AquaScope Theory Guide¶
A mathematical reference for every method in AquaScope. Each section states the equations, assumptions, and key references with DOI links so you can trace every result back to the peer-reviewed literature.
Table of Contents¶
- Flood Frequency Analysis
- Baseflow Separation
- Flow Duration Curves
- Hydrological Signatures
- Extreme Value Theory (Extended)
- Copula Analysis
- Change-Point Detection
- Bayesian Uncertainty Quantification
- Rating Curves
- Transfer Learning for Ungauged Basins
- Decision Trees — Choosing the Right Method
1. Flood Frequency Analysis¶
1.1 Generalized Extreme Value (GEV)¶
The GEV distribution unifies the Gumbel, Fréchet, and Weibull extreme-value families into a single three-parameter model.
CDF:
$$F(x) = \exp!\Bigl{-\Bigl[1 + \xi\,\frac{x - \mu}{\sigma}\Bigr]^{-1/\xi}\Bigr}, \qquad 1 + \xi\,\frac{x-\mu}{\sigma} > 0$$
where $\mu$ is the location, $\sigma > 0$ the scale, and $\xi$ the shape parameter. When $\xi \to 0$ the distribution reduces to the Gumbel.
Quantile function (return-period estimate):
$$x_p = \mu - \frac{\sigma}{\xi}\bigl[1 - (-\ln p)^{-\xi}\bigr], \qquad p = 1 - 1/T$$
where $T$ is the return period in years.
Parameter estimation: AquaScope uses maximum-likelihood estimation (MLE) via
scipy.stats.genextreme.fit(). Bootstrap confidence intervals are computed by
resampling the annual maximum series 1 000 times.
Reference: Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer. doi:10.1007/978-1-4471-3675-0
1.2 Log-Pearson Type III (LP3)¶
The US standard method (Bulletin 17C). The annual maximum series $Q$ is log-transformed:
$$Y = \log_{10}(Q)$$
A Pearson Type III distribution is fitted to $Y$ with moments:
- $\bar{Y} = \frac{1}{n}\sum Y_i$ (mean)
- $S_Y = \sqrt{\frac{1}{n-1}\sum(Y_i - \bar{Y})^2}$ (standard deviation)
- $G_s = \frac{n}{(n-1)(n-2)\,S_Y^3}\sum(Y_i - \bar{Y})^3$ (station skew)
Bulletin 17C weighted skew:
$$G_w = w_1\,G_s + w_2\,G_r$$
where $G_r$ is the regional (generalized) skew and the weights are inverse-variance:
$$w_1 = \frac{\text{MSE}{G_r}}{\text{MSE}, \qquad w_2 = 1 - w_1$$} + \text{MSE}_{G_r}
Station skew MSE is approximated as:
$$\text{MSE}_{G_s} \approx \frac{6}{n}\Bigl(1 + \frac{9}{6}G_s^2 + \frac{15}{48}G_s^4\Bigr)$$
The default regional skew MSE is 0.302 (USGS national value).
Return-period quantile:
$$Q_T = 10^{\bar{Y} + K_T \cdot S_Y}$$
where $K_T$ is the Pearson Type III frequency factor for exceedance probability $1/T$ and skew $G_w$.
Reference: England, J. F., Jr., et al. (2018). Guidelines for Determining Flood Flow Frequency — Bulletin 17C. USGS Techniques and Methods, Book 4, Chapter B5. doi:10.3133/tm4B5
1.3 Expected Moments Algorithm (EMA)¶
EMA extends LP3 to handle censored observations — zero-flow years, low outliers detected by the Multiple Grubbs-Beck test, and historical (paleoflood) data.
Each observation $i$ is associated with a perception threshold interval $[T_{L,i},\, T_{U,i}]$:
- Systematic record above threshold: $T_L = 0,\ T_U = \infty$
- Zero-flow year: $T_L = 0,\ T_U = Q_{\text{thresh}}$ (left-censored)
- Historical flood: $T_L = Q_{\text{perception}},\ T_U = \infty$
The algorithm iteratively adjusts the first three moments of $\log_{10}(Q)$ to account for the probability mass below/above the perception thresholds.
Censored-adjusted mean:
$$\bar{Y}{\text{adj}} = \frac{1}{n_c\Bigr]$$}}}\Bigl[\sum_{i \in \text{obs}} Y_i + n_c\,\hat{\mu
where $n_c$ is the number of censored observations and $\hat{\mu}c$ is the expected value of $Y$ conditional on $Y \leq \log(T_U)$ under the current LP3 fit.
Reference: Cohn, T. A., Lane, W. L., & Baier, W. G. (1997). An algorithm for computing moments-based flood quantile estimates when historical flood information is available. Water Resources Research, 33(9), 2089–2096. doi:10.1029/96WR03706
1.4 Multiple Grubbs-Beck Test¶
Identifies low outliers that should be treated as censored in the EMA.
For sample size $n$, the test statistic for the $k$-th smallest value is:
$$G_k = \frac{\bar{Y}{(-k)} - Y$$}}{S_{(-k)}
where $\bar{Y}{(-k)}$ and $S$ are the mean and standard deviation of the sample excluding the $k$ smallest values. Values exceeding the critical value at significance level $\alpha$ are flagged as low outliers.
Reference: Grubbs, F. E., & Beck, G. (1972). Extension of sample sizes and percentage points for significance tests of outlying observations. Technometrics, 14(4), 847–854. doi:10.1080/00401706.1972.10488981
1.5 L-Moments¶
L-moments are linear combinations of probability-weighted moments (PWMs), more robust than conventional moments for small samples.
Probability-weighted moments:
$$\beta_r = \frac{1}{n}\sum_{i=1}^{n} \frac{\binom{i-1}{r}}{\binom{n-1}{r}}\,x_{(i)}$$
First four L-moments:
$$\lambda_1 = \beta_0, \qquad \lambda_2 = 2\beta_1 - \beta_0$$ $$\lambda_3 = 6\beta_2 - 6\beta_1 + \beta_0, \qquad \lambda_4 = 20\beta_3 - 30\beta_2 + 12\beta_1 - \beta_0$$
L-moment ratios:
$$\tau = \lambda_2 / \lambda_1 \text{ (L-CV)}, \qquad \tau_3 = \lambda_3 / \lambda_2 \text{ (L-skewness)}, \qquad \tau_4 = \lambda_4 / \lambda_2 \text{ (L-kurtosis)}$$
GEV parameters from L-moments:
$$c = \frac{2}{3 + \tau_3} - \frac{\ln 2}{\ln 3}, \qquad \xi \approx 7.8590c + 2.9554c^2$$ $$\sigma = \frac{\lambda_2\,\xi}{(1 - 2^{-\xi})\,\Gamma(1+\xi)}, \qquad \mu = \lambda_1 - \frac{\sigma}{\xi}\bigl[\Gamma(1+\xi) - 1\bigr]$$
Reference: Hosking, J. R. M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society B, 52(1), 105–124. doi:10.1111/j.2517-6161.1990.tb01775.x
1.6 Non-Stationary GEV¶
For climate-affected flood series, the location parameter varies with time:
$$\mu(t) = \mu_0 + \mu_1\,t$$
The log-likelihood is maximised over $(\mu_0, \mu_1, \sigma, \xi)$ using Nelder-Mead optimisation. AquaScope reports trend significance and per-year quantile estimates.
1.7 Regional Frequency Analysis¶
The index-flood method (Hosking & Wallis, 1997) pools data from multiple sites to improve quantile estimates at each site.
- Discordancy measure $D_i$: identifies sites whose L-moment ratios are outliers relative to the regional average.
- Heterogeneity measure $H$: tests whether sites form a homogeneous region. $H < 1$: acceptably homogeneous; $1 \leq H < 2$: possibly heterogeneous; $H \geq 2$: definitely heterogeneous.
- Regional growth curve: a common dimensionless frequency curve scaled by each site's index flood (typically the sample mean).
Reference: Hosking, J. R. M., & Wallis, J. R. (1997). Regional Frequency Analysis: An Approach Based on L-Moments. Cambridge University Press. ISBN 978-0521019408.
2. Baseflow Separation¶
2.1 Lyne-Hollick Digital Filter¶
A one-parameter recursive filter applied to the total streamflow $Q(t)$.
Quickflow equation:
$$q_f(t) = \alpha\,q_f(t-1) + \frac{1+\alpha}{2}\bigl[Q(t) - Q(t-1)\bigr]$$
with the constraint $0 \leq q_f(t) \leq Q(t)$.
Baseflow:
$$q_b(t) = Q(t) - q_f(t)$$
The filter parameter $\alpha$ controls smoothness (typical value: 0.925). Multiple forward-backward passes improve separation quality.
Reference: Nathan, R. J., & McMahon, T. A. (1990). Evaluation of automated techniques for base flow and recession analyses. Water Resources Research, 26(7), 1465–1473. doi:10.1016/0022-1694(90)90259-2
2.2 Eckhardt Two-Parameter Filter¶
$$q_b(t) = \frac{(1 - \text{BFI}{\max})\,\alpha\,q_b(t-1) + (1-\alpha)\,\text{BFI}$$}\,Q(t)}{1 - \alpha\,\text{BFI}_{\max}
with $q_b(t) \leq Q(t)$.
The $\text{BFI}_{\max}$ parameter depends on aquifer type:
| Aquifer type | Recommended BFI_max |
|---|---|
| Perennial streams, porous aquifer | 0.80 |
| Ephemeral streams, porous aquifer | 0.50 |
| Perennial streams, hard rock | 0.25 |
Reference: Eckhardt, K. (2005). How to construct recursive digital filters for baseflow separation. Hydrological Processes, 19(2), 507–515. doi:10.1016/j.jhydrol.2005.07.017
3. Flow Duration Curves¶
The FDC shows the percentage of time a given discharge is equalled or exceeded.
Weibull plotting position:
$$p_i = \frac{i}{n + 1}$$
where $i$ is the rank (1 = largest) and $n$ is the sample size. This is an exceedance probability.
FDC slope (between the 33rd and 66th percentiles on a log scale):
$$S_{\text{FDC}} = \frac{\ln Q_{33} - \ln Q_{66}}{0.66 - 0.33}$$
A steep slope indicates a flashy regime; a flat slope indicates sustained baseflow.
Reference: Vogel, R. M., & Fennessey, N. M. (1995). Flow duration curves II: A review of applications in water resources planning. Water Resources Bulletin, 31(6), 1029–1039. doi:10.1111/j.1752-1688.1995.tb04392.x
4. Hydrological Signatures¶
AquaScope computes 22 catchment signatures from daily discharge. These characterise magnitude, variability, frequency, duration, timing, and rate of change of streamflow.
| Signature | Formula | Unit |
|---|---|---|
| Mean discharge ($\bar{Q}$) | $\frac{1}{n}\sum Q_i$ | m³/s |
| Median discharge ($Q_{50}$) | 50th percentile | m³/s |
| Coefficient of variation | $\text{CV} = S_Q / \bar{Q}$ | — |
| Skewness | $\frac{n}{(n-1)(n-2)}\sum\bigl(\frac{Q_i - \bar{Q}}{S_Q}\bigr)^3$ | — |
| Low flow (Q95) | 95th exceedance percentile | m³/s |
| High flow (Q5) | 5th exceedance percentile | m³/s |
| Baseflow index (BFI) | $\sum q_b / \sum Q$ | — |
| Runoff ratio | $\sum Q / \sum P$ | — |
| FDC slope | See §3 | — |
| Flashiness index | $\sum\lvert Q_i - Q_{i-1}\rvert / \sum Q_i$ | — |
| Rising limb density | (# rising days) / (# peaks) | d/peak |
| Falling limb density | (# falling days) / (# peaks) | d/peak |
| Zero-flow fraction | (# days with Q = 0) / n | — |
| High-flow frequency | (# days with Q > 9 × $Q_{50}$) / n | — |
| High-flow duration | mean consecutive days above 9 × $Q_{50}$ | d |
| Low-flow frequency | (# days with Q < 0.2 × $\bar{Q}$) / n | — |
| Low-flow duration | mean consecutive days below 0.2 × $\bar{Q}$ | d |
| Recession constant ($k$) | slope of $\ln Q$ during recessions | 1/d |
| Peak month | month with highest mean Q | month |
| Seasonality index | circular variance of monthly Q | — |
| Flow elasticity | $\text{median}\bigl(\frac{\Delta Q/\bar{Q}}{\Delta P/\bar{P}}\bigr)$ | — |
| Mean half-flow date | day of year at which 50 % of annual flow has passed | DOY |
Reference: McMillan, H. (2020). Linking hydrologic signatures to hydrologic processes: A review. WIREs Water, 7(6), e1481. doi:10.1002/wat2.1481
5. Extreme Value Theory (Extended)¶
5.1 Gumbel Distribution¶
A special case of GEV with $\xi = 0$:
$$F(x) = \exp!\bigl[-\exp!\bigl(-\frac{x - \mu}{\sigma}\bigr)\bigr]$$
$$x_p = \mu - \sigma\,\ln(-\ln p)$$
Appropriate when there is no evidence of a heavy or bounded upper tail.
5.2 Generalized Pareto Distribution (POT)¶
For peaks over a threshold $u$, exceedances $y = x - u$ follow:
$$F(y) = 1 - \Bigl(1 + \xi\,\frac{y}{\tilde\sigma}\Bigr)^{-1/\xi}$$
The threshold $u$ is selected as the empirical quantile at which the mean residual life plot becomes approximately linear.
Equivalence: If block maxima follow $\text{GEV}(\mu, \sigma, \xi)$, then exceedances above a high threshold follow $\text{GPD}(\tilde\sigma, \xi)$ with the same shape parameter.
Reference: Davison, A. C., & Smith, R. L. (1990). Models for exceedances over high thresholds. Journal of the Royal Statistical Society B, 52(3), 393–442. doi:10.1111/j.2517-6161.1990.tb01796.x
5.3 Goodness-of-Fit Tests¶
Anderson-Darling (AD):
$$A^2 = -n - \frac{1}{n}\sum_{i=1}^{n}(2i - 1)\bigl[\ln F(x_{(i)}) + \ln(1 - F(x_{(n+1-i)}))\bigr]$$
Emphasises tail behaviour more than the Kolmogorov-Smirnov test.
Cramér-von Mises (CvM):
$$W^2 = \sum_{i=1}^{n}\Bigl[F(x_{(i)}) - \frac{2i - 1}{2n}\Bigr]^2 + \frac{1}{12n}$$
Probability Plot Correlation Coefficient (PPCC):
$$r = \frac{\sum(x_{(i)} - \bar{x})(m_i - \bar{m})}{\sqrt{\sum(x_{(i)} - \bar{x})^2\sum(m_i - \bar{m})^2}}$$
where $m_i$ are the theoretical quantiles. Values close to 1 indicate good fit.
6. Copula Analysis¶
6.1 Sklar's Theorem¶
Any joint distribution $H(x, y)$ can be expressed via a copula $C$:
$$H(x, y) = C\bigl(F_X(x),\, F_Y(y)\bigr)$$
where $F_X, F_Y$ are the marginal CDFs. AquaScope uses pseudo-observations $u_i = R_i / (n + 1)$ (rank-based) to avoid boundary artefacts.
6.2 Implemented Families¶
| Family | CDF $C(u,v)$ | $\tau \to \theta$ | Tail dependence |
|---|---|---|---|
| Gaussian | $\Phi_2\bigl(\Phi^{-1}(u), \Phi^{-1}(v); \rho\bigr)$ | $\rho = \sin(\pi\tau/2)$ | None |
| Clayton | $\bigl(u^{-\theta} + v^{-\theta} - 1\bigr)^{-1/\theta}$ | $\theta = 2\tau/(1-\tau)$ | Lower: $2^{-1/\theta}$ |
| Gumbel | $\exp!\bigl[-\bigl((-\ln u)^\theta + (-\ln v)^\theta\bigr)^{1/\theta}\bigr]$ | $\theta = 1/(1-\tau)$ | Upper: $2 - 2^{1/\theta}$ |
| Frank | $-\frac{1}{\theta}\ln!\Bigl[1 + \frac{(e^{-\theta u}-1)(e^{-\theta v}-1)}{e^{-\theta}-1}\Bigr]$ | Numerical (Debye function) | None |
Model selection uses AIC:
$$\text{AIC} = 2k - 2\ln\hat{L}$$
where $k = 1$ (all families have one parameter) and $\hat{L}$ is the pseudo-log-likelihood evaluated over the copula density.
References: - Nelsen, R. B. (2006). An Introduction to Copulas. 2nd ed. Springer. ISBN 978-0387286785. - Genest, C., & Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12(4), 347–368. doi:10.1061/(ASCE)1084-0699(2007)12:4(347)
6.3 When to Use Which Copula¶
- Symmetric dependence (e.g., precipitation at two nearby gauges) → Gaussian or Frank
- Lower tail dependence (joint low-flow risk, drought co-occurrence) → Clayton
- Upper tail dependence (joint flood risk) → Gumbel
- Unsure → use
fit_copula(x, y, family="auto")— AquaScope fits all four and selects by AIC
7. Change-Point Detection¶
7.1 PELT (Pruned Exact Linear Time)¶
Minimises the penalised cost:
$$\sum_{j=0}^{m} \bigl[\mathcal{C}(y_{\tau_j+1:\tau_{j+1}})\bigr] + \beta\,m$$
where $\mathcal{C}$ is the segment cost (e.g., negative Gaussian log-likelihood), $m$ is the number of changepoints, and $\beta$ is the penalty. AquaScope uses the BIC penalty $\beta = \ln(n)$ by default.
The pruning rule discards candidate changepoints that provably cannot improve the optimum, achieving $O(n)$ expected complexity.
Reference: Killick, R., Fearnhead, P., & Eckley, I. A. (2012). Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association, 107(500), 1590–1598. doi:10.1080/01621459.2012.737745
7.2 Pettitt Test¶
A non-parametric test for a single changepoint. The test statistic:
$$U_T = \max_{1 \leq t < n} \Bigl|\sum_{i=1}^{t}\sum_{j=t+1}^{n} \text{sgn}(X_i - X_j)\Bigr|$$
The approximate p-value is:
$$p \approx 2\exp!\Bigl(\frac{-6\,U_T^2}{n^3 + n^2}\Bigr)$$
Reference: Pettitt, A. N. (1979). A non-parametric approach to the change-point problem. Journal of the Royal Statistical Society C, 28(2), 126–135. doi:10.2307/2346729
7.3 CUSUM¶
The cumulative sum chart detects shifts in the mean:
$$S_t = \max!\bigl(0,\, S_{t-1} + (x_t - \bar{x}) - k\bigr)$$
where $k$ is a slack parameter (default: $0.5\,\hat\sigma$). A changepoint is flagged when $S_t$ exceeds threshold $h$ (default: $4\,\hat\sigma$).
8. Bayesian Uncertainty Quantification¶
8.1 Conjugate Normal-Inverse-Gamma Model¶
For linear regression $y = X\beta + \varepsilon$, $\varepsilon \sim N(0, \sigma^2)$:
Prior:
$$\beta \mid \sigma^2 \sim N(\mu_0,\, \sigma^2 \Lambda_0^{-1}), \qquad \sigma^2 \sim \text{Inv-Gamma}(a_0, b_0)$$
Posterior (closed-form):
$$\Lambda_n = \Lambda_0 + X^T X$$ $$\mu_n = \Lambda_n^{-1}(\Lambda_0 \mu_0 + X^T y)$$ $$a_n = a_0 + n/2$$ $$b_n = b_0 + \tfrac{1}{2}\bigl(y^T y + \mu_0^T \Lambda_0 \mu_0 - \mu_n^T \Lambda_n \mu_n\bigr)$$
Predictive distribution (Student-t):
$$\tilde{y} \mid x_ \sim t_{2a_n}!\Bigl(\mu_n^T x_,\; \frac{b_n}{a_n}\bigl(1 + x_^T \Lambda_n^{-1} x_\bigr)\Bigr)$$
Reference: Gelman, A., et al. (2013). Bayesian Data Analysis. 3rd ed. Chapman & Hall/CRC. ISBN 978-1439840955.
8.2 Metropolis-Hastings MCMC¶
For non-conjugate models, AquaScope uses random-walk Metropolis-Hastings:
- Propose $\theta^* = \theta^{(t)} + \varepsilon$, $\varepsilon \sim N(0, \Sigma_{\text{prop}})$
- Compute acceptance ratio:
$$\alpha = \min!\Bigl(1,\; \frac{p(\theta^)\,L(y|\theta^)}{p(\theta^{(t)})\,L(y|\theta^{(t)})}\Bigr)$$
- Accept with probability $\alpha$; otherwise $\theta^{(t+1)} = \theta^{(t)}$
The proposal covariance $\Sigma_{\text{prop}}$ must be tuned for ~23 % acceptance rate (high dimension) or ~44 % (1-D).
8.3 Convergence Diagnostics¶
Gelman-Rubin $\hat{R}$ (split-chain method):
$$\hat{R} = \sqrt{\frac{\hat{V}}{W}}$$
where $W$ is the within-chain variance and $\hat{V}$ is the pooled variance estimate. Convergence is indicated by $\hat{R} < 1.05$.
Effective sample size:
$$n_{\text{eff}} = \frac{mn}{1 + 2\sum_{k=1}^{K} \hat\rho_k}$$
where $\hat\rho_k$ is the estimated autocorrelation at lag $k$.
DIC (Deviance Information Criterion):
$$\text{DIC} = \bar{D} + p_D, \qquad p_D = \bar{D} - D(\bar\theta)$$
WAIC (Widely Applicable Information Criterion):
$$\text{WAIC} = -2\sum_{i=1}^{n}\ln!\Bigl(\frac{1}{S}\sum_{s=1}^{S}p(y_i|\theta^{(s)})\Bigr) + 2\,p_{\text{WAIC}}$$
where $p_{\text{WAIC}} = \sum_i \text{Var}_s[\ln p(y_i|\theta^{(s)})]$.
9. Rating Curves¶
9.1 Power-Law Model¶
$$Q = a\,(h - h_0)^b$$
where $Q$ is discharge, $h$ is stage, $h_0$ is the effective zero-flow stage, $a$ is a coefficient, and $b$ is the exponent. Fitted in log-space via linear regression:
$$\ln Q = \ln a + b\,\ln(h - h_0)$$
9.2 Segmented Rating Curves¶
When channel geometry changes at a breakpoint stage $h_b$, two power-law segments are fitted with continuity enforced at $h_b$:
$$Q = \begin{cases} a_1(h - h_{0,1})^{b_1} & h \leq h_b \ a_2(h - h_{0,2})^{b_2} & h > h_b \end{cases}$$
9.3 Shift Detection¶
Temporal shifts in the rating curve (due to scour, deposition, or vegetation change) are detected by monitoring residuals over time. A significant trend in residuals triggers a shift warning.
Reference: Rantz, S. E., et al. (1982). Measurement and Computation of Streamflow. Volume 2: Computation of Discharge. USGS Water-Supply Paper 2175.
10. Transfer Learning for Ungauged Basins¶
10.1 Problem¶
Many basins lack sufficient streamflow data for calibration. Transfer learning borrows information from donor sites with similar hydrological behaviour.
10.2 Donor Selection¶
AquaScope selects donors by computing the similarity score between the target and candidate sites based on their hydrological signature vectors:
$$d(A, B) = \sqrt{\sum_{k=1}^{K} w_k \Bigl(\frac{s_k^A - s_k^B}{\hat\sigma_k}\Bigr)^2}$$
where $s_k$ are standardised signatures and $w_k$ are optional weights.
Spatial proximity weighting uses the Haversine distance:
$$d_{\text{geo}} = 2R\arcsin!\sqrt{\sin^2!\frac{\Delta\phi}{2} + \cos\phi_1\cos\phi_2\sin^2!\frac{\Delta\lambda}{2}}$$
$$w_{\text{spatial}} = \frac{1}{1 + d_{\text{geo}} / d_0}$$
10.3 Workflow¶
- Compute signatures for all candidate donors and the target
- Rank donors by similarity score (optionally weighted by spatial proximity)
- Pool training data from top-$k$ donors
- Train a model on pooled data
- Fine-tune on available target data (even if very short record)
Reference: Hrachowitz, M., et al. (2013). A decade of Predictions in Ungauged Basins (PUB) — a review. Hydrological Sciences Journal, 58(6), 1198–1255. doi:10.5194/hess-17-1893-2013
11. Decision Trees — Choosing the Right Method¶
Which Flood Frequency Method?¶
┌─ Yes ─→ EMA (§1.3)
Zero-flow years? ─┤
└─ No
┌─ Yes ─→ LP3 + Bulletin 17C (§1.2)
US regulatory context? ┤
└─ No
┌─ Yes ─→ Non-stationary GEV (§1.6)
Evidence of trend? ─┤
└─ No
┌─ < 25 years ─→ L-moments (§1.5)
Record length? ─┤
├─ 25–50 years ─→ GEV MLE (§1.1)
└─ Multiple sites ─→ Regional (§1.7)
Which Baseflow Method?¶
Which Copula?¶
Joint flood risk (upper tail) ─→ Gumbel
Joint drought risk (lower tail)─→ Clayton
Symmetric / unsure ─→ Gaussian or Frank
Don't know ─→ family="auto" (AIC selection)
When to Use Bayesian Methods?¶
Small sample (n < 30) ─→ Bayesian with informative prior (§8.1)
Uncertainty is the main output ─→ Always Bayesian
Regulatory context ─→ Frequentist primary, Bayesian as check
Complex non-linear model ─→ MH-MCMC (§8.2)
When to Use Transfer Learning?¶
Target has < 2 years of data ─→ Transfer from donors (§10)
Target has > 10 years ─→ Calibrate directly
2–10 years ─→ Transfer + fine-tune
References (Consolidated)¶
- Addor, N., et al. (2017). The CAMELS data set. HESS, 21, 5293–5313. doi:10.5194/hess-21-5293-2017
- Cohn, T. A., et al. (1997). Water Resources Research, 33(9), 2089–2096. doi:10.1029/96WR03706
- Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer. doi:10.1007/978-1-4471-3675-0
- Davison, A. C., & Smith, R. L. (1990). J. R. Statist. Soc. B, 52(3), 393–442. doi:10.1111/j.2517-6161.1990.tb01796.x
- Eckhardt, K. (2005). Hydrological Processes, 19(2), 507–515. doi:10.1016/j.jhydrol.2005.07.017
- England, J. F., Jr., et al. (2018). Bulletin 17C. USGS TM 4-B5. doi:10.3133/tm4B5
- Gelman, A., et al. (2013). Bayesian Data Analysis. 3rd ed. CRC Press.
- Genest, C., & Favre, A.-C. (2007). J. Hydrol. Eng., 12(4), 347–368. doi:10.1061/(ASCE)1084-0699(2007)12:4(347)
- Grubbs, F. E., & Beck, G. (1972). Technometrics, 14(4), 847–854. doi:10.1080/00401706.1972.10488981
- Hosking, J. R. M. (1990). J. R. Statist. Soc. B, 52(1), 105–124. doi:10.1111/j.2517-6161.1990.tb01775.x
- Hosking, J. R. M., & Wallis, J. R. (1997). Regional Frequency Analysis. Cambridge.
- Hrachowitz, M., et al. (2013). Hydrol. Sci. J., 58(6), 1198–1255. doi:10.5194/hess-17-1893-2013
- Killick, R., et al. (2012). JASA, 107(500), 1590–1598. doi:10.1080/01621459.2012.737745
- McMillan, H. (2020). WIREs Water, 7(6), e1481. doi:10.1002/wat2.1481
- Nathan, R. J., & McMahon, T. A. (1990). Water Resour. Res., 26(7), 1465–1473. doi:10.1016/0022-1694(90)90259-2
- Nelsen, R. B. (2006). An Introduction to Copulas. 2nd ed. Springer.
- Pettitt, A. N. (1979). J. R. Statist. Soc. C, 28(2), 126–135. doi:10.2307/2346729
- Rantz, S. E., et al. (1982). USGS Water-Supply Paper 2175.
- Vogel, R. M., & Fennessey, N. M. (1995). Water Resour. Bull., 31(6), 1029–1039. doi:10.1111/j.1752-1688.1995.tb04392.x