Scientific Background

This page provides detailed scientific context for the methods implemented in HyGCS.

Hysteresis in C-Q Relationships

Hysteresis occurs when the relationship between concentration (C) and discharge (Q) differs between rising and falling flow limbs, creating a loop in C-Q space.

Direction and Interpretation

Clockwise Hysteresis
Concentration peaks before discharge peaks. Indicates:

  • Flushing of readily available material

  • Transport-limited export

  • Proximal sources

  • High connectivity during rising limb

Counter-Clockwise Hysteresis
Concentration peaks after discharge peaks. Indicates:

  • Progressive mobilization

  • Source-limited export

  • Distal sources

  • Delayed connectivity

Figure-8 or Complex Patterns
Multiple loops or mixed patterns. Indicates:

  • Multiple source contributions

  • Changing transport pathways

  • Complex hydrological response

  • Need for multi-method analysis

Implemented Hysteresis Methods

HyGCS implements three complementary methods for hysteresis analysis.

HARP Method (Roberts et al., 2023)

Hysteresis Analysis of Rising and falling Peaks

Empirical classification based on:

  • Peak timing difference (ΔT = T_C - T_Q)

  • Loop area

  • Residual (end-state deviation)

Strengths:

  • Intuitive interpretation

  • Clear process identification

  • Named classification system

Limitations:

  • Requires clear peaks

  • Qualitative rather than quantitative

  • May struggle with complex patterns

Reference:

Roberts, M.E. et al. (2023). Hysteresis Analysis of Rising and falling Peaks (HARP): A new method to identify changing sediment and hydrological connectivity. Hydrological Processes.

Zuecco Index (Zuecco et al., 2016)

Integration-Based Hysteresis Index

Calculates h-index by integrating differential areas between rising and falling limbs:

\[h = \sum_{i} (A_{rise,i} - A_{fall,i})\]

where areas are computed between Q percentiles.

9-Class System:

Class

h-index Range

Description

0

Near zero

Linear/no hysteresis

1-4

Positive (varying magnitude)

Clockwise variants

5-8

Negative (varying magnitude)

Counter-clockwise variants
Strengths:

  • Quantitative magnitude assessment

  • Detects complex/mixed patterns

  • Robust to noise

Limitations:

  • Requires interpolation

  • Classification thresholds somewhat arbitrary

Reference:

Zuecco, G. et al. (2016). A versatile index to characterize hysteresis between hydrological variables at the runoff event timescale. Hydrological Processes, 30(9), 1449-1466.

Lloyd/Lawler Methods (2016, 2006)

Percentile-Based Indices

Samples C at 9 Q percentiles (0.1, 0.2, …, 0.9) on rising and falling limbs.

HInew (Lloyd 2016) - Recommended:

\[HI_{new} = \frac{C_{rise} - C_{fall}}{C_{mid}}\]

  • Symmetric range: [-1, 1]

  • C_mid = (C_rise + C_fall) / 2

HIL (Lawler 2006) - Original:

\[\begin{split}HI_L = \begin{cases} (C_{rise} / C_{fall}) - 1 & \text{if } C_{rise} > C_{fall} \\ (-1 / (C_{rise} / C_{fall})) + 1 & \text{otherwise} \end{cases}\end{split}\]

  • Asymmetric range

  • More sensitive at extremes

Strengths:

  • Standard in literature

  • Percentile-based (robust to outliers)

  • HInew facilitates comparison

Limitations:

  • Requires both rising and falling limbs

  • May miss complex patterns

References:

  • Lloyd, C.E.M. et al. (2016). Using hysteresis analysis of high-resolution water quality monitoring data. Hydrology and Earth System Sciences, 20, 2705-2719.

  • Lawler, D.M. et al. (2006). Turbidity dynamics during spring storm events. Science of the Total Environment, 360, 109-126.

CVc/CVq Framework

Coefficient of Variation Approach (Musolff et al., 2015)

Distinguishes chemostatic from chemodynamic behavior using variability ratios:

\[\frac{CV_c}{CV_q} = \frac{\sigma_C / \mu_C}{\sigma_Q / \mu_Q}\]

Interpretation:

CVc/CVq > 1 (Chemodynamic)
Concentration varies more than flow. Indicates:

  • Variable source contributions

  • Transport-dependent export

  • Event-driven dynamics

  • Hysteretic behavior likely

CVc/CVq < 1 (Chemostatic)
Concentration buffered relative to flow. Indicates:

  • Consistent source strength

  • Concentration-discharge equilibrium

  • Hysteresis less pronounced

  • Stable biogeochemical processes

Implementation in HyGCS:

Computed on rolling windows (typically 5-10 samples) to capture temporal dynamics.

Reference:

Musolff, A. et al. (2015). Catchment controls on solute export. Advances in Water Resources, 86, 133-146.

C-Q Relationships

Power-Law Model (Thompson et al., 2011)

The C-Q relationship is commonly modeled as:

\[C = aQ^b\]

Taking logarithms:

\[\log(C) = \log(a) + b \cdot \log(Q)\]
where:

  • a = intercept (baseline concentration)

  • b = slope (mechanistic indicator)

Slope Interpretation:

b > 0.15 (Dilution/Flushing)
Concentration increases with flow. Indicates:

  • Transport-limited export

  • Flushing of accumulated material

  • Proximal sources activated

  • Increasing connectivity

b < -0.15 (Enrichment/Loading)
Concentration decreases with flow. Indicates:

  • Dilution of point sources

  • Source-limited export

  • Groundwater contribution dominant

  • Decreasing connectivity

|b| < 0.1 (Chemostatic)
Weak C-Q relationship. Indicates:

  • Buffered system

  • Consistent source strength

  • Concentration-discharge equilibrium

  • Weak flow dependency

Reference:

Thompson, S.E. et al. (2011). Comparative hydrology across AmeriFlux sites: The variable roles of climate, vegetation, and groundwater. Water Resources Research, 47(10).

Geochemical Phase Classification

HyGCS implements a hierarchical 6-phase classification system developed by Sanchez et al. (2025, in review) that integrates:

  1. Window-scale hysteresis indices

  2. C-Q slope (power-law exponent)

  3. CVc/CVq variability ratios

  4. Flow dynamics (rising/falling, peaks)

  5. Temporal context (phase transitions)

The 6 Phases

F - Flushing

Characteristics:

  • Steep concentration decline during high flow

  • Positive C-Q slope (b > 0.15)

  • High CVc/CVq (chemodynamic)

  • Clockwise hysteresis common

  • Concentration in high percentile, declining

Process interpretation:

Rapid mobilization of accumulated material during high-flow events. Transport-limited export with strong connectivity.

L - Loading

Characteristics:

  • Concentration rising to maximum

  • Negative C-Q slope (b < -0.15)

  • Rising concentration trajectory

  • Counter-clockwise hysteresis possible

  • Concentration increasing with or before flow

Process interpretation:

Accumulation phase with progressive source mobilization. Enrichment before peak flow arrival.

C - Chemostatic

Characteristics:

  • Low hysteresis magnitude

  • Flat C-Q slope (|b| < 0.1)

  • Low CVc/CVq (< 1)

  • Stable concentration

  • Low variability

Process interpretation:

Buffered system with consistent source strength. Concentration-discharge equilibrium maintained.

D - Dilution

Characteristics:

  • Post-flush recovery

  • Declining flow

  • Declining concentration

  • Follows flushing phase

  • Medium to low connectivity

Process interpretation:

Recovery after flushing event. Sources depleted, system returning to baseflow conditions.

R - Recession

Characteristics:

  • Late cycle, both flow and concentration declining

  • Low CVc/CVq

  • Low hysteresis

  • Days since peak > threshold

  • Low connectivity

Process interpretation:

Baseflow-dominated conditions. Limited source availability and weak connectivity.

V - Variable

Characteristics:

  • Ambiguous patterns

  • Mixed signatures

  • Low confidence classification

  • Transitional behavior

Process interpretation:

Complex or mixed processes not fitting other categories. May indicate multiple overlapping processes or insufficient data.

Percentile-Based Thresholds

The classification uses percentile-based thresholds rather than absolute values, making it compound-agnostic and adaptable to different concentration ranges.

Advantages:

  • Works across different compounds (metals, nutrients, etc.)

  • Adapts to site-specific conditions

  • Robust to concentration scale differences

  • Reduces need for parameter tuning

Thresholds Computed:

  • Flow percentiles (33rd, 67th for low/medium/high)

  • Concentration change percentiles (25th, 75th)

  • C-Q slope absolute thresholds (±0.15, ±0.1)

  • CVc/CVq ratio (typically 1.0 threshold)

Window-Scale Hysteresis

Unlike traditional event-scale hysteresis, HyGCS computes hysteresis indices on moving windows (typically 10-20 points) around each classified segment.

Why window-scale?

  • Captures local temporal dynamics

  • Avoids artifacts from full time series loops

  • More appropriate for long-term monitoring data

  • Detects changing patterns over time

Hierarchical Rule System

Rules are checked in priority order to avoid ambiguity:

  1. Strong signatures (F, L) - checked first

  2. Moderate signatures (C, D, R) - checked next

  3. Default to V if no clear pattern

Confidence scoring based on:

  • Number of rules triggered

  • Agreement between indicators (hysteresis, slope, CVc/CVq)

  • Data quality (sufficient points, valid metrics)

  • Consistency with temporal context (previous phases)

Critical Perspective

Knapp & Musolff (2024) - Critical Assessment

Important considerations for C-Q analysis:

Multi-Method Validation

No single method is perfect. Convergent evidence from multiple methods increases confidence. When methods disagree, investigate further.

Contextual Interpretation

Hysteresis and C-Q slopes reveal patterns but not mechanisms directly. Must combine with:

  • Site knowledge (geology, land use, hydrology)

  • Process understanding (biogeochemistry, transport)

  • Additional data (tracers, high-frequency monitoring)

Data Requirements
Quality and quantity of data matter:

  • Temporal resolution affects pattern detection

  • Sampling bias can create artifacts

  • Outliers and measurement errors propagate

  • Need adequate coverage of flow range

Limitations of Classification
Automated classification is a tool, not truth:

  • Low confidence classifications need investigation

  • Phase boundaries are fuzzy, not discrete

  • Complex systems may not fit simple categories

  • Always validate against known reference periods

Reference:

Knapp, J.L.A. & Musolff, A. (2024). Mind the gap: A critical perspective on concentration-discharge relationships. Hydrological Processes. https://doi.org/10.1002/hyp.15328

Best Practices

  1. Use Multiple Methods

    Always compare HARP, Zuecco, and Lloyd/Lawler results. Agreement → confidence.

  2. Validate with Known Events

    Test on reference events with known behavior to calibrate interpretation.

  3. Check Data Quality

    • Remove obvious outliers

    • Ensure Q and C are synchronized

    • Verify sufficient temporal coverage

    • Examine gaps and missing data

  4. Consider Temporal Context

    • Seasonal patterns

    • Antecedent conditions

    • Long-term trends

    • Event sequencing

  5. Integrate Process Knowledge

    • Site characteristics (geology, land use)

    • Known sources and pathways

    • Historical behavior

    • Expert knowledge

  6. Report Uncertainty

    • Confidence scores

    • Method agreement/disagreement

    • Data limitations

    • Alternative interpretations

  7. Avoid Over-Interpretation

    Hysteresis reveals patterns, not mechanisms directly. Use as hypothesis-generating tool, not definitive proof.

See Also