Downstream hydraulic geometry relationships: Gathering reference reach-scale width values from LiDAR

Geomorphology 250 (2015) 236–248

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Downstream hydraulic geometry relationships: Gathering reference reach-scale width values from LiDAR G. Sofia a,⁎, P. Tarolli a, F. Cazorzi b, G. Dalla Fontana a a b

Department of Land, Environment, Agriculture and Forestry, University of Padova, Agripolis, viale dell'Università 16, 35020 Legnaro (PD), Italy Department of Agricultural and Environmental Science, University of Udine, Viale delle Scienze, 208-33100 Udine (UD), Italy

a r t i c l e

i n f o

Article history: Received 20 December 2014 Received in revised form 2 September 2015 Accepted 4 September 2015 Available online 9 September 2015 Keywords: LiDAR DTM Feature extraction Downstream hydraulic geometry relationship

a b s t r a c t This paper examines the ability of LiDAR topography to provide reach-scale width values for the analysis of downstream hydraulic geometry relationships along some streams in the Dolomites (northern Italy). Multiple reach-scale dimensions can provide representative geometries and statistics characterising the longitudinal variability in the channel, improving the understanding of geomorphic processes across networks. Starting from the minimum curvature derived from a LiDAR DTM, the proposed algorithm uses a statistical approach for the identification of the scale of analysis, and for the automatic characterisation of reach-scale bankfull widths. The downstream adjustment in channel morphology is then related to flow parameters (drainage area and stream power). With the correct planning of a LiDAR survey, uncertainties in the procedure are principally due to the resolution of the DTM. The outputs are in general comparable in quality to field survey measurements, and the procedure allows the quick comparison among different watersheds. The proposed automatic approach could improve knowledge about river systems with highly variable widths, and about systems in areas covered by vegetation or inaccessible to field surveys. With proven effectiveness, this research could offer an interesting starting point for the analysis of differences between watersheds, and to improve knowledge about downstream channel adjustment in relation, for example, to scale and landscape forcing (e.g. sediment transport, tectonics, lithology, climate, geomorphology, and anthropic pressure). © 2015 Elsevier B.V. All rights reserved.

1. Introduction

(e.g., Wolman and Leopold, 1957; Emmett, 1975; Andrews, 1980; Carling, 1988; Finnegan et al., 2005; Wohl and David, 2008):

1.1. Background and aims of research w ¼ aQ b Natural rivers develop channels in a broad range of forms (Knighton, 1987). As a consequence, the flow width (w) and the morphological characteristics of a stream are critical parameters in a wide range of hydrologic applications. Detailed topographic data are therefore a fundamental requirement (Biron et al., 2013; Cavalli et al., 2008; Charlton et al., 2003; Heritage and Hetherington, 2007; Hilldale and Raff, 2008; Jones et al., 2007; Lane et al., 2003; Tarolli, 2014). Analysing bankfull widths at a reach-scale can provide representative geometry and statistics characterising longitudinal variability in a channel (Wohl et al., 2004; Stewardson, 2005; Harman et al., 2008; Xia et al., 2014). Flow width generally changes depending on the bankfull discharge, as the conditions forming the shape and morphology of a channel with recurrence intervals ranging between one and two years (Wolman and Leopold, 1957). The reach-scale flow width and discharge (Q ) are related through a power law relationship

⁎ Corresponding author. E-mail address: giulia.sofi[email protected] (G. Sofia).

http://dx.doi.org/10.1016/j.geomorph.2015.09.002 0169-555X/© 2015 Elsevier B.V. All rights reserved.

ð1Þ

where a and b are empirically derived constants. Channel geometry also reflects the capacity of the stream to transport sediment (Vianello and D'Agostino, 2007), thus widths may also be related to the stream power, which defines the rate of energy expenditure per unit length of the channel (Bagnold, 1966; Brummer and Montgomery, 2003; Finnegan et al., 2005). Analysing the stream power in relation to the sediment grain sizes also allows identification of thresholds for the adjustment of flow width in response to changes in discharge (Wohl, 2004). The empirical parameters of the downstream hydraulic geometry relationships are normally estimated by regression, using values surveyed at multiple cross-sections having two or more discharges. The comparison between watersheds is made through field work by researchers, but it is limited to specific field-surveyed case studies. Although some countries have numerous investigated sites (i.e. Faustini et al., 2009), in Italy, hydraulic relationships are available only for few areas in the Dolomites (i.e. Lenzi, 2001; D'Agostino and Vianello, 2005; Vianello and D'Agostino, 2007; Comiti et al., 2007; Wilcox et al., 2011) and some rivers in central Italy (i.e.

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Orlandini and Rosso, 1998; Whittaker et al., 2007a, 2007b; Yanites and Tucker, 2010). LiDAR datasets, on the other hand, are nowadays widely available, including for public download (e.g. the NSF-EAR-funded data facility OpenTopography, or the LiDAR geoportal for the Alpine region Trentino Alto Adige in Italy). Questions still need to be asked, such as why rivers with similar drainage areas have different widths. Some field-related works underlined how the nature of channels affects this, but urbanisation (e.g. Hession et al., 2003), geology and climate should also be considered. The availability of a large number of downstream relationships would allow investigation of the differences between watersheds at multiple scales, and in areas subject to various landscape forcings, such as sediment transport, tectonics, lithology, climate, geomorphology, and anthropic pressure. Clearly, although it is theoretically possible to measure bankfull width manually, it becomes impractical if hundreds or thousands of river width values must be obtained. During the past two decades, advances in remote-sensing technology have allowed the fast, precise and effective acquisition of topographic information with high quality (see Tarolli, 2014 for a full review), and the field of fluvial geomorphology has seen increased application of high resolution surveying technologies to characterise river bathymetry and floodplain topography (Heritage et al., 2009; Milan et al., 2011; Marcus, 2012; Sofia et al., 2014a, 2014b). Different data-driven methods have been proposed for channel geometry (Pavelsky and Smith, 2008; McKean et al., 2009; Johansen et al., 2011; Biron et al., 2013; Fisher et al., 2013; Güneralp et al., 2014; Bangen et al., 2014); however, an automated method for continuously extracting reach-scale width values from raster-based imagery would provide valuable insight into many hydrologic studies (Pavelsky and Smith, 2008). Limits related to channel sizes compared to data resolution have been noted (e.g. McKean et al., 2009), and automation should be further analysed, especially concerning: 1) small channels or channels with great variability in width; 2) analysis of channels in complex landscapes or covered by vegetation; and 3) the choice of the scale of analysis. The most recent literature includes pioneering approaches to the automated characterisation of landscape components: several authors have shown that physical processes or anthropogenic activities leave important signatures in the statistics of the morphological parameters derived from DTMs, and they have pointed out that by quantifying these signatures in detail, statistics or objective indexes can be used at different scales to automatically detect thresholds through which to identify processes, extract particular features, or characterize landscapes (e.g. Molloy and Stepinski, 2007; Thommeret et al., 2010; Passalacqua et al. 2012; Pelletier, 2013; Clubb et al., 2014; Sofia et al., 2014c, 2015; Chen et al. 2015; Tarolli et al. 2015; Prosdocimi et al., 2015). Following this line of research, this paper examines the ability of LiDAR topography to automatically provide reach-scale width values for the analysis of downstream hydraulic geometry relationships along some streams in the Dolomites, northern Italy. The approach provides reach-scale width data, rather than exact channel width at each location of the network. The automatically-captured downstream channel morphology is then related to flow parameters including drainage area and stream power.

The drainage area (A) is the morphometric variable most directly correlated to the peak discharge (Strahler, 1964). The link between Q bkf and A is so clear that it enables Q bkf to be estimated from A (Wolman and Leopold, 1957; Leopold et al., 1964; Emmett, 1975; Dunne, 1987; Leopold, 1994; Rice, 1998; Whiting et al., 1999; Brummer and Montgomery, 2003):

1.2. Summary of equations and related literature

Ω ¼ γðgmÞAðh−θÞ :

This section briefly summarises the main downstream hydraulic geometry relationships that are considered in the research. Flow width varies with bankfull discharge (Q bkf), generally exhibiting a power law relationship: w ¼ dQ bk f e

ð2Þ

where d and e are empirically derived constants (Wolman and Leopold, 1957; Emmett, 1975; Andrews, 1980; Carling, 1988; Finnegan et al., 2005; Wohl and David, 2008).

Q bk f ¼ gAh

ð3Þ

where g and h are constants. As a consequence, various researchers have found a direct proportionality in different environments for channel flow width and basin drainage area, which follows a power law similar to Eqs. (2) and (3) (Leopold and Maddock, 1953; Harman et al., 1999; Ibbitt, 1997; Rodriguez-Iturbe and Rinaldo, 1997; Knighton, 1998; Whiting et al., 1999; Dutnell, 2000; Montgomery and Gran, 2001; Brummer and Montgomery, 2003; Vianello and D'Agostino, 2007): w ¼ sAt

ð4Þ

where s and t are constants. For many mountain streams, however, channel geometry reflects not only the downstream variation of bankfull discharge, but also the magnitude of bank erosion and the consequent changes in crosssection width (Vianello and D'Agostino, 2007). The main parameter responsible for the capacity to transport sediment, associated with the downstream coarsening of headwater channels, is the stream power (Ω) which defines the rate of energy expenditure per unit length of the channel (Bagnold, 1966; Brummer and Montgomery, 2003). Values of Ω, associated with the cross-sections at the bankfull condition, can be calculated from Q bkf and local channel slope (S) (Bagnold, 1966; Richards, 1976; Ferguson and Lewin, 1981; Keller and Brookes, 1983; Ferguson et al., 1987; Van den Berg, 1995), according to the equation: Ω ¼ γ Q bk f S

ð5Þ

where γ is the specific weight of the water. Using Eqs. (3) and (5), the stream power can be computed as a function of A. S generally varies empirically as a function of A, in the form S ¼ mA−θ

ð6Þ

where m and θ are empirical constants representing the steepness and concavity of the river profile, respectively (Hack, 1957; Seidl and Dietrich, 1992; Montgomery and Foufoula-Georgiou, 1993; Montgomery, 2001; Stock, 2003; Brummer and Montgomery, 2003; Sklar and Dietrich, 2013). Therefore, Ω can be re-written as: ð7Þ

The erosive power of flowing water can also be evaluated based on the assumption that discharge is proportional to a specific catchment area (As), which is the catchment area draining across a unit width of contour (m2 m−1), and the Stream Power Index (SPI) can be computed from digital topography according to the formulation SPI ¼ As tanβ

ð8Þ

where β is the slope gradient in degrees (Wilson and Gallant, 2001).

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Bankfull widths may then be related to Ω or SPI, according to the relationship (Knighton, 1999; Fonstad, 2003; Golden and Springer, 2006; Vianello and D'Agostino, 2007): w ¼ rSPI f

or

rΩ f

ð9Þ

where r and f are empirically derived constants. 2. Study areas and data In this work, we considered one main study area, the upper Cordevole basin at la Vizza-Arabba (Fig. 1a), and two secondary study sites, the Rio Vauz basin (a sub-basin of the Cordevole watershed, Fig. 1a) and the Rio Cordon basin (Fig. 1b). The Rio Cordevole and Rio Vauz are alluvial channels, and at the bankfull stage their flows reach the top of the banks (Vianello and D'Agostino, 2007). Parts of the channels and banks of the Rio Cordon, however, are frequently not alluvial, featuring bedrock outcrops and/ or large boulders derived from hillslope processes. Channel adjustment to water sediment flow is heavily reduced, becoming more similar to that seen in bedrock channels (Mao et al., 2005). The Rio Cordon at the bankfull stage presents flows that in some parts are below the bank line, and in others slightly overflow the channel (Mao et al., 2005). The Rio Cordevole watershed has been the subject of hydrological and geomorphological research since the 1990s (Cazorzi and Dalla Fontana, 1996; D'Agostino and Vianello, 2005). The Rio Vauz basin has also been considered a field laboratory for years, mainly in the analysis of landslide activities and soil moisture variability (Borga et al., 2002; Penna et al., 2013). Previous studies in the Rio Cordon mainly concerned the measurement and assessment of bedload (Lenzi, 2001), the morphological structure and sedimentology of the stream bed (Lenzi et al., 1999), and analysis of sediment sources (Marchi and Dalla Fontana, 2000). The Cordon basin has also been considered for morphological analysis based on LiDAR data (Passalacqua et al., 2010; Tarolli et al., 2012). LiDAR data were acquired for the three study sites using a helicopter with an ALTM 3100 OPTECH, and a Rollei H20 digital camera, flying at an average altitude of 1000 m above ground during snow-free and low discharge conditions in 2006. The flying speed was 80 knots, the scan angle 20° and the pulse rate 71 kHz. The survey design point density was specified as greater than 5 points m−2, recording up to four returns, including first and last. LiDAR point measurements were filtered into returns from vegetation and bare ground using the Terrascan® software

classification routines and algorithms. The LiDAR bare ground dataset was used to generate a Digital Terrain Model (DTM) of 1 m grid cell size, with vertical accuracy of about 0.3 m, an acceptable value for many LiDAR analyses in the field of geomorphology (McKean and Roering, 2004; Glenn et al., 2006; Frankel and Dolan, 2007). Table 1 shows the main characteristics of the basins and streams considered, as derived from field surveys, literature or from the LiDAR DTMs. For the Rio Cordevole site, field surveyed data for bankfull widths and morphological units are available for a total of 97 values of reachscale average widths, and have already been published (Vianello and D'Agostino, 2007). For the Cordon basin, Comiti et al. (2007) and Wilcox et al. (2011) present values of reach-scale widths as compared to discharge for 30, and 42 cross sections respectively. Finally, we recently conducted a field survey of bankfull widths for a total of 20 reach-scale width values for the Vauz basin and the connected average drainage areas. Measurements of bankfull widths in the Vauz basin were conducted following the work by Vianello and D'Agostino (2007). To obtain accurate bankfull widths, we identified the water level reached by the formative discharge (Wolman and Leopold, 1957), which corresponds to the top of the bank for the Vauz stream, as for the Cordevole river. Two or three width measurements per channel reach were made, and then a mean width per reach was calculated. 3. Methods 3.1. Quantification of topographic variability The use of a LiDAR DTM for width measurement requires a statement of some assumptions. The infrared signal from non-bathymetric LiDAR is largely absorbed by water, leading to data dropouts or missing values in the point cloud. These dropouts are usually interpolated from surrounding data points. The river water surface thus often appears as made by flat, or only slightly inclined, triangular facets in the resultant DTM. On the digital model, therefore, the minimum bankfull extent at each location of the stream reflects the width of the channel actually occupied by water. Having square regular pixels, the DTM simplifies the channel topography, but it allows an approximation of the channel width that corresponds at its minimum to the pixel size (Cazorzi et al., 2013). Giving these assumptions, an exact local characterisation of a channel is not expected; instead, the aim is the average characterisation of width at a reach scale.

Fig. 1. Investigated streams and their catchments, as displayed by the LiDAR DTMs. a) Rio Cordevole watershed and the sub-basin of the Rio Vauz stream. The black arrow indicates the channel investigated by Vianello and D'Agostino (2007), while the white dots show the locations surveyed in 2014. b) Rio Cordon catchment and investigated stream.

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Table 1 Characteristics of the watersheds and streams studied, derived from literature, field surveys, and LiDAR DTMs (slope and height). Watershed characteristics

Rio Cordevole Rio Vauz Rio Cordon

Stream characteristics

Area (km2)

Height (avg) (m a.s.l.)

Slope (avg) (%)

Length (m)

Width (avg) (m)

Width range (m)

7.0 1.9 5.0

2274 2454 2257

68.2 76.7 74.5

2886 1444 2455

2.67* 2.57** 5.70***

0.50–7.28* 1.60–4.10** Highly variable, with drastic changes after discharges of 50 years of Return Time****

*Vianello and D'Agostino (2007), **Field surveyed, ***Comiti et al. (2007), ****Lenzi (2001). Values in the width range column refer to localised widths, not to reach-scale ones.

The main parameter considered for the measurement of the width is the minimum curvature (Cmin) (Evans, 1972, 1979; Wood, 1999), generalised according to a moving window (kernel) as  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 C min ¼ k  g  − a−b− ða−bÞ þ c2

ð10Þ

where k is the size of the moving window, g is the DTM resolution, and a, b and c are coefficients derived after fitting a biquadratic equation to the original surface, and differentiating it (Evans, 1972; Wood, 1999; Sofia et al., 2011, 2013, 2014a; Tarolli et al., 2012). Cmin is a fundamental surface property whose application, for example in geomorphology and hydrology, has long been recognised (see e.g. Shary, 1995), because quadratic models are more reliable and resilient to data errors (Schmidt et al., 2003). The presented procedure can be divided into three steps: 1) identification of homogeneous reaches, and of the hydrologic floodplain; 2) characterisation of transects orthogonal to the thalweg; and 3) calculation of the width of each transect and calculation of the reach-scale width. Fig. 2 is a conceptual workflow of the procedure. Only two initial inputs are required by the procedure. The first is the DTM; the second is a raster map representing the thalweg. The main requirement for the thalweg map is that its location is robustly consistent with surface convergences: recent literature has pioneered numerous

methods for this type of extraction (Lashermes et al., 2007; Passalacqua et al., 2010; Sofia et al., 2011), although the details of such techniques are beyond the scope of this work. In the present work, we considered the procedure proposed by Sofia et al. (2011). This approach has been chosen because it provides a framework for the automatic definition of the scale of analysis, which will be used for the width extraction. This will be further described in the following sections. 3.2. Identification of homogeneous reaches and hydrologic floodplains The identification of homogeneous reaches is based on analysis of the drainage area map. Drainage area and discharge increase linearly, and abrupt changes in discharge (or drainage areas as a proxy) might result in variation of bankfull widths (Eqs. (2) and (4)). We considered those presenting a progressive increase of drainage area to be homogeneous reaches (Fig. 3a), since the discontinuities between the clusters of points are due to the intake of lateral reaches, and they might result in changes in discharge. This simple assumption provides a tool to automatically label reaches without requiring an a-priori database of field measurements. The identification of the hydrologic floodplain is an important step, since it is the dynamic plain within which the main channel develops: the bankfull flow width can be smaller than the hydrologic channel, but not larger. By focusing the analysis on this extent, it is possible to

Fig. 2. Workflow of the proposed approach.

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Fig. 3. Identification of homogeneous reaches and of the hydrologic floodplain. Homogeneous reaches are those presenting a progressively increasing drainage area (a). For each reach, the hydrologic floodplain (b) corresponds to a binary mask having a width of kstart around each side of the thalweg.

reduce biases due to noise and local morphologies not necessarily related to channelized processes. The extent of the hydrologic floodplain corresponds to an optimum watershed-scale of analysis. A review of recent literature underlined how the curvature of the landscape is the key component in the location of the channel network (Lashermes et al., 2007; Passalacqua et al., 2010; Pelletier, 2013; Sofia et al., 2011; Clubb et al., 2014). Sofia et al. (2011), as confirmed by Sofia et al. (2013), showed that in the absence of errors on the DTM, there is a strong relationship between the asymmetry (skewness) of the curvature distribution, and the size of the main feature in the analysed area: the optimum scale of analysis then corresponds to the higher asymmetry of Cmin. The identification of the hydrologic floodplain is achieved via the fitting-enforcing approach proposed by Sofia et al. (2011). Considering Cmin, the procedure works as follows: the topographic parameter is calculated using different moving windows or kernels (k in Fig. 4), from 3 to 33 cells, and the skewness of each curvature map is derived. The skewness varies as a function of the moving window size, and the fitting-enforcing method allows iterative identification of a polynomial that provides a curve representing this variation (Fig. 4a), forcing the polynomial degrees in order to have at least one stationary point (a point having the highest skewness in the negative domain) within the kernel range. Analysis of the polynomial derivative (Skew'k) allows

automatic identification of the stationary point, where the first derivative of the polynomial equals zero (Fig. 4b). This stationary point is assumed to be the optimum kernel size, kstart (Sofia et al., 2011). Once the optimum kernel size is defined, a binary dilation of the thalweg is applied using a disk-shaped structuring element (SE) with a radius of two pixels, performing the operation a number of times equal to kstart. For computational efficiency (Haralick and Shapiro, 1992; Lin et al., 1998), an SE of small size and simple shape is chosen, with a small number of hits, and the binary operation is computed considering successive dilations using this small structuring element, rather than computing the dilation once using a large SE. The result of this step is a binary mask having a width of kstart around each side of the thalweg (Fig. 3b). This mask indicatively corresponds to the hydrologic floodplain. 3.3. Reach-scale transect and curvature characterisation As already stated, the bankfull flow width can be smaller than or equal to that of the hydrologic floodplain. However, it generally increases in the downstream direction as discharge or drainage area increases (Eqs. (2) and (4)). For each homogeneous reach (Fig. 3a) the fitting-enforcing procedure is applied using the skewness of Cmin within

Fig. 4. Schematization of the fitting-enforcing procedure. (a) Skewness of Cmin at the increase in the kernel size, and the fitted polynomial with its stationary point highlighted by the arrow. (b) Derivative of the fitted polynomial (Skew'k) identifying the stationary point where the derivative equals zero.

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Fig. 5. Example of the power law relationship (Rio Cordevole) between automatically derived kernel width (ktemp) and mean drainage area (A) for each homogeneous reach. The derived power law is kopt = 4.9 A0.51 (R2 = 0.80). The 95% confidence bounds of the power law are also shown.

the hydrological floodplain, and obtaining an indicative reach-scale kernel size (ktemp). For the whole watershed, then, the average drainage area per reach is plotted against the ktemp values (Fig. 5). ktemp overall increases with the increase of the drainage area in the downstream direction. If there is a relationship between bankfull width and drainage area (Eq. (4)), we should consider a similar relationship between the drainage area and a reach-scale of analysis in terms of the optimum kernel size for curvature evaluation (kopt) and the transect

length for each reach. Following the downstream hydraulic geometry relationships, we define a power law between the drainage area and the optimum kernel size in the form kopt ¼ vAl

ð11Þ

where A is the average drainage area and v and l are constants derived through least squares fitting, using the average drainage area per

Fig. 6. Automatic measurement of bankfull width. (a) Minimum curvature map and considered transect, perpendicular to the thalweg. (b) QQ-plot approach to define the threshold. (c) Extracted width; in this case about 2 m.

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reach and the fitting-enforcing derived ktemp values. Applying this rule, outlier kernels that could be present due, for example, to specific natural or artificial morphologies, are eliminated and the characterisation is more regular (Fig. 5). The transect length, for each homogeneous reach, then, equals kopt according to Eq. (11) on each side of the thalweg. Cmin, for computational constraints, is instead calculated for each homogeneous reach with an optimum kernel equal to the odd number closest to kopt. 3.4. Bankfull width characterisation On the curvature map (Fig. 6a), the values of the index can be read along each transect. The hypothesis here is that the curvature can be considered as a proxy of the channel geometry, and the bankfull limit can be identified through a statistical threshold (Fig. 6b). The threshold that identifies the bankfull is given by the QQ-plot (quantile–quantile plot) approach (Fig. 6b). The QQ-plot displays the sample quantiles of Cmin read along the thalweg versus theoretical quantiles from a normal distribution. If the probability distribution of Cmin is normal, the plot will be close to linear. Lashermes et al. (2007) and Passalacqua et al. (2010) found that a sharp deviation in the tail of the probability distribution of curvature from a Gaussian distribution defined a critical threshold delineating well-organised valley axes. Sofia et al. (2011) and Tarolli et al. (2012) successfully applied the same threshold for channel and ridge extraction. Following a similar concept, for the proposed approach, the point where the distribution of Cmin values deviates from normality can thus be considered a threshold (Fig. 6b) that identifies the channel bankfull width. The deviation value is simply identified as the first point, in the negative tail, at which the curve diverges from the straight line. Bankfull width then corresponds to the width flanked by the two points of intersection between the curvature line and the line representing the QQ-plot threshold (Fig. 6c). In this way, the width can also have values that are not multiples of the pixel size. A unique average width value is then computed per homogeneous reach. 4. Results and discussion 4.1. Scale of analysis and downstream cross-section widening We computed Eq. (11) for the Rio Cordevole, using the fittingenforcing derived kernel values, and the average drainage area computed for each homogeneous reach (Fig. 3a). The intercept and exponent of Eq. (11) were obtained using least-squares fitting, and from these values, hypothetical kopt were computed considering the drainage areas for the Rio Cordevole according to Vianello and D'Agostino

(2007). The derived kopt values were then plotted versus the values of measured bankfull widths (wm, Fig. 7). In general, for a given drainage area, the derived kopt values range between twice and three times the actual bankfull width size, as indicated by the confidence bounds in Fig. 7. Other works (Pirotti and Tarolli, 2010; Sofia et al., 2011, 2014a; Tarolli et al., 2012; Lin et al., 2013) have explained how the optimum kernel size for curvature should be approximately twice or three times the spatial extent of the features of interest. The use of Eq. (11) seems reliable in this sense. Fig. 8 shows the extracted bankfull width for the three watersheds, considering both a transect-scale width and the reach-scale widths. The transect-scale widths show a general trend towards an overall increase in the downstream direction; however, numerous outlier widths are present, mainly due to local morphologies within or near the channels. Biases may also be due to the creation of transects orthogonal to a raster thalweg. This step implies a geometric simplification that can result in localised errors: a line perpendicular to a raster in the synthetic domain does not necessarily correspond to an ideal line perpendicular to the actual channel thalweg.Another source of bias can be the water extent at the time of the field survey (as expressed at the beginning of Chapt. 3.1). When considering a reach-scale width instead, these biases are overall reduced, and in general the procedure can capture overall channel width variability in the downstream direction: the downstream increase in reach-scale widths is captured for all the considered streams. The actual width of the Rio Cordevole ranges between 1 and 7 m, and the results with the transect widths are in line with this data. Vianello and D'Agostino (2007) found the last section of the channel to have an average reach-scale width of about 4.5 m for drainage area values greater than 5 km2. Overall, the extracted reach scale widths follow these values. The reach-scale extracted widths have an average value of 2.8 m, a measure comparable to the actual channel size (2.7 m according to Vianello and D'Agostino, 2007). Similar conclusions can be drawn for the Rio Vauz (Fig. 7c), which has reach scale widths from 1 to 4 m according to field surveys, and an overall average value of 2.6 m (Table 1). The extracted reach-scale widths cover a similar range, and present a slightly underestimated average value of 2 m. In both the Rio Cordevole and Rio Vauz, however, the overall error is smaller than the pixel resolution. The Rio Cordon (Fig. 7b) presents a slightly greater underestimation of width (~ 2 m). According to Comiti et al. (2007) and Wilcox et al. (2011), the lowest reach of the channel presents widths larger than 6 m, while our methodology shows widths below 5 m. The underestimation is of about 2 m when comparing the average width measured in the field (5.7 m, Table 1) with the estimated average width (3.5 m). One must note that, despite the good quality indicated by the final vertical accuracy of 0.3 m, the Rio Cordon DTM presents some striping

Fig. 7. Measured widths for the Rio Cordevole (Vianello and D'Agostino, 2007) compared to kopt values derived using the drainage area published by Vianello and D'Agostino (2007), and the coefficients of Eq. (11) calculated using the reach-scale drainage area and the kernels calculated with the fitting-enforcing approach.

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Fig. 8. Transect-scale widths and reach-scale widths for the Rio Cordevole (a), Rio Cordon (b), and Rio Vauz (c).

artefacts (Sofia et al., 2011, 2013), depending on the nature of the ground point survey and point density. The ability to perform surface analyses from this DTM is not hindered by these errors, as demonstrated in other works (Passalacqua et al., 2010; Sofia et al., 2011, 2013; Tarolli et al., 2012), but it should not be confused with the ability to calculate reliable, unbiased morphometric parameters (Albani et al., 2004). For this specific case study, if DTMs are without errors, the scale-dependent change in curvature distribution is primarily controlled by the real morphology, and the fitting-enforcing results in window sizes related to the size of the main features. When DTMs include errors, however, curvature distributions become controlled by the errors (Sofia et al., 2013), and the fitting-enforcing ‘forces’ the kernel size, giving values that might not be related to the actual feature size. Furthermore, the precipitation regime and the characteristics of the Cordon basin create a dynamic geomorphic setting in which infrequent floods reorganize the channel morphology Table 2 Constants of the best-fit equations of each power law relationship for the Rio Cordon, after correcting the extracted widths considering the overall underestimation (2 m). Values shown include the best-fit value and the 95% confidence bounds (in parentheses). Corrected value w=sA

t

w = r SPI

f

s t r f

3.45 (±0.60) 0.30 (±0.12) 0.44 (±0.95) 0.18 (±0.21)

(Lenzi, 2001; Wilcox et al., 2011). Depending on the time of the LiDAR survey and on the seasonal variation in flow dynamics before the survey, therefore, the morphology captured in the DTM could be slightly biased. The underestimation of widths in the Rio Cordon stream is mainly due to these facts. Table 2 reports the downstream hydraulic geometry relationships as computed after correcting the extracted widths according to the 2 m underestimation. 4.2. Rio Cordevole: comparison with field values Table 3 summarises the estimated coefficients of the downstream hydraulic geometry relationships compared to those reported by Table 3 Constants of the best-fit equations of each power law relationship, as automatically derived from the DTM, and published by Vianello and D'Agostino (2007) for the Rio Cordevole. Values shown include the best-fit value and, where available, the 95% confidence bounds (in parentheses).

w=sA

t

S=mA

−θ

w = r Ωf

s t m θ r f

Published

DTM-derived

2.66 0.28 0.13 0.12 0.42 0.31

2.30 (±0.24) 0.30 (±0.08) 0.11 (±0.01) 0.12 (±0.09) 0.39 (±0.13) 0.33 (±0.09)

244

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Vianello and D'Agostino (2007). Values shown include the best-fit value and, where available, the 95% confidence bounds for the coefficients of the equations (in parentheses). Considering the power law relationship between bankfull width and drainage area (with w in m and A in km2), the referenced work shows   w ¼ 2:66A0:28 R2 ¼ 0:62 :

ð12Þ

ð13Þ

The RMSE value of the computed relationship is less than the pixel dimension (1 m). The small difference between the exponent of the derived relationship and that by Vianello and D'Agostino (2007) is due to the slight underestimation of the extracted widths (~ 0.1 m, see Section 4.1). Differences in values are also influenced by data resolution: Vianello and D'Agostino (2007) considered the drainage area obtained from a 10 m Digital Elevation Model (DEM) derived from contour lines, while this research uses a 1 m LiDAR DTM. The automatic choice of homogeneous channel reaches according to the drainage area (Section 3.1) allows a definition of the relationship between local slope and drainage area (Eq. (6)). For these two variables, the published power law (with S dimensionless and A in km2) is in the form S ¼ 0:13A−0:12

  R2 ¼ unknown :

SPI ¼ 0:85Ω þ 138:4



 R2 ¼ 0:90 :

ð20Þ

Consequently, we can derive a relationship between the DTM derived widths and SPI. w ¼ 0:06SPI 0:29

Using the proposed procedure, the obtained power law reads w ¼ 2:30A0:30 ðRMSE ¼ 0:38mÞ:

Eq. (17) and the SPI evaluated from the LiDAR DTM (Eq. (8)), in the form

ð21Þ

ðRMSE ¼ 0:64 mÞ

Again, the exponent of the power law is similar to that in the field survey (Eq. (19)), and it is statistically significant (two-tailed Student's t-tests for statistical significance at α = 0.05). 4.3. Comparison between empirically derived downstream hydraulic geometry relationships and relationships referenced in literature Fig. 9 shows the relationship between the bankfull width and drainage area, the channel profile slope and drainage area, and bankfull width and the stream power relationships for the three streams considered. In the following, the best-fit equations are shown with the 95% confidence bounds (±) for the constants. For these equations, the results from analyses for the Rio Cordon are corrected to reflect the 2 m underestimation (the corrected data will be marked with *). Bankfull widths (in m) and drainage areas (in km2 Fig. 9a) follow the following relationships:

ð14Þ

w ¼ ð1:58  0:31ÞA0:630:46 ðRMSE ¼ 0:11mÞ

for the Rio Vauz

With the automatically identified reaches, we can calculate a comparable power law as

w ¼ ð3:45  0:60ÞA0:300:13 ðRMSE ¼ 0:69mÞ

for the Rio Cordon ð23Þ

S ¼ 0:11A−0:12

ðRMSE ¼ 0:02Þ :

ð15Þ

w ¼ ð2:30  0:24ÞA0:300:08 ðRMSE ¼ 0:38mÞ

ð22Þ

for the Rio Cordevole:ð24Þ

From the field survey data, the relationship for the Rio Vauz is: The constants of the power law are closely related to those in the field survey. With Eq. (15) and the bankfull width, it is possible to calculate the stream power according to Eq. (7). For this, g and h in Eq. (7) are from the empirical parameters of the relationship found by Vianello and D'Agostino (2007) between bankfull discharge and drainage area: Q bk f ¼ 0:162 A

ð16Þ

and m and θ are from Eq. (15). Eq. (7) then becomes: Ω ¼ γð0:162  0:11ÞAð1−0:12Þ

ð17Þ

With this constraint, we obtain a power law interaction between the automatically characterised bankfull widths and Ω. We considered A for each homogeneous reach, and derived Ω according to Eq. (17). Finally, we related Ω to the automatically measured width, thus obtaining the following power law: w ¼ 0:39Ω0:33

ðRMSE ¼ 0:38 mÞ :

ð18Þ

The published equation reads w ¼ 0:42Ω0:31

  R2 ¼ 0:68 :

ð19Þ

When using the automatically identified homogeneous reaches, we obtain a strong linear relationship between Ω calculated from

w ¼ ð1:91  0:30ÞA0:710:33 ðRMSE ¼ 0:52mÞ:

ð25Þ

Like the case of the Rio Cordevole (Section 4.2), Eq. (25) is similar to Eq. (22). For the lower reaches of the Rio Cordon stream, Comiti et al. (2007) found that widths and discharge are related with an exponent of 0.25 on average (standard deviation of 0.18), and the exponent of Eq. (23) is similar to that of Comiti et al. (2007). Eq. (23) suggests that there could be a linear or nearly linear relationship between the drainage area and discharge; h in Eq. (3) ~1 as found for the Cordevole watershed by Vianello and D'Agostino (2007), and in other catchments by Petit and Pauquet (1997) and Whiting et al. (1999). To confirm what Eq. (23) suggests, further field surveys are needed. Assuming that bankfull conditions become contemporary in the whole hydrographic network, and using discharges measured during different periods of observation, it is possible to estimate Q bk at specific locations (Vianello and D'Agostino, 2007), and to relate such discharge to A. This would allow us to better understand the changes in the power law exponent when considering w–A relationships instead of w–Q relationships. The Rio Cordevole and the Rio Cordon widths appear to be similarly influenced by the drainage area in terms of exponents, with the Rio Cordon reaching a wider course due to the larger value of s. The higher exponent of the w–A relationship for the Rio Vauz suggests a stronger modification of the channel geometries with a small increase in drainage area. Literature reports a large variation in t values, ranging from 0.1 (e.g. Hession et al., 2003) to 0.6 (e.g. Montgomery and Gran, 2001). Considering five mountain channel networks, Montgomery and Gran (2001) showed values of the exponent t for alluvial and bedrock streams ranging from 0.3 to 0.6. Using data for N 1000 sites in the U.S., Faustini et al.

G. Sofia et al. / Geomorphology 250 (2015) 236–248

245

Fig. 9. Downstream hydraulic geometry relationships for the three streams. (a) Bankfull width and drainage area. (b) Channel profile slope and drainage area. (c) Bankfull width and the stream power index.

(2009) found that t ranges from 0.22 to 0.38. More in detail, considering site specific studies, numerous authors showed t values around 0.5 for alluvial channels (Leopold and Maddock, 1953; Rodriguez-Iturbe and Rinaldo, 1997; Ibbitt, 1997; Knighton, 1998; Whiting et al., 1999). Harman et al. (1999) in North Carolina found values of the exponent of the power law around 0.3. In North Carolina piedmont rivers, Doll et al. (2002) estimated t values around 0.3 for both rural and urban watersheds. Hession et al. (2003), on the other hand, found that t can be influenced by the degree of urbanisation: in piedmont watersheds of Pennsylvania urban watersheds had t of about 0.1, but nonurban watersheds had values N0.3. A value of t ~ 0.5 was observed by Brummer and Montgomery (2003) for small basins (A b 10 km2). Moody et al. (2003) found that t value was influenced by the location of the rivers, with t ~ 0.36 for plains and ~ 0.41 for mountains. D'Agostino and Vianello (2005) and Vianello and D'Agostino (2007) found t to be ~ 0.3 for alluvial channels, but slightly increased (~ 0.4) when including colluvial channels in the analysis. Golden and Springer (2006) and Wohl and David (2008) found t ~0.33 for bedrock channels. For channel slope and drainage area (Fig. 9b), we obtained the following relationships:

S ¼ ð0:11  0:01ÞA−0:120:09 ðRMSE ¼ 0:02Þ; for the Rio Cordevole ð27Þ S ¼ ð0:16  0:12ÞA−0:180:57 ðRMSE ¼ 0:05Þ; for the Rio Cordon:

ð28Þ

For the Rio Cordevole and Rio Vauz, the automatically identified homogeneous reaches allowed the definition of a statistically significant S–A relationship (α = 0.05), which is consistent with values from field surveys (Rio Cordevole) and the literature: exponent θ can range from less than 0.3 in steep headwater channels to greater than 1.0 in some alluvial channels (Hack, 1957; Willgoose et al., 1991; Seidl and Dietrich, 1992; Montgomery and Foufoula-Georgiou, 1993; Montgomery, 2001; Stock, 2003; Brummer and Montgomery, 2003; Golden and Springer, 2006; Sklar and Dietrich, 2013). Fonstad (2003) noted that slope is negatively related to Q (θ~0.6) and consequently, to A. For the Rio Cordon stream, the relationship is within the range reported in the literature; however, the relationship is not statistically significant. The stream power seems to be strongly associated with channel width (Fig. 9c) as seen in other studies (Lecce, 2013): w ¼ ð0:06  0:002ÞSPI0:290:22 ðRMSE ¼ 0:64mÞ; for the Rio Cordevole

−0:600:51

S ¼ ð0:24  0:12ÞA

ðRMSE ¼ 0:05Þ; for the Rio Vauz

ð26Þ

ð29Þ

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w ¼ ð0:44  0:95ÞSPI0:180:21 ðRMSE ¼ 0:61mÞ; for the Rio Cordon ð30Þ

w ¼ ð0:09  0:32ÞSPI 0:250:73 ðRMSE ¼ 0:32 mÞ; for the Rio Vauz: ð31Þ w shows a significant positive correlation with SPI for all the considered streams (p b 0.01, α = 0.05), suggesting that hydraulic geometry is dynamically balanced with the expenditure of energy, and channels are dominated by sediment transport processes. The grain sizes of bed sediments may play an additional role in the channel adjustment of alpine streams. In the upper Cordevole river, the effects of downstream coarsening and local slope variation can account for 38% of downstream width variation, with the remaining 62% being explained only by the drainage area (Vianello and D'Agostino, 2007). Brummer and Montgomery (2003) and Fonstad (2003) also reached similar results for drainage areas b10 km2. Eqs. (29) to (31) indicate that the channels studied underwent limited vertical incision, and therefore the stream power is mostly expended in width adjustment. Graf (1983) also inferred that erosive floods resulted in a total increase of the stream power in the downstream direction, for channels at the bottom of arroyos with confined discharge. The relationship for the Rio Cordon suggests a stronger effect of stream power on the channel width than the Rio Cordevole and Rio Vauz. Lenzi (2001) noted strong modifications in channel widths of Rio Cordon after strong floods with high sediment discharge. For all watersheds, the exponent of the power function is similar to that from different environments reported by Golden and Springer (2006): w ¼ ð9:51  0:71ÞSPI 0:280:04 :

ð32Þ

Van den Berg (1995) and Van den Berg and Bledsoe (2003) also showed that channel width increases with increasing stream power. 5. Conclusions The availability of high resolution DTMs has created great opportunity for researchers to analyse channel forms and processes. This work explores the ability of LiDAR to derive morphological relationships between reach-scale channel widths and parameters related to flow, i.e., drainage area and stream power. The proposed approach , when applied to river morphologies having a well defined bankfull condition, produces results that are comparable with the relationships identified through field surveys, and it allows comparison of different watersheds. Uncertainties in the procedure are generally less than the resolution of the DTMs, or to the time of flight of the LiDAR survey for channels having bankfull widths highly modified by floods. The higher underestimation in the procedure is mainly related to errors in the DTM. However, even with a rough measurement of channel widths, it is possible to adjust the downstream hydraulic relationships using the differences between the actual average channel width and the overall DTM-derived average width. Although additional field measurements are needed in order to broaden the findings, a clean input DTM with no outliers or striping artefacts together with the proposed automatic approach improve the knowledge about river systems particularly in areas covered by vegetation or inaccessible areas. Analysis of more watersheds in future will provide more implications about channel adjustment in relation to spatial scale and geomorphic processes such as sediment transport, tectonics, and anthropic pressure. Acknowledgements Resources for data analysis were provided by the Interdepartmental Research Center of Geomatics CIRGEO at the University of Padova. We thank Prof. Vincenzo D'Agostino for his review of the earlier version of this manuscript, and Dr Alessandro Vianello for his constructive insights

about the field data of the Rio Cordevole and Rio Vauz catchments. The research has benefited from the helpful suggestions about the Rio Cordon catchment given by Prof. Mario Aristide Lenzi, Dr Luca Mao and Dr Francesco Comiti. We would like to express our gratitude to Prof. Ellen Wohl, the anonymous reviewer, and the Editor for their insightful suggestions and careful revision of our manuscript.

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