In many locations, our ability to study the processes which shape
the Earth are greatly enhanced through the use of high-resolution digital
topographic data. However, although the availability of such datasets has
markedly increased in recent years, many locations of significant geomorphic
interest still do not have high-resolution topographic data available. Here,
we aim to constrain how well we can understand surface processes through
topographic analysis performed on lower-resolution data. We generate digital
elevation models from point clouds at a range of grid resolutions from 1 to
30 m, which covers the range of widely used data resolutions available
globally, at three locations in the United States. Using these data, the
relationship between curvature and grid resolution is explored, alongside the
estimation of the hillslope sediment transport coefficient (

Geomorphologists have always made use of topographic data, from initial
qualitative observations of surface morphology and its link to process

Presently, lidar data coverage is predominantly focused around locations of
particular scientific interest or infrastructural importance, as can be seen
on many lidar data portals

As a consequence of this data availability it is crucial to understand the
limitations of lower-resolution data when performing topographic analysis for
geomorphic research. Extracting channels from topography is a common
requirement of many analyses, and it is expected that the accuracy of
extracted channel networks will be affected by increasing grid resolution

Here, we grid topographic data at a range of resolutions in order to test the sensitivity of these techniques to decreasing grid resolution, with the aim of placing constraints on the estimation of common geomorphic parameters when lidar topographic data are unavailable. Through an analysis of one-dimensional curvature and topographic gradient approximations, the changes in fidelity as grid resolution decreases for both curvature and topographic gradient are examined and placed within the context of the two-dimensional results of this study and the wider literature.

It has long been recognized that the scale of topographic data used in an
analysis or model will have an impact on the scale of the processes which can
be measured

Many attempts to constrain the error content of topographic measurements have
focused on comparisons between elevation values taken from differing
resolution data products, often in conjunction with field survey data, with
the aim of discriminating between DEM generation methods.

The advent of lidar-derived topographic data provided a new technique and
increased the range of possible grid resolutions to evaluate.

Topographic gradient (or slope) is one of the most fundamental topographic
derivatives across the disparate disciplines which utilize topographic data.
This measurement has been used in geomorphology

Numerous authors have considered the impact of grid resolution on
hydrological applications, which often require slope calculation as a
fundamental processing step. It has been demonstrated across many landscapes
and scales that as grid resolution is decreased the upslope contributing area
will increase and the local slope will decrease, which will have a
significant impact on any hydrological analysis

Topographic wetness index (TWI), calculated as

The accuracy of channel network extraction from topographic data was tested
by

As models of agricultural soil loss depend heavily on topographic variables
such as slope, work has been carried out to understand the influence of grid
resolution on calculated rates of soil loss.

Although considerable work has been carried out on the sensitivity of various
factors to grid resolution, much of it has been focused on a specific
application (e.g.,

Previous studies that have explored the impact of changing grid resolution on
topographic or geomorphic parameters have typically produced coarser-resolution topographic data by downsampling the highest-resolution data
product available for their study sites (e.g.,

Lidar point cloud metadata.

As a consequence of these constraints we have generated topographic data for
our three study sites without downsampling or re-gridding high-resolution
data products, as is commonly performed

Example shaded reliefs of the same section of Santa Cruz Island at
increasing grid resolutions. All coordinates are in UTM Zone 11

The point clouds are gridded using Points2Grid, which employs a local binning
algorithm, searching for points within a circular window of radius defined by

An inverse distance-weighted averaging approach is then performed to assign
an elevation value to each grid cell. This approach, which has been employed
in previous studies

The topographic data used in this study have been gridded at 20 resolutions,
and Fig.

Landscape curvature has long been recognized as a key geomorphic
characteristic of landscapes, from Gilbert's
(

This is particularly important with the proliferation of high-resolution
topographic data from lidar, allowing the analysis of curvature on increasingly fine scales. Recent developments in channel extraction
techniques

The measured curvature of a topographic surface depends on the orientation of
the measurement. Here, we consider two common types of curvature, with the
following definitions: (1) total curvature (

Work by

We calculate curvature using a circular window passed across the landscape,
with a radius defined by identifying scaling breaks in the standard deviation
and interquartile range of curvature calculated at increasing window sizes,
consistent with the length scales of individual hillslopes

From the measure of

For our data on hilltop curvature,

Extracting channel networks from digital topographic data remains a
fundamental challenge for many areas of topographic analysis. Without the
ability to discriminate between fluvial and hillslope domains, it is not
possible extract many topographic metrics such as hillslope length

Many authors have made use of field-mapped channel heads both as a basis for
geomorphic analysis and as a method for evaluating channel extraction methods

Several methods have been proposed to identify channel heads from high-resolution topography. Typically these methods exploit the high-resolution
nature of topographic data to resolve morphometric or process-based
signatures of channel initiation or the transition between the hillslope and
fluvial domain

The DrEICH method was selected for evaluation as the technique on which it is
based has been shown to operate successfully in lower-resolution data

The geometric method, used by

Channels were extracted from the 5, 10, 20, and 30 m DEMs generated in
Sect.

To assess the accuracy of the channel networks extracted using both methods,
we employ two measures of quality described by

We follow

The reliability,

The sediment transport coefficient,

The nonlinear model proposed by

At low hillslope gradients (e.g., on hilltops), the term within brackets in
Eq. (

Hillslope length (

Many studies have attempted to calculate hillslope length through the
inversion of drainage density

Using the 20 topographic datasets generated in
Sect.

Three study sites from the United States have been selected for this study:
Santa Cruz Island, California; Gabilan Mesa, California; and the Oregon Coast
Range, Oregon. The first two sites have regularly spaced valleys at a range
of length scales, particularly Gabilan Mesa, which has been the focus of
previous work in this context

Gabilan Mesa, a section of the Central Coast Ranges in California, USA
(Fig.

A series of large, linear canyons running northeast to southwest are fed by
parallel tributaries which flow perpendicular to the main trunk channel.
These regularly spaced valleys present two distinct length scales in the
landscape which have been observed both qualitatively

Maps showing the spatial variation in total curvature measurements
as grid resolution is decreased for the same section of Santa Cruz Island as
displayed in Fig.

Santa Cruz Island (Fig.

The island has a Mediterranean climate similar to that of Gabilan Mesa

Plots of the distribution of

The Oregon Coast Range in Oregon (Fig.

Figure

Plots of the reduction in range between the 2nd and 98th percentiles
(blue triangles) and the interquartile range (red circles) of

Representative sections of each landscape's channel network
displaying the extent of each network as grid resolution is decreased. Panels

Figure

In the Oregon Coast Range for both measurements of curvature, there is little variation between the 1, 2, and 3 m datasets, with a broad range of measurements shown in the probability distributions. Beyond this point the mean and median do not significantly change, but as in Santa Cruz Island, the overall distribution of measurements compresses towards the average value for the landscape. The Gabilan Mesa data show similar trends to those of Santa Cruz Island but exhibit less variability at lower resolutions. The probability distributions of each measurement also exhibit less change with resolution than the other two datasets, indicating a reduced sensitivity to grid resolution at this location.

The variations in reliability (Eq.

Figure

Figure

In Santa Cruz Island the geometric method's reliability index is similar to Gabilan Mesa; however, the sensitivity index is not as high, which indicates that a large number of channel heads are being missed, but where a prediction is made, it is typically accurate. The DrEICH method exhibits a similarly large reliability initially but again shows more rapid degradation in the index value as grid resolution is decreased. The sensitivity values again decline more rapidly and reach a 0 value at 20 m grid resolution.

Changes in the estimated sediment transport coefficient,

Published parameters used to calculate diffusivity.

Reliability and sensitivity metrics for the DrEICH method of channel extraction.

Reliability and sensitivity metrics for the geometric method of channel extraction.

The data for the Oregon Coast Range show similar patterns for both methods, although the geometric method exhibits systematically larger index values. In each case the reliability increases slightly from 5 to 10 m resolution and then declines gradually towards 30 m resolution. The sensitivity indexes for both methods begin at a larger value than the reliability indexes and steadily decline towards 0. A sensitivity value exceeding the reliability value suggests that in this landscape there are fewer missed channel heads in the 5 m data but at the expense of too many predicted channel heads in locations where there are none predicted in the 1 m data.

Plots of the distribution of hillslope length

Using the values for hilltop curvature generated in Sect.

The hillslope length measurements for Santa Cruz Island calculated using 1 m
channel heads (Fig.

In Gabilan Mesa the hillslope length measurements calculated using 1 m
channel heads (Fig.

Plots of the distribution of hillslope length

Plots of the distribution of hillslope length

The hillslope length measurements for the Oregon Coast Range with channel
heads from the 1 m data (Fig.

Across the three landscapes the variance of the distributions of both total
and tangential curvature values are systematically reduced as resolution is
decreased, an effect that is particularly notable after the grid resolution
exceeds 3–4 m (Fig.

In producing a DEM, we are sampling a complex two-dimensional elevation
signal, in which spatial variations in geomorphic processes drive variations in
topographic amplitude at different wavelengths

We can explain some of the observed behavior in Figs.

We will examine the spectral behavior of a simplified one-dimensional system. We acknowledge that a 1-D approach cannot fully describe complex two-dimensional topography of real landscapes, but a one-dimensional system is amenable to mathematical treatment that can at least give us qualitative insight into trends observed in our data. In addition, some of the features of interest, for example ridgelines and channels, can be roughly approximated as one-dimensional structures within a two-dimensional landscape.

Curvature in one dimension,

In the case of our curvature filter (Eq.

Plot of fidelity (

Again, fidelity is a measure of how closely our discrete filter (here
curvature measured at discrete points in the landscape) reflects the true
curvature (that is, the curvature measured if we had a perfectly continuous
dataset). Fidelity is a function of the ratio between the grid interval and
the wavelength (Fig.

As the frequency approaches the Nyquist wave number, defined as

What does this mean in practical terms? In our simple, one-dimensional example, if we use 1 m resolution data we can only capture the curvature of a one-dimensional ridgeline that had a wavelength of 3–4 m (one does not need the entire wave to capture the peak of the waveform) but with a loss of fidelity on the magnitude of the curvature. Or, in other words, we would underestimate the magnitude of the curvature.

Another landscape metric that is widely measured is topographic gradient. In
our study we have not computed how topographic gradient varies as a function
of grid resolution because this has been examined by many previous authors
(e.g.,

Equation (

Equation (

Having explored simplified one-dimensional filters, we now return to our two-dimensional results. Although real landscapes are two-dimensional and we use
polynomial fitting rather than simple differencing as in
Eq. (

The loss of fidelity predicted by the simple one-dimensional system
(Eq.

Santa Cruz Island and the Oregon Coast Range have the highest tangential
curvature at 1 m resolution. High tangential curvature at Santa Cruz Island
corresponds to observations of extensive gullying and hillslope erosion

It is intuitive to consider that when extracting channel networks at any data resolution, regardless of method, the higher-order, larger channels will be more accurately constrained than lower-order channels. This pattern is observed in each of the study landscapes, with the majority of the variations in channel locations occurring in first- and second-order channels. Such loss of low-order channels from datasets has implications for studies focusing on upland areas, in particular where detailed measurements which depend on channel network position are performed.

The contrast between the extent of channel networks and their indexes of
quality for the two methods outline that a geometric method of channel
extraction outperforms the process-based DrEICH algorithm. Due to the
relative simplicity of the geometric method of channel extraction, errors
inherent in the DEM are not compounded on the same scale as the DrEICH
algorithm, which performs more operations on topographic data. As the
geometric method identifies channels based on their tangential curvature,
although channel head features may be smoothed out of the DEM as resolution
is decreased, the channel will still express some positive curvature in lower-resolution data. The initiation point may be located downslope of the true
channel head but even in this worst case most of the channel network will be
extracted correctly. This is observed in Fig.

The indexes of quality defined by

This assessment of high-resolution methods with degraded-quality data
demonstrates the ongoing challenges that channel extraction poses to the
geomorphology community.

The predicted values of the sediment transport coefficient (

The sediment transport coefficients calculated at the Oregon Coast Range and
Santa Cruz Island locations both increase with grid resolution, reflecting
the sensitivity of

These data suggest that estimating

Measurements of hillslope length and relief have been used to test sediment
flux laws

The relief measurements for each landscape, however, show more sensitivity to
grid resolution, with a systematic increase in the median values in each
location beyond 10 m grid resolution. As decreasing grid resolution acts as
a low-pass filter on the landscape, the elevation of ridges are expected to
be reduced, whilst the elevation of channel beds are raised, producing a net
reduction in topographic relief. However, the increased relief observed with
decreasing grid resolution is produced by the decrease in drainage density
with decreasing resolution observed in Fig.

By contrasting the

Through the generation of topographic data spanning the range of grid resolutions
currently used in much of geomorphic research, a number of key metrics have
been evaluated for their sensitivity to grid resolution. We have demonstrated
the reduction in the range of total and tangential curvature values as grid
resolution is decreased, across three test landscapes. These curvature
measurements are important in the estimation of the hillslope sediment
transport coefficient (

The extraction of channel networks from digital topographic data is a
significant challenge on all spatial scales, as the definition of a channel
network is integral in the execution of many analyses (e.g.,

Average values of hillslope length and relief for each landscape are shown to be broadly insensitive to grid resolution up to grid resolutions which correspond to the highest-resolution topographic data globally available. This indicates that these measurements can be used to identify landscape transience in locations where lidar data are unavailable. The accuracy of these measurements is dependent on the accuracy of the channel network used, however, as using a geometric method of channel extraction from the 1 m DEM still provides robust measurements of hillslope length and relief.

The relationships between decreasing grid resolution and the geomorphic parameters explored here demonstrate the influence of the spatial scale of the topographic expression of process on the quality of results which can be extracted from lower-resolution topography. From these analyses it is challenging to identify a clear threshold below which data become unsuitable for use in geomorphic analysis. Rather, it is important to highlight the influence of landscape morphology and the dominant processes acting upon it in the selection of an appropriate data resolution for a study. Using this work as a framework, it is now possible to place constraints on the accuracy of results derived from coarse-resolution topographic data, particularly where non-topographic or field data can be used to provide insight into general landscape morphology.

All of the code used in this analysis is open source and the topographic
analysis routines are available at

The topographic data used in this study are freely available from

This table provides the parameters used to generate channel networks both
using the geometric method and the DrEICH method. The drainage area value is
used to thin the initial extracted network by removing channels which have a
drainage area below the threshold value. The connected-components value
defines the point at which a group of contiguous channel pixels are
considered to be connected. The

Parameters used by the geometric and process-based techniques in the extraction of channel networks.

Stuart W. D. Grieve, Simon M. Mudd, David T. Milodowski, and Fiona J. Clubb wrote the software. Stuart W. D. Grieve performed the analysis. David J. Furbish and Simon M. Mudd resurrected the spectral filtering analysis from an unpublished 2002 manuscript because they are lovers of the long game. Stuart W. D. Grieve wrote the paper with contributions from the other authors.

Simon M. Mudd and Stuart W. D. Grieve are funded by NERC grant NE/J009970/1 and Simon M. Mudd is funded by US Army Research Office contract number W911NF-13-1-0478. Fiona J. Clubb is funded by the Carnegie Foundation for the Universities of Scotland. David T. Milodowski was funded by a NERC Doctoral Training Grant NE/152830X/1 and NE/J500021/1. David J. Furbish was funded by US National Science Foundation grant EAR-1420831. We thank Marie-Alice Harel, Kristin Sweeney, Wolfgang Schwanghart, and two anonymous reviewers for comments on earlier versions of this manuscript. Edited by: R. Gloaguen Reviewed by: K. Sweeney, W. Schwanghart, and two anonymous referees