5  Metrics

Level 1

📐 Theory ⚡ Quick Reference

Drainage Area

Definition: The total upstream land area that contributes surface water flow to a specific point along a stream channel.

Technical Specification:

  • Symbol: \(A_d\) or DA
  • Units: km² (square kilometers) or mi² (square miles)
  • Data Type: Float/Double
  • Typical Range: 0.01 - 10,000+ km²

Calculation Method:

Drainage area is typically derived from Digital Elevation Models (DEMs) using flow accumulation algorithms:

\[A_d = \frac{\text{Flow\_Accumulation} \times \text{Cell\_Area}}{1,000,000}\]

Where:

  • Flow_Accumulation = number of upstream cells
  • Cell_Area = DEM cell size² (in m²)
  • Division by 1,000,000 converts m² to km²

Implementation Notes:

  • Extracted at reach pour points or cross-section locations
  • Requires hydrologically conditioned DEM
  • Critical input for regional curve development
  • Primary independent variable for Level 2 metric predictions
NoteCross-Reference

See User Manual Section on Watershed Delineation for workflow guidance.

Data Sources:

  • USGS StreamStats
  • NHDPlus catchment attributes
  • Custom watershed delineation from DEM

Slope

Definition: The change in elevation per unit length along the stream channel, representing the gradient of energy dissipation.

Technical Specification:

  • Symbol: \(S\) or \(S_0\)
  • Units: m/m (dimensionless) or ft/ft, often expressed as percentage or ratio
  • Data Type: Float/Double
  • Typical Range: 0.0001 - 0.10 (0.01% - 10%)

Calculation Methods:

1. Reach-Average Slope:

\[S = \frac{Z_{\text{upstream}} - Z_{\text{downstream}}}{L_{\text{reach}}}\]

Where:

  • \(Z_{\text{upstream}}\) = elevation at upstream extent (m)
  • \(Z_{\text{downstream}}\) = elevation at downstream extent (m)
  • \(L_{\text{reach}}\) = channel length between points (m)

2. Water Surface Slope:

Derived from surveyed water surface elevations at bankfull stage:

\[S_{ws} = \frac{\Delta Z_{ws}}{L_{\text{channel}}}\]

3. Energy Slope:

From Manning’s equation, back-calculated from known discharge and channel geometry.

Implementation Notes:

  • Reach length should be 20-30× bankfull width minimum
  • Use channel centerline length, not valley length
  • Water surface slope preferred over bed slope for hydraulic calculations
  • Critical parameter for shear stress and stream power calculations

Quality Control:

  • Compare with regional slope-drainage area relationships
  • Flag values outside expected range for stream type
  • Verify against LiDAR-derived profiles where available
NoteCross-Reference

See User Manual Section on Reach Delineation for slope measurement protocols.

Sinuosity

Definition: The ratio of channel length to valley length, quantifying the degree of channel meandering.

Technical Specification:

  • Symbol: \(K\) or SI
  • Units: Dimensionless ratio
  • Data Type: Float/Double
  • Typical Range: 1.0 - 4.0+ (1.0 = straight, >1.5 = meandering)

Calculation Method:

\[K = \frac{L_{\text{channel}}}{L_{\text{valley}}}\]

Where:

  • \(L_{\text{channel}}\) = length along channel centerline (m)
  • \(L_{\text{valley}}\) = straight-line distance between endpoints (m)

Classification Thresholds:

Table 5.1: Classification of channel sinuosity
Sinuosity Classification Description
K < 1.05 Straight Minimal meandering
1.05 ≤ K < 1.5 Sinuous Moderate meandering
K ≥ 1.5 Meandering Well-developed meanders

Implementation Notes:

  • Measure over consistent reach lengths (typically 20× bankfull width)
  • Channel length follows thalweg or centerline
  • Valley length is straight-line distance between reach endpoints
  • Sensitive to reach length selection; longer reaches yield more stable values

Alternative Metrics:

  • Meander Wavelength Ratio: \(\lambda/W\) (see Level 3 metrics)
  • Bend Curvature: More detailed planform characterization

Geomorphic Significance:

  • Indicates channel adjustment to slope and sediment transport
  • Higher sinuosity = lower energy gradient
  • Key discriminator in stream classification systems (Rosgen, River Styles)
NoteCross-Reference

See User Manual Section on Planform Assessment for sinuosity measurement procedures.

Level 2

Discharge

Definition: The volumetric flow rate of water passing through a channel cross-section, typically referenced to bankfull or other recurrence interval flows.

Technical Specification:

  • Symbol: \(Q\) (\(Q_{bf}\) for bankfull, \(Q_2\), \(Q_{1.5}\), etc. for recurrence intervals)
  • Units: m³/s (cms) or ft³/s (cfs)
  • Data Type: Float/Double
  • Typical Range: 0.01 - 10,000+ m³/s (highly variable)

Calculation Methods:

1. Regional Curve Prediction:

\[Q_{bf} = a \times (A_d)^b\]

Where:

  • \(a\), \(b\) = regional regression coefficients
  • \(A_d\) = drainage area (km²)

2. Manning’s Equation:

\[Q = \frac{1}{n} \times A \times R^{2/3} \times S^{1/2}\]

Where:

  • \(n\) = Manning’s roughness coefficient
  • \(A\) = cross-sectional area (m²)
  • \(R\) = hydraulic radius (m)
  • \(S\) = energy slope (m/m)

3. Gage Analysis:

  • Frequency analysis of annual peak flows
  • Bankfull discharge typically \(Q_{1.5}\) to \(Q_{2.3}\)

Recurrence Intervals:

Table 5.2: Common discharge recurrence intervals
Symbol Return Period Description
\(Q_{1.5}\) 1.5-year Common bankfull proxy
\(Q_2\) 2-year Standard bankfull reference
\(Q_{10}\) 10-year Small flood
\(Q_{25}\) 25-year Moderate flood
\(Q_{50}\) 50-year Large flood
\(Q_{100}\) 100-year Design flood

Implementation Notes:

  • Bankfull discharge is the channel-forming flow
  • Regional curves vary by physiographic province
  • Uncertainty increases with extrapolation beyond calibration range
  • Critical input for all hydraulic geometry calculations

Data Sources:

  • USGS StreamStats regional equations
  • Gage analysis (USGS NWIS data)
  • Hydraulic modeling (HEC-RAS, SRH-2D)

Quality Control:

# Check against Dunne & Leopold (1978) approximation
Q_check <- 2.3 * A_d^0.7  # for A_d in km², Q in m³/s
NoteCross-Reference

See User Manual Section on Bankfull Identification and Regional Curves.

XS Area

Definition: The cross-sectional area of flow perpendicular to the direction of flow, measured at bankfull stage.

Technical Specification:

  • Symbol: \(A\) or \(A_{bf}\)
  • Units: m² or ft²
  • Data Type: Float/Double
  • Typical Range: 1 - 1,000+ m²

Calculation Method:

From Survey Data:

\[A = \sum_{i=1}^{n-1} \left[(x_{i+1} - x_i) \times \frac{(z_i + z_{i+1})}{2}\right]\]

Where:

  • \(x_i\) = horizontal station (m)
  • \(z_i\) = depth below bankfull elevation (m)
  • Summation from left to right bank

Trapezoidal Rule Integration:

\[A = \sum_{i=1}^{n-1} \left[0.5 \times (d_i + d_{i+1}) \times \Delta x_i\right]\]

Regional Curve Prediction:

\[A_{bf} = c \times (A_d)^d\]

Where:

  • \(c\), \(d\) = regional regression coefficients
  • \(A_d\) = drainage area (km²)

Implementation Notes:

  • Measured perpendicular to flow direction
  • Extends from bankfull elevation on left to right bank
  • Excludes overbank areas
  • Should represent active channel geometry

Hydraulic Geometry Relationship:

\[A = W \times D_{\text{mean}}\]

Where:

  • \(W\) = bankfull width (m)
  • \(D_{\text{mean}}\) = mean depth (m)

Quality Control:

  • Compare measured vs. predicted from regional curves
  • Check against continuity equation: \(Q = A \times V\)
  • Verify bankfull elevation identification
  • Flag cross-sections with unusual geometry

Data Processing:

  • Remove survey errors (spikes, gaps)
  • Interpolate between survey points if needed
  • Apply consistent bankfull datum
NoteCross-Reference

See User Manual Section on Cross-Section Surveys and Bankfull Delineation.

XS Width

Definition: The top width of the channel measured at bankfull stage, perpendicular to flow direction.

Technical Specification:

  • Symbol: \(W\) or \(W_{bf}\)
  • Units: m or ft
  • Data Type: Float/Double
  • Typical Range: 1 - 200+ m

Calculation Method:

From Survey Data:

\[W = x_{\text{right}} - x_{\text{left}}\]

Where:

  • \(x_{\text{right}}\) = station at right bankfull indicator (m)
  • \(x_{\text{left}}\) = station at left bankfull indicator (m)

Regional Curve Prediction:

\[W_{bf} = e \times (A_d)^f\]

Where:

  • \(e\), \(f\) = regional regression coefficients (typically \(f \approx 0.5\))
  • \(A_d\) = drainage area (km²)

Hydraulic Geometry:

\[W = a \times Q^b\]

Where:

  • \(a\), \(b\) = at-a-station or downstream coefficients
  • \(Q\) = discharge (m³/s)
  • Typical \(b \approx 0.5\) for downstream relations

Implementation Notes:

  • Measured at water surface at bankfull stage
  • Perpendicular to flow direction
  • Excludes vegetated benches unless inundated at bankfull
  • Most stable and easily measured hydraulic geometry parameter

Measurement Considerations:

  • Use consistent bankfull indicators (topographic break, vegetation line, etc.)
  • Multiple cross-sections per reach for averaging
  • Avoid locations with local controls (bedrock, structures)

Quality Control:

# Width-to-depth ratio check
WD_ratio <- W / D_mean
# Typical range: 10-40 for most alluvial channels
NoteCross-Reference

See User Manual Section on Bankfull Width Measurement and Field Indicators.

XS Depth

Definition: Vertical distance from the water surface to the channel bed, measured at bankfull stage.

Technical Specification:

  • Symbol: \(D\) (\(D_{\text{max}}\) for maximum, \(D_{\text{mean}}\) for mean)
  • Units: m or ft
  • Data Type: Float/Double
  • Typical Range: 0.1 - 10+ m

Maximum Depth

Calculation Method:

\[D_{\text{max}} = \max(z_{bf} - z_{\text{bed}})\]

Where:

  • \(z_{bf}\) = bankfull elevation (m)
  • \(z_{\text{bed}}\) = bed elevation at each station (m)

Characteristics:

  • Typically occurs near thalweg
  • Location varies with channel morphology
  • Less stable than mean depth for regional relationships

Mean Depth

Calculation Method:

\[D_{\text{mean}} = \frac{A}{W}\]

Where:

  • \(A\) = cross-sectional area (m²)
  • \(W\) = bankfull width (m)

Alternative Calculation:

\[D_{\text{mean}} = \frac{1}{W} \times \sum_{i=1}^{n} \left[d_i \times \Delta x_i\right]\]

Where:

  • \(d_i\) = depth at station \(i\) (m)
  • \(\Delta x_i\) = horizontal distance increment (m)

Regional Curve Prediction:

\[D_{\text{mean}} = g \times (A_d)^h\]

Where:

  • \(g\), \(h\) = regional regression coefficients (typically \(h \approx 0.3-0.4\))
  • \(A_d\) = drainage area (km²)

Hydraulic Geometry:

\[D_{\text{mean}} = c \times Q^f\]

Where:

  • \(c\), \(f\) = coefficients (typical \(f \approx 0.3-0.4\))
  • \(Q\) = discharge (m³/s)

Implementation Notes:

  • Mean depth preferred for regional curves and ratios
  • Maximum depth useful for habitat assessment
  • Hydraulic depth (\(A/W\)) equivalent to mean depth for rectangular channels
  • More variable than width; requires multiple cross-sections

Quality Control:

  • Verify reasonable width-to-depth ratios
  • Check against regional expectations
  • Compare maximum to mean depth (typically \(D_{\text{max}} \approx 1.5-2.0 \times D_{\text{mean}}\))
NoteCross-Reference

See User Manual Section on Depth Measurement and Cross-Section Analysis.

XS Width to Depth Ratio

Definition: The ratio of bankfull width to mean depth, indicating channel shape and lateral vs. vertical adjustment potential.

Technical Specification:

  • Symbol: \(W/D\) or \(F\) (form ratio)
  • Units: Dimensionless
  • Data Type: Float/Double
  • Typical Range: 5 - 100+ (highly variable by stream type)

Calculation Method:

\[\frac{W}{D} = \frac{W_{bf}}{D_{\text{mean}}}\]

Where:

  • \(W_{bf}\) = bankfull width (m)
  • \(D_{\text{mean}}\) = mean bankfull depth (m)

Classification Ranges:

Table 5.3: Width-to-depth ratio classification
W/D Ratio Classification Channel Form
< 12 Narrow/Deep Incised or confined
12 - 40 Moderate Typical alluvial
> 40 Wide/Shallow Laterally active

Geomorphic Significance:

  • Incised or confined channels
  • Resistant banks (cohesive, vegetated)
  • Higher shear stress on bed
  • Vertical adjustment dominant
  • Unconfined, laterally active channels
  • Erodible banks
  • Lower unit stream power
  • Lateral adjustment dominant

Implementation Notes:

  • Key discriminator in stream classification (Rosgen types)
  • Indicates channel stability and adjustment mode
  • Varies systematically with stream power and bank strength
  • Use mean depth, not maximum depth

Relationships:

Schumm (1960) relationship with silt-clay content:

\[\frac{W}{D} = 255 \times M^{-1.08}\]

Where \(M\) = weighted mean percent silt-clay in channel perimeter

Quality Control:

  • Compare with regional values for similar stream types
  • Check consistency across multiple cross-sections
  • Verify depth measurement accuracy (primary source of error)
NoteCross-Reference

See User Manual Section on Stream Classification and Channel Form.

Entrenchment Ratio

Definition: The ratio of flood-prone width to bankfull width, indicating the degree of vertical containment and access to floodplain.

Technical Specification:

  • Symbol: ER
  • Units: Dimensionless
  • Data Type: Float/Double
  • Typical Range: 1.0 - 10.0+

Calculation Method:

\[ER = \frac{W_{fpa}}{W_{bf}}\]

Where:

  • \(W_{fpa}\) = flood-prone area width (m)
  • \(W_{bf}\) = bankfull width (m)

Flood-Prone Area Definition:

  • Width of area inundated at 2× maximum bankfull depth
  • Measured perpendicular to valley orientation
  • Represents approximate 50-year floodplain

Measurement Protocol:

  1. Establish bankfull elevation (\(z_{bf}\))
  2. Calculate flood-prone elevation: \(z_{fpa} = z_{bf} + (2 \times D_{\text{max}})\)
  3. Measure horizontal width at \(z_{fpa}\) elevation
  4. Calculate ratio

Classification Thresholds:

Table 5.4: Entrenchment ratio classification
ER Value Entrenchment Level Description
< 1.4 Entrenched Minimal floodplain access
1.4 - 2.2 Moderately Entrenched Limited floodplain access
> 2.2 Slightly Entrenched Good floodplain connectivity

Geomorphic Significance:

  • Indicates vertical stability and incision history
  • Controls sediment storage and flood attenuation
  • Key parameter in stream classification systems
  • Affects riparian habitat and nutrient cycling

Implementation Notes:

  • Measure from valley cross-section, not channel cross-section
  • Use topographic maps or LiDAR for consistent measurement
  • May vary significantly along reach; use representative locations
  • Sensitive to local topography and valley confinement

Alternative Metrics:

  • Incision Ratio: Depth of incision below historical floodplain
  • Confinement Ratio: Valley width to bankfull width
NoteCross-Reference

See User Manual Section on Entrenchment Assessment and Valley Confinement.

Shear Stress

Definition: The tangential force per unit area exerted by flowing water on the channel bed, driving sediment transport.

Technical Specification:

  • Symbol: \(\tau\) (tau) or \(\tau_0\)
  • Units: N/m² (Pascals) or lb/ft²
  • Data Type: Float/Double
  • Typical Range: 0.1 - 100+ N/m²

Calculation Method:

Boundary Shear Stress:

\[\tau_0 = \gamma \times R \times S\]

Where:

  • \(\gamma\) = specific weight of water (N/m³)
  • \(R\) = hydraulic radius (m)
  • \(S\) = energy slope (m/m)

With Density:

\[\tau_0 = \rho \times g \times R \times S\]

Where:

  • \(\rho\) = water density (1000 kg/m³ at 20°C)
  • \(g\) = gravitational acceleration (9.81 m/s²)
  • \(R\) = hydraulic radius (m)
  • \(S\) = energy slope (m/m)

Density

Water Density Values:

Table 5.5: Water density and specific weight by temperature
Temperature Density (kg/m³) Specific Weight (N/m³)
4°C 1000.0 9810
10°C 999.7 9807
20°C 998.2 9789
30°C 995.7 9764

Temperature Correction:

\[\rho(T) = 1000 \times \left[1 - (T - 4)^2 \times 6.8 \times 10^{-6}\right]\]

for \(T\) in °C

Implementation Notes:

  • Use bankfull hydraulic radius and water surface slope
  • Represents average bed shear stress
  • Local shear stress varies with bed topography
  • Critical for sediment transport calculations

Hydraulic Radius:

\[R = \frac{A}{P}\]

Where:

  • \(A\) = cross-sectional area (m²)
  • \(P\) = wetted perimeter (m)

For wide channels (\(W \gg D\)):

\[R \approx D_{\text{mean}}\]

Critical Shear Stress:

\[\tau_c = \theta_c \times (\rho_s - \rho) \times g \times D_{50}\]

Where:

  • \(\theta_c\) = Shields parameter (≈ 0.045 for coarse sediment)
  • \(\rho_s\) = sediment density (2650 kg/m³ for quartz)
  • \(D_{50}\) = median grain size (m)

Geomorphic Significance:

  • Indicates sediment transport capacity
  • Threshold for bed mobility
  • Controls channel morphology and stability
  • Key input for stream restoration design

Quality Control:

  • Verify slope measurement accuracy
  • Check against expected values for stream type
  • Compare with critical shear stress for bed material
NoteCross-Reference

See User Manual Section on Sediment Transport and Hydraulic Analysis.

Stream Power

Definition: The rate of energy expenditure per unit length of channel, representing the capacity to perform geomorphic work.

Technical Specification:

  • Symbol: \(\Omega\) (omega) for total, \(\omega\) (omega) for unit
  • Units:
    • Total: W/m (watts per meter)
    • Unit: W/m² (watts per square meter)
  • Data Type: Float/Double
  • Typical Range:
    • Total: 1 - 10,000+ W/m
    • Unit: 1 - 300+ W/m²

Calculation Methods:

Total Stream Power:

\[\Omega = \gamma \times Q \times S\]

Or:

\[\Omega = \rho \times g \times Q \times S\]

Where:

  • \(\gamma\) = specific weight of water (9810 N/m³)
  • \(\rho\) = water density (1000 kg/m³)
  • \(g\) = gravitational acceleration (9.81 m/s²)
  • \(Q\) = discharge (m³/s)
  • \(S\) = energy slope (m/m)

Unit Stream Power

Definition: Stream power per unit bed area, normalizing for channel size.

Calculation Method:

\[\omega = \frac{\Omega}{W} = \frac{\gamma \times Q \times S}{W}\]

Or:

\[\omega = \tau_0 \times V\]

Where:

  • \(W\) = bankfull width (m)
  • \(\tau_0\) = boundary shear stress (N/m²)
  • \(V\) = mean flow velocity (m/s)

Simplified Form:

\[\omega = \rho \times g \times D \times S \times V\]

Geomorphic Significance:

Classification Thresholds (Nanson & Croke, 1992):

Table 5.6: Stream power classification
Unit Stream Power (W/m²) Energy Level Process Regime
< 10 Low Energy Deposition-dominated, fine sediment
10 - 50 Medium Energy Balanced transport, mixed sediment
50 - 300 High Energy Erosion-dominated, coarse sediment
≥ 300 Very High Energy Bedrock control, limited alluvium

Implementation Notes:

  • Use bankfull discharge and water surface slope
  • Unit stream power preferred for comparing channels of different sizes
  • Critical parameter for stream classification and restoration design
  • Correlates with channel pattern and sediment transport regime

Specific Stream Power:

Alternative formulation per unit mass of water:

\[\omega_s = g \times S \times V \quad \text{(in m²/s³)}\]

Relationships:

With Shear Stress:

\[\omega = \tau_0 \times V = \tau_0 \times \frac{Q}{A}\]

With Slope and Discharge:

\[\omega = \frac{\rho \times g \times S \times Q}{W}\]

Quality Control:

  • Verify discharge and slope accuracy
  • Compare with regional values for similar stream types
  • Check consistency with observed channel morphology
  • Validate against sediment transport observations

Applications:

  • Stream classification (River Styles, Rosgen)
  • Restoration design (reference reach selection)
  • Sediment transport prediction
  • Channel evolution assessment
  • Habitat characterization
NoteCross-Reference

See User Manual Section on Stream Power Analysis, Channel Classification, and Restoration Design.

Level 3

Bend Radius of Curvature

Definition: The radius of the circular arc that best fits the centerline of a meander bend, quantifying bend tightness.

Technical Specification:

  • Symbol: \(R_c\)
  • Units: m or ft
  • Data Type: Float/Double
  • Typical Range: 10 - 1000+ m (varies with channel size)

Calculation Methods:

1. Three-Point Circle Method:

\[R_c = \frac{a \times b \times c}{4 \times K}\]

Where:

  • \(a\), \(b\), \(c\) = distances between three points on bend
  • \(K\) = area of triangle formed by three points

2. Curvature from Coordinates:

\[\kappa = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}\]

\[R_c = \frac{1}{\kappa}\]

Where:

  • \(x'\), \(y'\) = first derivatives of centerline coordinates
  • \(x''\), \(y''\) = second derivatives
  • \(\kappa\) = curvature

3. Best-Fit Circle:

Least-squares fit of circular arc to bend apex points:

\[(x - x_c)^2 + (y - y_c)^2 = R_c^2\]

Solve for center \((x_c, y_c)\) and radius \(R_c\)

Implementation Notes:

  • Measure at bend apex (point of maximum curvature)
  • Use channel centerline or thalweg
  • Requires high-resolution planform data (aerial imagery, LiDAR)
  • Multiple bends per reach for statistical analysis

Measurement Extent:

  • Inflection point to inflection point
  • Minimum 3 points for calculation
  • More points improve accuracy

Geomorphic Significance:

  • Indicates bend tightness and migration potential
  • Controls secondary circulation and bank erosion
  • Affects sediment sorting and bar formation
  • Key parameter in meander evolution models

Quality Control:

  • Visual inspection of fitted circle
  • Compare with field observations
  • Check for consistency along reach
  • Flag irregular or compound bends
NoteCross-Reference

See User Manual Section on Planform Analysis and Meander Geometry.

Meander Length

Definition: The along-channel distance between successive inflection points, representing one complete meander wavelength.

Technical Specification:

  • Symbol: \(\lambda\) (lambda) or \(L_m\)
  • Units: m or ft
  • Data Type: Float/Double
  • Typical Range: 50 - 5000+ m (scales with channel size)

Calculation Method:

Direct Measurement:

\[\lambda = L_{\text{channel}} \text{ (inflection to inflection)}\]

Measured along channel centerline between successive inflection points (crossover points where curvature changes sign).

From Planform:

  1. Identify inflection points on both sides of meander
  2. Measure centerline distance between corresponding points
  3. Average multiple meanders for reach-scale value

Empirical Relationships:

Leopold & Wolman (1960):

\[\lambda = 10.9 \times W_{bf}^{1.01}\]

Simplified:

\[\lambda \approx 11 \times W_{bf}\]

Where \(W_{bf}\) = bankfull width (m)

Alternative (with discharge):

\[\lambda = 4.7 \times Q_{bf}^{0.5}\]

Where \(Q_{bf}\) = bankfull discharge (m³/s)

Components:

  • Meander Wavelength (\(\lambda\)): Full wavelength (inflection to inflection)
  • Meander Arc Length: Distance along one bend
  • Meander Amplitude: Perpendicular distance from valley axis to bend apex

Implementation Notes:

  • Requires well-developed meander pattern
  • Measure multiple meanders for statistical reliability
  • Use consistent definition of inflection points
  • Scales predictably with channel width

Geomorphic Significance:

  • Fundamental property of meandering channels
  • Reflects balance between flow momentum and bank resistance
  • Controls floodplain width and reworking rate
  • Used in channel restoration design

Quality Control:

# Check against empirical relationship
lambda_predicted <- 11 * W_bf
ratio <- lambda_measured / lambda_predicted
# Typical range: 0.7 - 1.5
NoteCross-Reference

See User Manual Section on Meander Characterization and Planform Metrics.

Meander Bend Width

Definition: The straight-line distance from the meander apex to the opposite inflection point, perpendicular to the valley axis.

Technical Specification:

  • Symbol: \(A_m\) or \(W_m\)
  • Units: m or ft
  • Data Type: Float/Double
  • Typical Range: 20 - 2000+ m

Calculation Method:

Direct Measurement:

\[W_m = \text{perpendicular distance from valley axis to bend apex}\]

Or:

\[W_m = 2 \times A_m\]

Where \(A_m\) = meander amplitude (distance from valley centerline to apex)

From Coordinates:

  1. Define valley axis (straight line between inflection points)
  2. Identify bend apex (point of maximum curvature)
  3. Measure perpendicular distance from axis to apex
  4. Multiply by 2 for full bend width

Empirical Relationships:

Typical Ratio:

\[W_m \approx 0.5 \times \lambda\]

Where \(\lambda\) = meander wavelength

With Bankfull Width:

\[W_m \approx 18 \times W_{bf}\]

(Derived from \(\lambda \approx 11 \times W_{bf}\) and \(W_m \approx 0.5 \times \lambda\))

Implementation Notes:

  • Measure perpendicular to valley trend, not channel
  • Use bend apex (point of maximum deflection)
  • Represents lateral extent of channel migration
  • Important for floodplain width and easement determination

Meander Belt Width:

Related but distinct metric:

\[W_{\text{belt}} = \text{maximum width of meander belt}\]

Encompasses multiple meanders and historical channel positions.

Geomorphic Significance:

  • Indicates lateral migration potential
  • Controls floodplain reworking rate
  • Defines meander belt for land use planning
  • Key parameter for restoration corridor width

Quality Control:

  • Verify valley axis orientation
  • Check consistency across multiple bends
  • Compare with meander wavelength (\(W_m \approx 0.5 \times \lambda\))
  • Validate against historical channel positions
NoteCross-Reference

See User Manual Section on Migration Assessment and Corridor Delineation.

Radius of Curvature to Bankfull Width Ratio

Definition: Dimensionless ratio quantifying bend tightness relative to channel size, indicating meander development and stability.

Technical Specification:

  • Symbol: \(R_c/W\) or \(r^*\)
  • Units: Dimensionless
  • Data Type: Float/Double
  • Typical Range: 2 - 20+

Calculation Method:

\[\frac{R_c}{W} = \frac{R_c}{W_{bf}}\]

Where:

  • \(R_c\) = radius of curvature at bend apex (m)
  • \(W_{bf}\) = bankfull width (m)

Classification Thresholds:

Table 5.7: Radius of curvature to width ratio classification
\(R_c/W\) Bend Tightness Characteristics
< 2 Very Tight High erosion risk, unstable
2 - 3 Tight Active migration, high shear
3 - 5 Moderate Balanced, typical meanders
5 - 10 Gentle Stable, lower migration rates
> 10 Very Gentle Approaching straight

Geomorphic Significance:

  • High near-bank velocities and shear stress
  • Rapid bank erosion and migration
  • Strong secondary circulation
  • Point bar development
  • Potential for cutoff formation
  • Lower erosion rates
  • More stable banks
  • Weaker secondary flow
  • Slower migration
  • Less sediment sorting

Implementation Notes:

  • Calculate at bend apex (maximum curvature)
  • Average multiple bends for reach characterization
  • Use consistent bankfull width measurement
  • Critical for assessing bend stability

Empirical Relationships:

Hickin & Nanson (1984) - Migration Rate:

Maximum migration rate occurs at \(R_c/W \approx 2-3\)

Stability Threshold:

\[\begin{align} R_c/W < 2 &: \text{High instability risk} \\ R_c/W > 10 &: \text{Minimal lateral activity} \end{align}\]

Applications:

  • Bank erosion hazard assessment
  • Restoration design (target \(R_c/W = 2-3\) for natural appearance)
  • Channel evolution prediction
  • Infrastructure risk evaluation

Quality Control:

# Check against typical values
if (Rc_W < 1.5) {
  flag <- "Unusually tight"
} else if (Rc_W > 15) {
  flag <- "Nearly straight"
}
NoteCross-Reference

See User Manual Section on Bend Stability and Migration Rates.

Meander Bend Width to Bankfull Width Ratio

Definition: Dimensionless ratio of meander bend width to bankfull width, indicating the lateral extent of meandering relative to channel size.

Technical Specification:

  • Symbol: \(W_m/W\) or \(A_m/W\)
  • Units: Dimensionless
  • Data Type: Float/Double
  • Typical Range: 10 - 30+

Calculation Method:

\[\frac{W_m}{W} = \frac{W_m}{W_{bf}}\]

Or using amplitude:

\[\frac{A_m}{W} = \frac{A_m}{W_{bf}}\]

Where:

  • \(W_m\) = meander bend width (m)
  • \(A_m\) = meander amplitude (m) [\(W_m = 2 \times A_m\)]
  • \(W_{bf}\) = bankfull width (m)

Empirical Relationships:

From Leopold & Wolman:

Since \(\lambda \approx 11 \times W\) and \(W_m \approx 0.5 \times \lambda\):

\[\frac{W_m}{W} \approx 18\]

Typical Range:

\[\frac{W_m}{W} = 15 - 25 \text{ (for freely meandering channels)}\]

With Sinuosity:

Higher sinuosity generally correlates with higher \(W_m/W\) ratios.

Implementation Notes:

  • Measure at representative bends (avoid anomalies)
  • Average multiple bends for reach value
  • Requires well-developed meander pattern
  • Sensitive to valley confinement

Geomorphic Significance:

  • Confined or constrained channel
  • Limited lateral migration space
  • Valley controls on planform
  • Reduced floodplain connectivity
  • Freely meandering
  • Typical alluvial channel
  • Active floodplain reworking
  • Natural lateral mobility
  • Highly sinuous
  • Extensive lateral migration
  • Wide meander belt
  • Low valley gradient

Applications:

Restoration Design:

# Target meander geometry
W_m_design <- 18 * W_bf_design
lambda_design <- 11 * W_bf_design
R_c_design <- 2.5 * W_bf_design

Corridor Width Estimation:

W_corridor <- W_m + (2 * migration_buffer)
W_corridor_min <- 20 * W_bf  # minimum

Channel Evolution:

  • Increasing \(W_m/W\) indicates lateral expansion
  • Decreasing \(W_m/W\) suggests confinement or straightening
  • Temporal changes indicate adjustment processes

Quality Control:

# Consistency check with wavelength
lambda_W <- lambda_measured / W_bf  # Should be ~11
Wm_W <- W_m_measured / W_bf  # Should be ~18

# Ratio should be approximately 0.5
ratio_check <- Wm_W / lambda_W
# Expected: ratio_check ≈ 0.5 ± 0.1

Comparison with Historical Data:

  • Overlay historical channel positions
  • Calculate \(W_m/W\) for different time periods
  • Assess trends in lateral activity
  • Identify reaches with changing planform
NoteCross-Reference

See User Manual Section on Planform Design, Meander Geometry, and Channel Migration Zones.

Summary

Metric Hierarchy and Relationships

Table 5.8: Summary of FluvialGeomorph metrics by level
Level Metric Primary Inputs Typical Use
1 Drainage Area DEM, pour point Regional curves, scaling
1 Slope Elevation, length Energy gradient, classification
1 Sinuosity Channel/valley length Pattern classification
2 Discharge Drainage area, regional curves Hydraulic design
2 XS Area Width, depth Flow capacity
2 XS Width Survey, bankfull elevation Hydraulic geometry
2 XS Depth Survey, bankfull elevation Hydraulic geometry
2 W/D Ratio Width, depth Channel form, stability
2 Entrenchment Flood-prone width, bankfull width Vertical stability
2 Shear Stress Depth, slope, density Sediment transport
2 Stream Power Discharge, slope, width Geomorphic work capacity
3 Bend \(R_c\) Planform coordinates Bend tightness
3 Meander Length Inflection points Planform scaling
3 Bend Width Amplitude, valley axis Lateral extent
3 \(R_c/W\) Radius, width Bend stability
3 \(W_m/W\) Bend width, bankfull width Planform development

Implementation Workflow

Level 1 → Level 2 → Level 3

TipWorkflow Sequence
  1. Level 1 Metrics (watershed-scale)
    • Derive from GIS analysis
    • Independent variables for predictions
    • Minimal field data required
  2. Level 2 Metrics (reach/cross-section scale)
    • Predicted from Level 1 using regional curves
    • Validated with field surveys
    • Core hydraulic geometry parameters
  3. Level 3 Metrics (planform scale)
    • Calculated from Level 2 dimensions
    • Require detailed planform mapping
    • Characterize lateral dynamics
NoteCross-Reference

See User Manual Section on Metric Calculation Workflow and Data Collection Protocols.