and terminal channel centerlines To ensure accuracy, meander bend cutoffs were manually identified and filtered out from relevant measurements prior to further analysis Although the length of river filtered as cutoffs increased with measurement intervals, this represented a negligible proportion of the overall 120 km study reach.
Different geomorphic conditions can lead to unique channel responses, prompting the classification of migration rates into three distinct geomorphic zones based on slope and valley confinement Our analysis revealed that lognormal distributions predominated in migration measurements, leading us to apply nonparametric statistics, specifically the Mann-Whitney Wilcoxon and Kolmogorov-Smirnov tests, to assess significant increases in medians, extremes, and distributions of each image pair We employed one-tailed tests with an alpha value of 0.05 to test the hypothesis that migration rates have increased with flow, ensuring a 95% confidence level to minimize Type I errors However, it is important to note that such stringent confidence levels may not always be applicable in water resource and environmental risk assessments.
(Johnson, 1999; Belmont et al., 2016b), we also evaluated significance at alpha values of 0.1 and 0.2 (Appendix C, Table A1a & A1b)
Measurement intervals differed for image pairs between 2003-2013 (Δt ~ 1-3 years) and those prior to 1991 (Δt ~ 6-23 years), possibly confounding results Thus, we also compared pre-1991 rates with ensemble rates measured from 2003-2013 (Δt = 10, n
The analysis of migration rates from 2003 to 2013 offers crucial evidence regarding potential changes in these rates If fewer years show significant differences, it suggests that timescale bias could have affected the interpretations of how channels respond to hydrologic changes.
We analyzed Geary’s C correlograms to assess the spatial measurement autocorrelation in river migration data, revealing that significant autocorrelation persisted at length scales of 50-200 meters, while it diminished beyond 200 meters (C-values > 0.8) To maintain the integrity of our statistical tests, we averaged migration rates over 400-meter increments, confirming that varying length scales had minimal impact on the timescale dependency results This approach allowed us to model migration accurately, as migration rates were not autocorrelated beyond 400 meters, enabling random sampling of empirical data distributions without concerns of spatial autocorrelation.
The article presents migration measurements through a series of images, including 1937 and 1947 imagery with channel centerlines overlaid Additionally, it features a depiction of 10-meter increments used for calculating migration distances, utilizing the Planform Statistics Toolbox (Lauer, 2007) Furthermore, it illustrates 12 channel centerlines derived from images spanning from 1937 to 2013, marked at 10-meter intervals.
3.3 Quantifying uncertainty from georeferencing and digitization error
We assessed uncertainty by calculating the sum of squares from both spatially variable georeferencing uncertainty and consistent user delineation and digitization, aiming to estimate the minimum level of detection (LoD) The georeferencing uncertainty was determined for a minimum of.
In this study, we utilized 185 georeferenced control points (GCPs) annually and applied interpolation methods to assess uncertainty for each raster cell, as outlined by Lea and Legleiter (2016) To estimate digitization uncertainty, we compared centerlines obtained from four repeat digitizations of an 11-km streambank reach Additionally, migration measurements that fell below the minimum limit of detection (LoD) were assigned a value of zero, as illustrated in the flat portions of Figures 3-4a to k.
The study assessed channel migration by plotting the mean migration distance (Δx) against various time intervals (Δt) in log-log space, following established methods (Gardner et al., 1987; Sadler and Jerolmack, 2015; Ganti et al., 2016) Trends were compared to a 1:1 line, with 98% confidence intervals used to determine if longer averaging time scales introduced systematic bias Significant deviations from the 1:1 line could indicate measurement-scale dependence or shifts in migration rates over time To mitigate potential confounding factors, the analysis focused on the process rate (Δx/Δt) using linear axes, given the aerial images covered less than a century A comparison of historical and contemporary migration rates for seven specific reaches with short measurement intervals (Δt ≤ 6 years) was conducted to evaluate whether observed timescale dependence stemmed from systematic rate changes or sample bias, thus providing independent evidence for the findings.
1) migration rates changed systematically over the period of study, and 2) observed timescale dependence actually reflected a dearth of short-Δt measurements for historical data, rather than an actual change in migration rates We further examined how sampling bias may affect timescale dependence using a statistical model (described later)
Past literature indicates that process reversals and hiatuses significantly contribute to timescale dependence and bias in river channel studies Although we anticipated low reversal rates in the Root River due to its wide valley and meander belt, we conducted manual measurements of channel reversals during our study period to enhance our statistical model addressing measurement bias Our criteria required that reversals be sustained over multiple years to exclude temporary offsets caused by digitization errors This necessity for manual measurement allowed us to apply expert judgment that automated methods might miss Additionally, we excluded data post-1991, as the shorter time intervals made it difficult to differentiate genuine reversals from noise.
We created a statistical model to analyze the impact of migration hiatuses and reversals on river migration patterns, focusing on their effects without delving into complex underlying mechanisms The model aims to determine the extent to which these hiatuses, channel reversals, and changes in migration rates contribute to measurement bias in migration data To facilitate this analysis, we compiled a comprehensive dataset that captures annual migration measurements.
We created 100-year synthetic annual migration rates for 100 reaches, each measuring 400 meters in length These rates were randomly selected based on lognormal distributions derived from empirical data collected from the Root River, with a time interval (Δt) of three years or less, utilizing seven years of observations.
The model script generated migration rates by randomly selecting mean values from a range of 0.31 to 1.42 m/yr for years with a time interval (Δt) of three years or less, incorporating 0-values that account for 50 to 75% of the data All initial migration rate values were positive, indicating movement in either direction, as the specific direction was not critical for the analysis The model calculated standard deviation from the selected average migration rate using an empirical linear relationship, reflecting the significant direct correlation between standard deviation and mean migration rates, as demonstrated in the accompanying figures.
Due to the high likelihood that the occurrence of channel reversals leads to underestimating measured migration rates, we evaluated the effect of reversal frequency using four model scenarios In the absence of literature quantifying the temporal frequency or probability of channel reversals, we evaluated a range of plausible reversal frequencies (0%, 1%, 2%, 5%, and 10%), supported by observations for the Root River The frequency of reversals explored in our model reflects a reasonable range of what we expect to occur in natural systems; reversal frequency varied from 1-6% across the definitive geomorphic zones of the Root River Highest reversal frequency lie in the confined upper reaches and decreased downstream, which supports intuition that reversal frequency is inversely related to valley width and our decision to model reversal frequencies from 1 to 10% Reversal frequency was implemented by probabilistically reversing simulated migration (i.e., multiplying migration rates for individual reaches by -1) until the end of the 100-year model run, or until chance (1%, 2%, 5%, or 10%) reversed the 400-m segment back to its original direction (i.e., a positive value) The model tracked cumulative migration distance for each 400-m reach, and thus, negative values (i.e., reversals) reduced the modeled cumulative migration distance and rate Similar to our analysis of empirical data, we calculated the mean for all 400-m segments to represent the ensemble mean annual river migration We plotted all possible Δt combinations of average (mean) migration rate to evaluate how increasing reversal frequency affected timescale dependence
In addition to hiatuses and reversals, systematic changes in migration rates may also cause trends in timescale dependence to diverge from a 1:1 relation, especially if recent photos dominate shorter timescales (Δt) and longer timescales are dominated by older photos acquired at lower frequencies We conducted an additional set of model runs to explore the effect of older photo sets typically dominating longer timescales, coupled with the impacts of systematic changes in migration rates We generated scenarios wherein contemporary migration rates (i.e., years 51-100) were increased and decreased by factors of 1.25, 2, 5, and 10 relative to historical rates (i.e., years 1-50, Fig 3-3) These scenarios also had a 10% chance for channel reversals Outputs from these eight scenarios of change allowed us to evaluate whether temporal changes in migration rates cause a false-positive timescale dependence, indicated by a shift/translation to the trends in Fig 3-7a We implemented a second test to verify or refute these results in which we
Results and Discussion
4.1 Does timescale dependence exist for river migration measurements?
The entire 120 km dataset of migration rates for adjacent time intervals are illustrated in Fig 3-4a-k, where channel cutoffs and measurements below the LoD are plotted as zeros Measurements of mean channel migration exhibit a visual timescale dependence for each zone of the Root River (Fig 3-5a) Loss of a record due to channel reversals would be similar to vertical reversals (e.g., sediment aggradation vs erosion) that cause bias in other measurements by erasing historical records (Sadler, 1981;
Reversals in migration direction were observed over 23 km (17%) of the Root River, suggesting a potential mechanism for timescale dependence The frequency and length of these reversals decreased from upstream to downstream, with rates of 66%, 32%, and 0.5% for Zones 3, 2, and 1, respectively This trend aligns with expectations, as upstream reaches demonstrate higher migration rates and are situated within narrower valleys.
Post-hoc correlations and regressions indicate a significant indirect relationship (p < 0.001, r² = 0.98) between the frequency of reversals and valley width (see Appendix A, Fig A1) It appears that long-term migration rates may seem systematically low due to longer Δt values being influenced by historical air photos from periods of slower migration However, when comparing a subset of historical reaches (n = 7, 3-29 km, Appendix C, Fig A3) with short measurement intervals (Δt ≤ 6 years) to contemporary measurements with similar Δt, no systematic shifts were found This observation will be further investigated using a statistical model of migration.
4.2 How does timescale dependence vary with degrees of channel dormancy and reversals?
Numerical simulations using a statistical model revealed the impact of channel dormancy and reversal frequency on migration measurements In the absence of reversals, channel dormancy led to only a minor timescale dependence, underestimating migration by approximately 1% However, as reversal frequency increased from 1% to 10%, the underestimation of migration distance and rate over a century rose significantly, ranging from 4% to nearly 30% compared to channels without reversals Notably, the bias diminished with shorter measurement intervals, becoming negligible at a measurement interval of Δt = 1 This indicates that the observed decay in empirical migration rates with longer measurement timescales may be influenced by the presence of reversals.
Fig 3-4a (A–E) Longitudinal profiles of migration rates for five measurements made between 1937 and 1991
Longitudinal profiles of migration rates from six measurements taken between 1991 and 2013 reveal significant trends, with vertical black bars marking the boundaries of Zones 3, 2, and 1 The data indicates an increase in reversals and dormancy periods, alongside noticeable rate convergence and asymptotic trends in the synthetic and modeled migration data, as detailed in Appendix C, Figures A4a and A4b.
The black circles in Fig 3-5 illustrate the average migration rates for zones spanning 34–48 km across various aerial photo pairs, such as those from 1937 to 1947 (refer to Fig 3-4) Short-term migration rates exhibit significant variability, while longer measurement intervals lead to a convergence of rates, diminishing short-term fluctuations As the time interval (Δt) increases, the measured migration rates consistently decline and align, suggesting that longer temporal scales are primarily influenced by channel dormancy and reversals.
4.3 How do actual changes in channel migration influence observed timescale dependence?
We conducted simulations to determine if actual temporal changes can be distinguished from timescale dependence and to assess the impact of the magnitude and direction of these changes Our study emulated various scenarios of migration rate alterations, implementing changes of 1.25, 2, 5, or 10 times (both increases and decreases) at the midpoint of a 100-year simulation To ensure consistency, all simulations incorporated a 10% probability of reversals.
The analysis indicates that changes in modeled migration correspond to trends based on a baseline scenario of 10% reversals This observation aligns with the empirical trends observed in Zone 2, which show an upward shift due to accelerated migration rates compared to Zones 1 and 3.
Between 1937 and 1991, a section of the Root River migrated approximately 40 meters southwest, followed by an additional 85 meters to the northeast from 1991 to 2013 This results in a total observed net migration of 45 meters, averaging 0.6 meters per year, if considering only the start and end dates without photographic evidence from 1937 to 2013.
125 m (1.6 m/yr) The LiDAR hillshade confirms southwestward migration followed by a reversal to its location in 2013 (dark-pink line)
Model simulations indicate that consistent trends in river channel migration cannot fully capture timescale dependence without accounting for channel reversals When rate changes are combined with biased sampling—favoring short-term measurements over historical data—timescale bias can be exacerbated This leads to artificially skewed measurements for low- to mid-range time intervals (1-30 years), which distort the migration slope when analyzed over Δt Consequently, if contemporary data dominates short-term records while historical data prevails in long-term assessments, conclusions about changes in channel behavior remain inconclusive without further independent evidence.
The model results indicate that the high variability of short-term migration rates (Δx/Δt) tends to converge towards a long-term average, mirroring trends observed in empirical migration data Mean rates are represented by black circles for each measurement timescale (Δt = 1, 2, 3, …, 100), while colored circles include reversals in the model simulations Notably, measurement bias escalates with increased reversal frequency and measurement timescale, as evidenced by progressively lower modeled migration rates (Δx/Δt) compared to scenarios without reversals.
The observed migration rates (Δx/Δt) were analyzed across various measurement timescales (Δt) under different scenarios of temporal change In this study, black and green points represent scenarios without and with reversals, respectively, while red and blue points indicate scenarios that included reversals along with a two-fold increase and decrease in migration rates after a specified year.
Temporal changes in migration alone do not adequately replicate or amplify timescale dependence without the influence of reversals, as evidenced by a shift in the trend slope The scenarios presented are variations of the simple reversal scenario, highlighting that both upward and downward changes maintain the same foundational structure without alterations.
4.4 Predicting and adjusting measurements for timescale bias
The combined empirical and model findings indicate that migration measurement is subject to timescale bias, which fluctuates based on reversal frequency, measurement duration, and migration rate variations Although inconsistent short-timescale empirical data hinder the removal of timescale dependence, our model reveals that estimates of reversal frequency can help identify the percentage of bias or underestimation in specific migration rate measurements.
Boxplots illustrating migration rates across different geomorphic zones of the Root River are presented in Fig 3-9 The rightmost boxplot indicates cumulative migration data from 2003 to 2013, allowing for a comparable measurement timeframe to data collected prior to 1991 To assess prediction bias, we evaluated four linear models based on reversal frequency, time interval, and their combined effects.
Conclusions
Migration rates are influenced by the measurement interval, with short-term assessments (under 10 years) showing significant variability due to sporadic migration bursts In contrast, long-term measurements (over 25 years) tend to stabilize, reflecting a more consistent trend in migration patterns.
Understanding the 'characteristic timescale' is crucial for accurately assessing channel migration and sediment flux, as it highlights the variability in measurements and the potential for long-term changes Long-term migration measurements often underestimate sediment flux due to channel reversals that obscure parts of the erosional record Consequently, the timescale of these measurements directly influences the questions they can effectively address Insufficient short-term data can lead to distorted projections of long-term sediment remobilization and fluvial changes To accurately determine significant long-term changes, researchers need intervals greater than 20-25 years, which allow for comparisons with similarly extensive measurements Additionally, multiple short-term measurements are essential for capturing the episodic nature of channel migration and understanding responses to flow and sediment flux changes These findings emphasize the importance of using consistent measurement intervals and caution when interpreting aerial-based assessments of channel activity and fluvial changes.
Recent empirical and modelled data indicate that migration rate measurements are increasingly underestimated due to the frequency of channel reversals, with minimal influence from channel dormancy This measurement bias suggests that contemporary channel migration rates appear to have risen, stemming from discrepancies in sampling intervals between modern and historical aerial photographs Our analysis reveals that long-term migration rates fail to accurately reflect sediment contributions from streambanks unless bias is corrected by considering the frequency of reversals Additionally, we found no empirical evidence that the Root River has significantly altered its migration patterns in response to increased flow over recent decades This underscores the complexity of the relationship between discharge and migration rates, highlighting the need for a more nuanced understanding that incorporates factors such as sediment supply, transport, and hydraulic structures in meander bends.
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EVALUATING THE RELATIONSHIP BETWEEN MEANDER-BEND CURVATURE, SEDIMENT SUPPY, AND MIGRATION RATES
SEDIMENT SUPPY, AND MIGRATION RATES
1.1.Background- River meander migration and curvature
River meander migration is a fundamental process that shapes and redistributes mass across the Earth's surface Since the early 20th century, scientists have been captivated by the forms and patterns of river meander development, with notable contributions from researchers such as Davis, Brice, Leopold, and Wolman Even Albert Einstein was intrigued, suggesting that meandering could be influenced by the Coriolis effect The complexity of meander migration is evident in numerous studies that examine various scales, from individual meander bends to the geological evolution of floodplains and valleys.
Research by Howard (1996) and Gran et al (2013) enhances predictive models for migration timing and locations, offering valuable insights for environmental and agricultural management, sediment load assessments for downstream habitats, stream restoration, and riparian/watershed management The use of remotely-sensed imagery is essential for analyzing changes in river planform due to factors such as land use alterations, urbanization, deforestation, and the construction or removal of dams (Hickin and Nanson, 1984; Gurnell et al., 1994; Gaeuman et al., 2005; Lauer and Parker, 2008; Constantine et al., 2014; Donovan et al., 2015, 2016; Morais et al., 2016).
Brice (1974) identified seven generalized classes of meander development using aerial imagery and USGS topographic maps, highlighting the predictable patterns influenced by sediment loads and water flow (Constantine et al., 2014) The helical flow patterns around meander bends create asymmetries in centrifugal forces and shear stresses on the outer banks, which drive sediment erosion, transport, and deposition (Leopold and Wolman, 1960; Dietrich et al., 1979) As the curvature of a meander bend increases, so do the centrifugal forces and shear stresses, suggesting that migration rates are directly related to the curvature (Howard and Knutson, 1984; Furbish).
Curvature (C) measures how much a segment or surface deviates from a straight line or plane, and it is inversely related to the radius of curvature (R) While centrifugal force and bank stress increase with greater bend curvature, empirical data show that migration rates peak at a radius of curvature that is 2 to 3 times the channel width (R/W ~2-3) when averaged across the scale of a meander bend This relationship has been consistently observed in various studies over the years.
Bends with identical average curvature can display varying levels of asymmetry, indicating that a single bend-averaged curvature value may correspond to multiple shear stress patterns (Furbish, 1988) For instance, in Fig 4-1b, two bends share the same curvature but differ significantly in flow asymmetry and shear stress due to their lengths, with longer bends experiencing greater shear stresses along the outer bank, leading to faster migration rates Migration paths are influenced not only by local curvature but also by cumulative upstream curvature, which changes with bend length Therefore, linking bend-averaged migration rates to bend-averaged curvature results in a single curvature value representing a range of migration rates Despite being published nearly three decades ago, Furbish's works have received only about 20% of the citations of Hickin (1974) and 30% of Hickin & Nanson (1975), illustrating how popular beliefs can overshadow rigorous scientific findings While Hickin & Nanson (1975) provided a significant breakthrough in understanding curvature-migration rate dynamics, later studies largely ignored Furbish's concerns, favoring a simplified approach that connects bend-averaged radius of curvature with migration rate.
Models that link bank erosion to local curvature effectively illustrate the relationship between local migration and curvature, as noted by Begin (1981) and Crosato (2009) However, some researchers argue that relying solely on local curvature for modeling meander development fails to capture the observed asymmetry and spatial heterogeneity found in complex planform changes, as highlighted by Carson and Lapointe (1983) and Güneralp and Rhoads (2011) A comparative study by Howard and Knutson (1984) demonstrated that using a model that incorporates both local and upstream curvature, weighted by distance upstream, successfully simulates the asymmetrical growth, downstream translation, and typical cutoffs of natural meandering rivers, unlike models based solely on local curvature.
Figure 4-1 illustrates competing theories on the relationship between curvature and meander-bend migration On the left, meander-bend averaged migration is shown against the normalized radius of curvature (R/W), based on findings from Hickin and Nanson (1975), indicating an inverse relationship between curvature and the normalized radius On the right, a conceptual diagram from Furbish (1988) demonstrates that two meander bends can share the same bend-averaged radius of curvature while exhibiting different shear stresses along the outer bank Consequently, even with identical radii, bend R2 migrates more rapidly due to its higher shear stress The transposition of *R1 and *R2 from each curve serves to confirm the equality of the radii.
Measuring migration and curvature at a bend's scale can overlook sub-meander flow dynamics that influence meander migration variability High-velocity flow filaments direct water toward the outer bank, often reaching downstream of the bend apex, while shear stress and erosion intensify along the outer bank due to centrifugal forces and secondary flow acceleration In contrast, inner bends experience lower velocities and deposition, leading to point bar formation that further directs high-velocity flows outward This relationship, known as bar push, links point bar development to outer bank erosion The spatial lag between bend apices and peak migration rates indicates a delay in secondary flow acceleration, suggesting that peak migration occurs downstream of the bend apex This lag's length is affected by factors such as meander arc length, width-to-depth ratio, flow resistance, flow depth, inner-bank bar angle, and suspended sediment concentration.
Sylvester et al (2019) provided empirical evidence of a direct relationship between channel curvature and downstream migration rates in seven Amazonian rivers by measuring migration rates and channel curvature at sub-meander bend scales Their findings revealed that migration rates consistently increased with channel curvature, without exhibiting a peak at intermediate curvature values Deviations from this trend were linked to reduced bank erodibility The authors concluded that previous studies showing peaked curvature-migration relationships, such as Hickin and Nanson (1975), were based on bend-averaged values rather than spatially explicit and lagged measurements of curvature and migration rates.
Channel migration is influenced by local shear stress patterns and the interplay between sediment loads and water flow When sediment supply surpasses a channel's transport capacity, it results in steeper slopes and the growth of point bars As these bars develop, the resulting asymmetry in the channel bed enhances flow velocities, depths, and shear stresses, increasing the likelihood of lateral migration through bar push Conversely, in areas lacking sufficient sediment supply, these dynamics are diminished, preventing bars from exerting significant influence on the flow The Amazon River, known for its high sediment transport rates, likely experiences migration driven by bar push feedbacks However, the relationship between curvature and migration in low-sediment conditions remains uncertain Investigating the effects of sediment supply and bar geometry on curvature-migration dynamics is crucial for advancing our understanding of fluvial geomorphology Comprehensive studies on process-form feedbacks in the meander morphodynamics of natural systems are essential for future theoretical and experimental research.
This study examines the relationship between channel curvature, migration rates, and bar geometry in the Root and Minnesota rivers over extensive temporal and spatial scales, utilizing repeated aerial imagery collected over 76 years We analyze the correlation between averaged meander-bend curvature and migration rates, comparing these findings to spatially explicit and lagged curvature measurements at sub-meander scales This approach allows us to determine whether the length scale of measurement influences the relationship between channel curvature and migration rates, specifically whether migration rates increase continuously with curvature or peak at intermediate curvature values.
This study investigates the magnitude and variability of the spatial lag between curvature and migration rate, focusing on how these factors interact in areas with high bank erodibility and low sediment supply It aims to determine whether the lag and the nature of the relationship between curvature and migration rate are affected by these conditions Additionally, sediment supply is examined for its potential influence on point bar geometry and growth, highlighting its significance in the curvature-migration relationship.
Study Area and Data
We evaluate curvature-migration relations using channel change along centerlines derived from aerial photographs spanning approximately 25 km of the Root River,
The Minnesota River is a single-threaded, winding river characterized by its sand- and gravel-bedded composition, ultimately draining into the Mississippi River The selected 25-km stretch of the river showcases the most dynamic meander bends and has been the focus of extensive research, as documented by Stout and Belmont (2013), Stout et al (2014), Souffront (2014), and Belmont et al.
In this section, meander bends are occasionally confined laterally by both natural and human-made structures The combination of channel confinement and varying riparian conditions creates enough irregularity in erosivity to evaluate the effectiveness of a straightforward curvature-migration model under diverse conditions To conduct this analysis, we utilized eight sets of images.
(1937, 1947, 1953, 1976, 1981, 1991, 2003, and 2013) with sufficiently similar time intervals to encompass significant channel adjustment (Donovan and Belmont, 2019)
The Minnesota River stretches 180 kilometers from Mankato to historical Fort Snelling, where it meets the Mississippi River This segment has been the subject of various detailed geomorphic studies, supported by six sets of images taken in 1937, 1951, 1964, 1980, 1991, and 2013.
Fig 4-2 Overview of Root River within the North American continent and state of
The Root River in Minnesota flows into the Mississippi River, with a specific focus on a 25-kilometer segment selected for analysis This study includes centerlines derived from delineations based on eight sets of images spanning from 1937 to the present.
The Minnesota River Valley has a unique geomorphic history, shaped significantly by the outpouring of glacial Lake Aggasiz approximately 13,400 years ago, which led to a 70-meter incision of the mainstem river This incision created multiple knickpoints and exposed highly erodible glacial sediments Over the past 80 years, changes in land use and precipitation have resulted in flow increases of 50-250%, amplifying rates of lateral channel migration and increasing channel width Within the 180-km study reach, sediment grain size, channel-bar geometry, and slope show abrupt changes about 100 km downstream near Belle Plaine, distinguishing high-supply and low-supply reaches Upstream reaches feature large, wide channel bars that promote sediment dynamics, while downstream areas have narrow, steep point bars with limited sediment supply, providing a unique opportunity to explore the relationship between sediment supply, channel curvature, and meander migration rates.
3.1 Measuring curvature and channel planform
Each year, channel banks were defined according to the methodology outlined by Donovan et al (2019), with bank lines interpolated to channel centerlines and converted into 10-meter coordinate points Channel width was calculated at each increment using the Planform Statistics Toolbox (Lauer and Parker, 2008) For the Root River and Minnesota River, bank migration was assessed at each 10-meter increment along the channel for seven and five sequential image pairs, respectively, utilizing a dynamic time warping algorithm (DTW).
DTW was originally developed to correlate time series (e.g., Lisiecki and
Lisiecki, 2002) and has been shown to greatly reduce computation time while improving
The study area along the Minnesota River, illustrated in Fig 4-3, extends from Mankato to Fort Snelling, with distinct upstream and downstream sections marked in blue and red Notably, the riverbed and bars in the northern part of the reach are emphasized for further analysis.
The downstream region of Belle Plaine is characterized by fine sands, silts, and clays, in contrast to the coarse sands and gravels found in the upstream area Unlike traditional nearest-neighbor algorithms, Dynamic Time Warping (DTW) employs a cost matrix and cosine similarity to optimize the alignment of trajectories between signals, focusing on minimizing the overall trajectory sum rather than individual distances This approach considers both the magnitude and spatial orientation of points, allowing for a more nuanced comparison that accommodates variations in signal length due to migration effects By leveraging cosine similarity, DTW effectively mitigates issues related to trajectory measurement discrepancies, such as gaps or bunching at the ends of signals Consequently, as the distance between centerlines increases, DTW demonstrates enhanced performance compared to nearest-neighbor methods.
After conducting DTW computations, we manually identified and excluded measurements taken within meander bend cutoffs prior to further analysis (see Fig 4-3) The curvature, measured in units of m -1, was calculated using the x and y components of the Cartesian coordinates for each point.
(𝑥 ′2 +𝑦 ′2 ) 3 2 ⁄ , Eq (1) where x’ and x’’ are the first and second-order derivatives of the x coordinate Curvature is the reciprocal of the radius of curvature, R (Eq 2):
The relationship between curvature and migration rates is inversely proportional to the width-normalized radius of curvature (R/W), as indicated by Eq (2) To enhance data clarity, both curvature and migration rates were processed using a Savitzky-Golay filter, effectively minimizing signal noise (Motta et al., 2012; Sylvester et al.).
2019) Savitzky-Golay filtering retains local precision without distorting the signal by fitting low-degree polynomials to successive subsets of data points (Savitzky and Golay,
3.2 Discerning spatial relationships in migration and curvature
We utilized a signal processing algorithm (scipy.signal.find_peaks) in Python to identify local maxima and minima, referred to as ‘peaks’, in our continuous profiles of migration and curvature A point was classified as a peak if it exceeded adjacent values within a 40-meter range By implementing straightforward criteria for peak detection, we minimized false negatives and manually filtered out false positives, ensuring that only curvature peaks corresponding to migration rate peaks were retained The lag distance between these paired peaks was measured along the channel centerline and normalized by the average channel width between the peaks.
In the context of analyzing channel characteristics, L* represents the dimensionless lag, while Loc Cpk indicates the location of peak curvature Additionally, Loc Mpk denotes the location of peak migration rate, and 𝑊̅ refers to the ensemble mean channel width between these two peaks.
We assessed the extent and variability of lags through summary statistics and histograms of offsets Additionally, we calculated the derivatives of curvature and migration, employing a similar approach to pinpoint paired inflections in both curvature and migration.
Inflections indicated the highest rates of change in curvature and migration along the profile, serving as a unique dataset for assessing spatial lags To ensure consistency in magnitude and variability, the distances between paired inflections were normalized to the average channel width, offering an independent evaluation method for spatial lags.
Meander migration peaks occur downstream from points where curvature is zero, marking the start of the current meander bend This leads to the development of asymmetrical flow, which increases shear stress along the outer bank.
Results
4.1 Basic data attributes and descriptions
The curvature values for the Root and Minnesota Rivers are normally distributed around zero, ranging from approximately -1 to 1 Migration rates exhibit a long-tailed right-skewed distribution, with median values between 0.5 and 1.5 m/yr and maximum rates reaching about 15 m/yr for both rivers In the 25-km study reach of the Root River, the mean channel width fluctuates between 47 and 55 meters annually, with the narrowest and widest cross-sections measuring 19 and 125 meters, respectively Meanwhile, the mean width of the Minnesota River increased significantly from 70 meters to 102 meters over the study period from 1937 to 2013.
Fig 4-5 (left) Distribution of dimensionless curvature along the Root River, derived from imagery obtained in 1981 (right) Distribution of Root River migration rates measured between 1981 and 1991
In our comprehensive analysis, we identified 371 paired peaks in migration and curvature for the Root River and 873 for the Minnesota River These totals were calculated after excluding cutoffs and values that fell below the detection threshold.
The study analyzed 585 paired inflections along the Root River and 873 along the Minnesota River to assess the offset between migration and curvature Utilizing cross-correlation analyses, which consider all measurements rather than just peaks or inflections, the research incorporated approximately 7,200 measurements for the Root River and 86,000 for the Minnesota River, excluding cutoffs and measurements below the detection level.
4.2 Optimizing search radius of cross-correlation analyses
For cross-correlation analysis of the Root River, the ideal window size is 600 meters, roughly 12 times the average channel width Beyond this size, further increases do not alter the results Conversely, narrower windows fail to adequately capture the optimized lag distance, as indicated by a consistent lag distance.
Fig 4-6 A range of window sizes were tested as input for the cross-correlation analysis
The optimal search window sizes for the Root and Minnesota Rivers were determined to be 600 meters and 800 meters, respectively, ensuring a balance between capturing relevant signals and minimizing excess computation time These distances correspond to approximately 9-12 channel widths for the Minnesota River, allowing for efficient cross-correlation analysis while accurately identifying the optimal lag distance This approach aligns with observed lag distances between curvature and migration signals, as verified through manual peak and inflection matching.
4.3 Magnitude and variability of lags between signals of curvature and migration
Cross-correlation analysis reveals that optimizing curvature signals for the Root River occurs by shifting them downstream by 2.3 (± 1.2) channel widths, closely aligning with the Minnesota River's signal offset of 2.2 (± 1.3) To ensure accurate mean calculations, all cross-correlation coefficients below 0.25 were excluded from the analysis, as these sub-optimal values could distort the mean and hinder the identification of an optimal phase lag based on strong correlation coefficients.
In the Minnesota River, we identified cross-correlation data from the downstream reach characterized by low-sediment supply, represented as red points in Figure 4-6c This analysis demonstrated that an impressive 94% of the cross-correlations occurred in the segment with minimal bedload sediment supply.
Downstream of Belle Plaine, low signal matching (