Everything Comes Down to Timing: Optimal Green Infrastructure Placement and the Effect of Within-Storm Variability

Abstract

Urban flood peak mitigation by green infrastructure (GI) is fundamentally a timing problem. Because GI storage is finite, interception occurs only within a brief active window; whether it reduces the outlet peak depends on GI placement in the network, routing lags, and rainfall timing. Here, we develop a timescale-based framework that links outlet peak reduction to the alignment among within-storm temporal structure, network response, and GI filling dynamics, providing a compact way to interpret when different network positions become most effective under a fixed GI design. Starting from a general convolution representation of runoff generation, interception, and routing, we show that peak reduction efficiency and location ranking can be organized by two nondimensional ratios—comparing storm duration and network response time to a characteristic GI filling time—plus simple descriptors of within-storm temporal structure. Under uniform rainfall, these ratios yield an interpretable regime diagram with analytical transition curves between downstream-, mid-network-, and upstream-optimal placement for a generic dispersive routing representation. Relaxing the uniform-rainfall assumption shows that within-storm variability can substantially reorganize these regimes because storm timing controls both how long GI storage remains available before it fills and which routed contributions overlap to form the outlet peak. Highly concentrated storms and storms with early internal peaks are especially likely to reorder the ranking of candidate locations relative to the uniform-rainfall baseline. Using 2351 observed hourly storm events evaluated across virtual catchments spanning fast to slow network responses, we quantify how often realistic event structure alters the optimal location and the regret associated with adopting a uniform design storm. The results motivate robustness-oriented placement strategies based on ensembles of plausible storm temporal structures, organized within the proposed timescale diagram rather than reliance on a single design hyetograph.

1. Introduction

Urban drainage systems are under increasing stress from both intensifying rainfall and expanding impervious cover. The IPCC Sixth Assessment Report emphasizes that a warming climate is changing the character of weather and climate extremes, while cities and critical infrastructure face growing exposure and vulnerability to these hazards [1,2]. In parallel, continued urbanization increases imperviousness and accelerates runoff production, and the limits of conventional conveyance expansion have been widely recognized in modern urban stormwater practice [3]. Against this backdrop, green infrastructure (GI) has become a central strategy for urban stormwater management. In its broadest sense, GI refers to “an interconnected network of green space that conserves natural ecosystem values and functions” [4], encompassing ecological, social, and hydrological services—including stormwater management as one of its core functions. GI overlaps with related concepts—such as LID, SuDS, WSUD, and BMPs—whose evolution and convergence Fletcher et al. [5] trace. Within stormwater practice specifically, GI has been moving steadily from isolated site-level retrofits toward catchment-scale portfolios [6], a shift that foregrounds a persistent planning question: where in a drainage network should the finite GI capacity be placed to most effectively reduce peak discharge at the outlet?

Studies tackling this question have shifted from site-specific simulation toward generalizable frameworks, but gaps remain. Site-specific simulation studies, generally employing design storms or limited sets of observed events, have demonstrated that GI effectiveness depends on placement location and its interaction with storm magnitude and network structure [7,8,9,10], and that distributed source-control approaches generally outperform centralized designs for small-to-moderate events [11,12]. Zhang and Chui [13], in a comprehensive review of spatial allocation optimization tools (SAOTs), catalogued over 140 such studies, confirming the field’s reliance on site-specific computational optimization of GI type, sizing, and location. Growing catchment-scale evidence further shows that GI performance depends on how storage and routing interact across scales [6,14,15]. Whereas site-specific simulations demonstrate that placement matters, they do not reduce these dependencies to a parsimonious set of governing controls. To identify such controls, Fiori and Volpi [16] and Hung et al. [17] sought to distill this scale dependence into generalizable, analytically tractable placement frameworks organized by nondimensional parameters—but both assume uniform rainfall. More broadly, the influence of within-storm temporal variability on optimal placement remains largely unexamined across both approaches.

This matters because the placement mechanism is fundamentally at the system scale and timing-driven. Outlet peak flow emerges as a superposition of routed contributions generated across the catchment, arriving with heterogeneous travel times and overlapping to varying degrees. Consequently, the leverage of a given GI location depends not on how much runoff it captures but on when that capture occurs relative to peak formation. Within-storm temporal variability enters this mechanism directly: the temporal distribution of rainfall intensity controls how quickly GI storage fills—and therefore how long GI can continue capturing runoff—while also shifting the timing of the outlet peak. Urban drainage systems are particularly exposed to this effect: their short response times [18] ensure that within-storm variability directly shapes peak formation.

Empirical studies confirm the broader importance of temporal variability for urban drainage response: Ochoa-Rodriguez et al. [19] show that temporal resolution of rainfall inputs exerts a stronger influence on urban hydrodynamic response than spatial resolution across seven European catchments, Cristiano et al. [20] identify within-storm variability as a critical but under-represented control, and recent work shows that temporal profiles of the same total depth produce markedly different local GI performance [21,22,23,24]. Whether this temporal sensitivity also affects the spatial placement of GI—potentially reordering which location is optimal—remains unexamined. Yet, current design practice often relies on a single design storm or a prescribed hyetograph pattern, e.g., [25].

This paper develops a timescale-based framework to assess the role of storm temporal structure in optimal GI placement. Extending the physical reasoning argument of Hung et al. [17] to a formal derivation from the governing equations, we show that placement is governed by two nondimensional timing ratios and the within-storm rainfall pattern. Under uniform rainfall, the two ratios alone determine optimal placement; non-uniform storm structure is an additional factor that can reverse this ordering. We first derive a nondimensional formulation from a convolution-based runoff model, then construct regime diagrams—mapping when downstream, mid-network, or upstream placement is optimal in the two-ratio space—under both uniform and non-uniform rainfall, and finally evaluate an ensemble of observed storms to quantify the decision risk of relying on a single design hyetograph.

2. Theoretical Framework and Methods

This section develops a compact timing framework for organizing the optimal placement of green infrastructure (GI). We formulate outlet peak reduction using a convolution-based representation of rainfall forcing, finite-storage interception, and network routing. This representation provides a common analytical basis for comparing candidate installation subcatchments. The problem is then nondimensionalized using a characteristic GI filling time, yielding two timing ratios that define a shared plane across storms and catchments. This framework serves as the organizing coordinate system for the analyses in Section 3, where it is used to construct a uniform-rainfall baseline regime diagram and to interpret how within-storm temporal variability and observed events reorganize optimal-location rankings and associated decision risk.

We first motivate the timing perspective (Section 2.1) and introduce a separable rainfall representation (Section 2.2). We then build the spatial framework in two stages—a continuum formulation (Section 2.3.1) followed by a three-subcatchment coarse-graining (Section 2.3.2)—and specify the routing representations that carry runoff to the outlet (Section 2.4). Section 2.5 formulates the GI-induced peak reduction and optimal-location criterion; we then nondimensionalize the problem into two timing ratios that define the coordinate system used throughout the analysis (Section 2.6). Section 2.7 describes the evaluation design—virtual catchments, the observed rainfall ensemble, and the regret metric.

2.1. Overview: Timing as the Organizing Principle

Peak flow mitigation by GI is fundamentally a timing problem. The outlet hydrograph is a superposition of contributions generated across the catchment and routed to the outlet with heterogeneous travel times. GI is a passive, finite-storage interceptor: it diverts inflow only while storage remains available, creating a finite active window within each event. Because GI cannot choose when to act, storm temporal structure determines how quickly storage fills, and the drainage network determines when intercepted runoff would have arrived at the outlet. Peak reduction is therefore controlled by the alignment among (i) the within-storm timing of intense rainfall, (ii) the travel-time structure of the network, and (iii) the duration of the GI-active window. A location is most effective when the runoff it can intercept is generated while the GI is still active and would otherwise contribute strongly to the outlet peak-forming interval.

These observations motivate several deliberate methodological choices, each designed to isolate the timing mechanism while keeping the framework analytically tractable. Rainfall is assumed spatially uniform, so that temporal structure alone governs the filling dynamics of each GI unit. Because the timing mechanism depends on how rainfall accumulates within the event, we adopt the Beta density for the within-storm temporal structure; it provides a flexible, parsimonious two-parameter representation (Section 2.2). The three-subcatchment coarse-graining is the minimal system that retains the essential placement decision—downstream, intermediate, or upstream—while keeping the nondimensional framework interpretable (Section 2.3.2). Linear, time-invariant routing—a simplification widely adopted in catchment-scale analytical models—enables the convolution-based framework and the nondimensionalization (Section 2.6). The GI is modeled as a storage-only interceptor with no within-event release, isolating the timing mechanism from underdrain complexity. The limitations of these choices are discussed in Section 4.5.

2.2. Separable Rainfall Representation

To separate storm magnitude from within-storm structure, we represent effective rainfall in a separable form:

$$r(t;t_r,{\mathbf{p}}_r)=\overline{i}f\left({\displaystyle \frac{t}{t_r}};{\mathbf{p}}_r\right),0\le t\le t_r,$$

(1)

where $\overline{i}$ is the mean event intensity, $t_r$ is storm duration, and $f(·;{\mathbf{p}}_r)$ is a dimensionless pattern function satisfying ${\int }_0^{1}f(u;{\mathbf{p}}_r)du=1$. Parameters ${\mathbf{p}}_r$ encode within-storm structure.

To span a simple yet flexible family of within-storm hyetographs, we parameterize the dimensionless pattern $f(u;{\mathbf{p}}_r)$ on $u=t/t_r\in [0,1]$ using a Beta density [26] as follows:

$$f(u;a,b)={\displaystyle \frac{u^{a-1}{(1-u)}^{b-1}}{B(a,b)}}.$$

(2)

We reparameterize $(a,b)$ by (i) the mode $m\in (0,1)$ (peak position within the event) and (ii) a concentration parameter $\kappa =a+b$ controlling sharpness [27], using $a=1+m(\kappa -2)$ and $b=1+(1-m)(\kappa -2)$ for $\kappa >2$, so larger $\kappa$ yields more concentrated storms while m shifts the peak.

2.3. Spatial Framework: From Continuum to Coarse-Grained Representation

2.3.1. General Convolution Formulation

Domain We represent the drainage system by a 1-D routing network $\mathsf{\Omega }\left(x\right)$, where x is a coordinate along $\mathsf{\Omega }$, and $a\left(x\right)$ denotes the contributing area density (contributing area per unit network length); see Figure 1A. The total contributing area is then

$$A={\int }_{\mathsf{\Omega }}a\left(x\right)dx.$$

(3)

Runoff generation prior to network routing. We work with area-normalized fluxes: $q(t,x)$ is the inflow to the routing network at location x per the area density $a\left(x\right)$, and $Q\left(t\right)$ is the outlet discharge per total area. Let $r\left(t\right)$ denote the event hyetograph interpreted as effective rainfall intensity, and let $p(t,x)$ denote the local pre-network response at x (e.g., small translation/delay before entering the routing network). The inflow to the routing network at x without GI is

$$q_0(t,x)={\int }_0^{t}r\left(\tau \right)p(t-\tau ,x)d\tau .$$

(4)

GI as finite storage and the GI active window. At location x, suppose a fraction $\eta \left(x\right)$ of the contributing area (density) is treated by GI. Let $\alpha$ be the GI loading ratio (area drained to GI relative to GI-installed area), and define the diverted fraction (while GI storage is available) as

$$\lambda \left(x\right)=\alpha \eta \left(x\right),0\le \lambda \left(x\right)\le 1.$$

(5)

GI is represented as a finite storage element with effective event-scale capacity $S_{\mathrm{GI}}$ and no within-event release to the drainage network. We further assume that the local delivery time from the GI-drained area is negligible compared to both the storm duration and network travel times, so that inflow to GI can be treated as instantaneous. Under these assumptions, the cumulative depth delivered to GI by time t is $\alpha P\left(t\right)$, where $P\left(t\right)={\int }_0^{t}r\left(\tau \right)d\tau$ is the cumulative rainfall depth. If GI fills completely during the event, the filling time is the earliest moment when diverted cumulative depth equals storage:

$$t_{\mathrm{fill}}=inf\left\{t\ge 0:\alpha P\left(t\right)\ge S_{\mathrm{GI}}\right\},$$

(6)

Because GI ceases to intercept runoff once either the storage is exhausted or rainfall ends, the effective capture period—termed the GI active window—spans the interval [0,$t_{\mathrm{cap}}$], where

$$t_{\mathrm{cap}}=min(t_r,t_{\mathrm{fill}}).$$

(7)

Then, the GI-modified inflow is

$$q(t,x)={\int }_0^t\left[1-\lambda \left(x\right){\mathbf{1}}_{[0,t_{\mathrm{cap}})}\left(\tau \right)\right]r\left(\tau \right)p(t-\tau ,x)d\tau ,$$

(8)

where ${\mathbf{1}}_{[a,b)}(·)$ is the indicator function.

Routing to the outlet. Let $g(t;t_n,x)$ denote the routing response from location x to the outlet, with characteristic network timescale $t_n$. Assuming g is causal and normalized (${\int }_0^{\infty }g(t;t_n,x)dt=1$), the area-normalized outlet discharge is

$$Q\left(t\right)={\displaystyle \frac{1}{A}}{\int }_{\mathsf{\Omega }}\left[{\int }_0^{t}q(\theta ,x)g(t-\theta ;t_n,x)d\theta \right]da\left(x\right).$$

(9)

2.3.2. Three-Subcatchment Coarse-Graining

The continuum formulation (9) is general, but its full spatial detail obscures a simple interpretation: peak formation is governed by the overlap of contributions arriving with different travel times. To expose this mechanism transparently, we coarse-grain $\mathsf{\Omega }$ into three representative subdomains ${\mathsf{\Omega }}_k$ ($k=1,2,3$), interpreted as lower, middle, and upper network zones (Figure 1A). For each zone k, let $A_k$ denote the total contributing area associated with ${\mathsf{\Omega }}_k$, where $A_k={\int }_{{\mathsf{\Omega }}_k}da\left(x\right)$, and define the corresponding area weight $w_k=A_k/A$. By construction, ${\sum }_{k=1}^3{w}_k=1$.

We then define the subcatchment-averaged inflow to the routing network from subcatchment k as the contributing area–weighted mean:

$$q_k\left(t\right)={\displaystyle \frac{1}{A_k}}{\int }_{{\mathsf{\Omega }}_k}q(t,x)da\left(x\right),$$

(10)

We then approximate routing from each subcatchment to the outlet by a representative response $g_k(t;t_n)$. The outlet hydrograph becomes

$$Q\left(t\right)\approx \sum _{k=1}^3{w}_k{\int }_0^t{q}_k\left(\theta \right)g_k(t-\theta ;t_n)d\theta .$$

(11)

To encode a simple travel-time hierarchy across the three subcatchments with a single network timescale $t_n$, we parameterize $g_k$ such that its mean travel time $t_k$ satisfies $t_k=\{t_n/2,t_n,3t_n/2\}$ for $k=\{1,2,3\}$, so that the mean travel-time separation between the upper and lower zones is $t_n$.

2.4. Network Routing Representations

Our network routing representations play two complementary roles in the analysis. We first use two idealized kernels (no dispersion and weak dispersion) to isolate the role of timing and overlap (synchronization) among subcatchment contributions without confounding effects of strong dispersion. We then adopt a single dispersive routing model as a consistent reference for the remainder of this study.

2.4.1. No Dispersion and Weak Dispersion Kernels

To isolate the role of timing and synchronization without confounding effects of dispersion, we first consider two idealized routing kernels.

Pure lag. The simplest limiting case is a pure translation with no dispersion (see the left panels in Figure 1B), as follows:

$$g_k^{\left(\mathrm{lag}\right)}\left(t\right)=\delta (t-t_k).$$

(12)

Under this kernel, each subcatchment contributes a rectangular pulse of duration $t_r$ shifted by the mean travel time $t_k$, so peak formation depends solely on whether and how these pulses overlap in time.

Weakly dispersive routing. To examine how small amounts of dispersion perturb the pure-lag picture without changing its timing structure, we also consider a weakly dispersive kernel (see the middle panels in Figure 1B), as follows:

$$g_k^{\left(\mathrm{wd}\right)}\left(t\right)={\displaystyle \frac{1}{K_s}}exp\left(-{\displaystyle \frac{t-t_k}{K_s}}\right),$$

(13)

where $K_s\ll t_k$. This form introduces mild attenuation and spreading after the lag, breaking exact symmetries of the pure-lag case.

2.4.2. Reference Dispersive Kernel

All quantitative analyses in this study employ a single, consistent dispersive routing model: a Nash cascade response function (see the right panels in Figure 1B). This model represents distributed travel times while retaining analytical tractability, and serves as the reference routing framework for regime mapping, within-storm variability experiments, and event-wise analyses (see also [16] for application in GI studies).

Motivated by the diffusive limit of Muskingum-type linear routing [28], a cascade of k identical linear reservoirs for subcatchment k yields the following Gamma-distributed response function [29,30]:

$$g_k(t;k,K)={\displaystyle \frac{t^{k-1}}{K^k\mathsf{\Gamma }\left(k\right)}}exp\left(-{\displaystyle \frac{t}{K}}\right),$$

(14)

where $K=t_n/2$.

2.5. GI-Induced Peak Reduction and Optimal Installation Location

We quantify the influence of GI in subcatchment k by comparing the routed runoff to the catchment outlet with and without interception. As in Fiori and Volpi [16] and Hung et al. [17], we neglect within-subcatchment pre-network delays relative to network routing and approximate

$$p(t,x)\approx \delta \left(t\right).$$

(15)

The GI-induced discharge reduction can be formulated using $Q_0\left(t\right)$, the no-GI outlet hydrograph, and $Q_k\left(t\right)$, the outlet hydrograph when GI is installed only in subcatchment k. The discharge reduction $R_k\left(t\right)=Q_0\left(t\right)-Q_k\left(t\right)$ is the routed image of rainfall diverted into GI during the GI active window to the catchment outlet, as follows:

$$R_k\left(t\right)\equiv Q_0\left(t\right)-Q_k\left(t\right)=w_k{\lambda }_k{\int }_0^{min(t_{\mathrm{cap}},t)}r(\theta ;t_r,{\mathbf{p}}_r)g_k(t-\theta ;t_n)d\theta ,$$

(16)

where ${\lambda }_k$ is the effective diverted fraction for the treated subcatchment.

We evaluate reductions at the no-GI peak time using

$$t_p=arg\underset{t\ge 0}{max}{Q}_0\left(t\right),$$

(17)

and define the peak reduction for installation in k as follows: $R_k^{*}=R_k\left(t_p\right)$.

To remove storm-magnitude dependence and isolate timing, we define the intensity-normalized reduction

$${\widehat{R}}_k\equiv {\displaystyle \frac{R_k^{*}}{\overline{i}}}=w_k{\lambda }_k{\int }_0^{min(t_{\mathrm{cap}},t_p)}f\left({\displaystyle \frac{\theta }{t_r}};{\mathbf{p}}_r\right)g_k(t_p-\theta ;t_n)d\theta .$$

(18)

The optimal GI location $k^{*}$ is

$$k^{*}=arg\underset{k\in \{1,2,3\}}{max}{\widehat{R}}_k.$$

(19)

2.6. Nondimensionalization and the $(T_r,T_n)$ Framework

To compare optimal GI locations across storms and catchments in a transferable way, we reformulate the peak-reduction objective in a low-dimensional timing coordinate system. This is achieved by reducing the problem to its minimal set of independent timing controls through nondimensionalization.

Equation (18) defines the intensity-normalized outlet peak reduction and suggests dependence on multiple timescales $(t_r,t_n,t_p,t_{\mathrm{cap}})$ as well as the within-storm temporal structure ${\mathbf{p}}_r$. However, for a fixed GI design $(\alpha ,S_{\mathrm{GI}})$ and a prescribed rainfall temporal structure ${\mathbf{p}}_r$, neither $t_p$ nor $t_{\mathrm{cap}}$ constitutes an independent control in the optimal-location problem. The no-GI peak time $t_p$ is an emergent property of the rainfall–routing convolution. Because the mapping from rainfall to discharge is linear in event intensity, $t_p$ is determined solely by $(t_r,t_n,{\mathbf{p}}_r)$ and is independent of the mean event intensity $\overline{i}$.

In contrast, the effective GI active window duration $t_{\mathrm{cap}}=min(t_r,t_{\mathrm{fill}})$ is set by the filling condition and therefore depends on storm magnitude and GI storage capacity. The filling time $t_{\mathrm{fill}}$ is defined implicitly by $\alpha P\left(t_{\mathrm{fill}}\right)=S_{\mathrm{GI}}$, where $P\left(t\right)={\int }_0^{t}r\left(\tau \right)d\tau$ is the cumulative rainfall depth. Under the separable rainfall representation $r\left(t\right)=\overline{i}f(t/t_r;{\mathbf{p}}_r)$, this condition can be written as

$$\alpha \overline{i}{t}_{r}F\left({\displaystyle \frac{t_{\mathrm{fill}}}{t_r}};{\mathbf{p}}_r\right)=S_{\mathrm{GI}},$$

(20)

where $F(u;{\mathbf{p}}_r)={\int }_0^{u}f(s;{\mathbf{p}}_r)ds$. This expression shows that the parameters $(\overline{i},\alpha ,S_{\mathrm{GI}})$ enter the timing problem only through the combined timescale

$$t_{\mathrm{GI}}\equiv {\displaystyle \frac{S_{\mathrm{GI}}}{\alpha \overline{i}}},$$

(21)

which represents the characteristic time required to fill the GI storage under the event mean intensity.

Using $t_{\mathrm{GI}}$ as the reference scale, the remaining independent dimensional times reduce to $(t_r,t_n,t_{\mathrm{GI}})$. Because these all have units of time, dimensional homogeneity implies that the peak-reduction problem can depend on at most two independent nondimensional ratios, which we choose as

$$T_r={\displaystyle \frac{t_r}{t_{\mathrm{GI}}}},T_n={\displaystyle \frac{t_n}{t_{\mathrm{GI}}}}.$$

(22)

The resulting two independent ratios are identical to the timescales identified by Hung et al. [17], who proposed $T_r=t_r/t_{\mathrm{GI}}$ and $T_n=t_n/t_{\mathrm{GI}}$ as key controls on GI efficacy from physical reasoning under simplified storm conditions. Our contribution is to derive these ratios from the structure of ${\widehat{R}}_k$ itself. Because $t_p$ emerges from the rainfall–routing convolution and $t_{\mathrm{cap}}$ from the rainfall–storage-filling condition, neither is an independent input. Dimensional homogeneity of the governing equations implies that ${\widehat{R}}_k$ depends on $(t_r,t_n,t_{\mathrm{GI}})$ only through $(T_r,T_n)$ and the rainfall pattern ${\mathbf{p}}_r$, which enters as a separate descriptor of within-storm temporal variability. Each storm maps to a single point in the $(T_r,T_n)$ plane through $t_{\mathrm{GI}}=S_{\mathrm{GI}}/\left(\alpha \overline{i}\right)$, which absorbs the GI design parameters $(S_{\mathrm{GI}},\alpha )$ and the mean event intensity $\overline{i}$ into a single timescale. The regime diagram is therefore invariant to the individual values of $S_{\mathrm{GI}}$, $\alpha$, and $\overline{i}$: changing any of them shifts where a storm maps, but leaves the plane unchanged.

The physical meaning of $(T_r,T_n)$ is most transparent under uniform rainfall, where storm intensity is constant. In that limit, $T_r$ measures storm duration relative to the GI filling time: $T_r<1$ implies GI remains active throughout the storm, whereas $T_r>1$ implies storage fills early and much of the storm bypasses GI. The ratio $T_n$ compares network routing time to GI filling time, indicating whether travel-time separation is fast ($T_n\ll 1$) or slow ($T_n\gg 1$) relative to the period over which GI can actively intercept runoff. Building on $(T_r,T_n)$, Hung et al. [17] delineated GI-efficacy regimes in the $(T_r,T_n)$ plane; for example, when $T_n\ll 1$, peak-reduction efficacy is only weakly sensitive to GI location (for details, see [17]).

The reduction in dimension indicates that the optimal location $k^{*}$ can be written as a function of ($T_n$, $T_r$, ${\mathbf{p}}_r$). Defining ${\tau }_p=t_p/t_{\mathrm{GI}}$, ${\tau }_{fill}=t_{\mathrm{fill}}/t_{\mathrm{GI}}$, and ${\tau }_{\mathrm{cap}}=t_{\mathrm{cap}}/t_{\mathrm{GI}}$ and rescaling the routing response by $t_{\mathrm{GI}}$, the dimensionless kernel becomes

$${\tilde{g}}_k(u;T_n)=t_{\mathrm{GI}}{g}_k(u{t}_{\mathrm{GI}};t_n).$$

(23)

With this scaling, the intensity-normalized peak reduction can be written as

$${\widehat{R}}_k(t_r,t_p,t_{cap},t_n;{\mathbf{p}}_r)=w_k{\lambda }_k{\tilde{R}}_k(T_r,T_n;{\mathbf{p}}_r),$$

(24)

where the reduced dimensionless performance metric, which is used to construct the $(T_r,T_n)$ optimal-location maps, is

$${\tilde{R}}_k(T_r,T_n;{\mathbf{p}}_r)={\int }_0^{min({\tau }_{\mathrm{cap}},{\tau }_p)}f\left({\displaystyle \frac{\tau }{T_r}};{\mathbf{p}}_r\right){\tilde{g}}_k({\tau }_p-\tau ;T_n)d\tau .$$

(25)

Accordingly, the optimal location satisfies

$$k^{*}=arg\underset{k\in \{1,2,3\}}{max}\left(w_k{\lambda }_k{\tilde{R}}_k\right),$$

(26)

and if $w_k{\lambda }_k$ is identical across candidate locations, this reduces to $k^{*}=arg{max}_{k\in \{1,2,3\}}{\tilde{R}}_k$.

2.7. Evaluation Design: Virtual Catchments, Rainfall Ensembles, and Performance Metrics

2.7.1. Virtual Catchments and GI Design

To quantify how within-storm variability translates into siting consequences across network scales, we evaluate three virtual catchments spanning fast to slow responses: $t_n=0.5$ h (fast), $t_n=2$ h (moderate), and $t_n=5$ h (slow); see Figure 1C for examples of the fast and slow catchments. These values fall within typical ranges of time of concentration reported for urban drainage systems and small-to-medium watersheds [31,32,33], and are used here as proxies for increasing catchment response time rather than as mappings to specific drainage areas.

Across all subcatchments, we assume identical GI design parameters: $S_{\mathrm{GI}}=300$ mm, $\alpha =6$, and a uniform treated area fraction corresponding to $\lambda =0.1$, so that $w_k{\lambda }_k$ is identical for all k and the optimal-location criterion reduces to the comparison of ${\tilde{R}}_k$ values alone (limitations of this equal-weight simplification are discussed in Section 4.5). These specific values determine where each observed storm maps in the $(T_r,T_n)$ plane but do not affect the regime diagram or regret values at any given point, which depend only on the nondimensional ratios (Section 2.6). These parameter values lie within ranges commonly adopted in GI/LID studies and designs [5,34,35].

2.7.2. Observed Rainfall Events

Hourly rainfall records from the Busan meteorological station (Busan, Republic of Korea) were used to construct the event set for the event-wise analysis. Busan rainfall is strongly seasonal, with most heavy events occurring during the East Asian summer monsoon and occasional typhoon passages [36]. This warm-season rainfall regime is characterized by pronounced within-event intensity fluctuations [37], providing a natural setting to evaluate how event-to-event temporal structure can shift the optimal GI location.

We extracted continuous records from 1961–2024 and identified individual rainfall events using an inter-event dry time (IETD) of 1 h; that is, a new event begins whenever at least one hour of zero rainfall separates two storms. Applying this criterion and excluding events with total depth below 5 mm or duration shorter than 3 h yielded 2351 events. For each event, the observed hyetograph was normalized by total event depth and duration to obtain the dimensionless temporal pattern $p_r$. The Beta parameters $(a,b)$ were then estimated by minimizing the sum of squared errors between $p_r$ and the Beta density, using the Nelder–Mead algorithm. The fitted parameters serve as descriptors of within-storm shape; all convolutions use the observed hourly hyetograph directly. The statistical characteristics of the events are summarized in Figure 2.

For each event j, we computed its duration $t_{r,j}$ and mean intensity $i_j$, then evaluated

$$T_{r,j}={\displaystyle \frac{t_{r,j}}{t_{\mathrm{GI},j}}},T_{n,j}={\displaystyle \frac{t_n}{t_{\mathrm{GI},j}}},$$

(27)

for each of the three catchment routing times $t_n$, where $t_{\mathrm{GI},j}=S_{\mathrm{GI}}/\left(\alpha i_j\right)$ depends on event magnitude. For every event–catchment pair, the optimal GI location $k^{*}$ was determined using (26).

2.7.3. Peak-Reduction Fraction and Regret

Performance is evaluated using the fractional reduction in outlet peak discharge at the no-GI peak time. For observed event j and GI installed in subcatchment k, the fractional reduction can be written as

$${\rho }_{k,j}={\displaystyle \frac{Q_{0,j}\left(t_{p,j}\right)-Q_{k,j}\left(t_{p,j}\right)}{Q_{0,j}\left(t_{p,j}\right)}}={\displaystyle \frac{R_{k,j}^{*}}{Q_{0,j}\left(t_{p,j}\right)}}.$$

(28)

Introducing the intensity-normalized peak reduction ${\tilde{R}}_{k,j}$ and the dimensionless no-GI peak magnitude ${\mathsf{\Phi }}_{p,j}=Q_{0,j}\left(t_{p,j}\right)/i_j$, we can write

$${\rho }_{k,j}={\displaystyle \frac{{\tilde{R}}_{k,j}}{{\mathsf{\Phi }}_{p,j}}},$$

(29)

so that the performance measure remains interpretable within the $(T_r,T_n)$ framework for a given rainfall pattern.

Beyond event-wise performance, we assess how much peak-reduction efficiency is lost when the GI location is chosen based on a uniform design storm rather than the event-specific optimal location. We summarize this loss using an event-wise regret metric. Let ${\rho }_k^U$ denote the corresponding peak-reduction fraction under a uniform rainfall pattern, and define the uniform-optimal location $k_U^{*}=arg{max}_k{\rho }_k^U$. For each observed event j, the regret associated with adopting this uniform-optimal location is

$${\mathrm{Regret}}_j=\underset{k}{max}{\rho }_{k,j}-{\rho }_{k_U^{*},j},$$

(30)

which measures the loss in achievable peak reduction due to neglecting within-storm temporal variability.

3. Results

We analyze optimal GI placement in the nondimensional plane $(T_r,T_n)$. Our analysis proceeds in three steps: (i) a constant-rainfall baseline to isolate timing effects, (ii) controlled within-storm variability using parameterized hyetographs, and (iii) an event-wise evaluation of 2351 observed storms across virtual catchments spanning fast to slow network response.

3.1. Uniform Rainfall: Baseline

3.1.1. Uniform Rain with Pure Lag and Weak Dispersion

We begin with a controlled three-subcatchment thought experiment that isolates the timing-based organizing principle of GI placement. Routing is first reduced to a pure translation (pure lag) to expose synchronization among subcatchment contributions without dispersion, and then weak dispersion is introduced as a small perturbation that attenuates and broadens responses while preserving the same travel-time ordering. This sequence clarifies which subcatchments can participate in peak formation in different parts of the $(T_r,T_n)$ plane (Figure 3(A-1)), and therefore which locations can in principle reduce the outlet peak under a fixed GI design.

The plane splits into the following three synchronization regimes bounded by $T_r=1/2T_n$ and $T_r=T_n$, corresponding to whether routed contributions from adjacent (or all) subcatchments overlap at the outlet.

(1) No synchronization: $T_r<\frac{1}{2}{T}_n$ ($t_r<\frac{1}{2}{t}_n$). Here, storm duration is shorter than the separation of arrival times between neighboring branches, so the three contributions reach the outlet largely separated in time (Figure 3(A-1), green region). In the pure-lag limit, constant rainfall produces three non-overlapping rectangular pulses. In the symmetric setup used to expose timing (equal branch weights and identical translated pulses), attenuating a single pulse by installing GI in one subcatchment simply leaves another unmodified pulse attaining the same original maximum; hence, the outlet peak magnitude cannot be reduced in this limit (${\tilde{R}}_k=0$ for all k). Introducing weak dispersion breaks this symmetry: later-arriving contributions experience more attenuation and spreading, so the earliest (lower) contribution forms the dominant isolated peak before substantial middle/upper flow arrives. Consequently, only lower-zone GI can reduce the outlet peak in this regime; upslope GI affects only later, subdominant portions of the hydrograph.

(2) Half synchronization: $\frac{1}{2}{T}_n<T_r<T_n$ ($\frac{1}{2}{t}_n<t_r<t_n$). Now, routed responses from adjacent subcatchments overlap, but the most distant pair (lower and upper) remains separated (Figure 3(A-1), blue region). In the pure-lag limit, the lower–middle overlap and, later, the middle–upper overlap form two peaks of identical height. Both are controlled by the middle contribution: reducing the overall maximum requires attenuating the middle pulse, and this is possible only if GI remains active through the storm ($t_{\mathrm{GI}}>t_r$). GI installed in the lower (upper) subcatchment modifies only the earlier (later) overlap peak, leaving the other peak unchanged and therefore not reducing the global maximum.

With weak dispersion, travel-time-dependent attenuation breaks the two-peak symmetry: the earlier lower–middle overlap becomes the global maximum, while the later middle–upper overlap is diminished. In this case, upper-zone GI cannot influence the dominant peak because its effect arrives too late, whereas lower- or middle-zone GI can reduce the peak. The storage requirement differs, however. Middle-zone GI can reduce the dominant peak even for a relatively small $t_{\mathrm{GI}}$, because attenuating only the early portion of the middle response can already depress the peak-forming lower–middle overlap. Lower-zone GI requires a longer active window so that interception persists into the peak-forming overlap interval; when this condition is met, lower placement can become competitive—and may in some cases outperform middle placement—because reductions applied to the shorter-travel-time (less attenuated) contribution translate more directly into outlet peak reduction.

(3) Full synchronization: $T_r>T_n$ ($t_r>t_n$). Here, all three branch responses overlap substantially at the outlet (Figure 3(A-1), yellow region). In the pure-lag limit, the outlet peak is generated when the delayed upper pulse arrives on top of the already elevated lower and middle contributions. All three locations can therefore affect the peak, but the storage requirement is smallest for the upper subcatchment (which contributes latest), larger for middle, and largest for lower, which must remain active long enough to suppress the lower contribution during the peak-forming three-way overlap.

Adding weak dispersion does not change the peak-forming instant—the maximum still occurs when the upper response joins the composite peak—but dispersion acts more strongly on the longer travel paths (upper most, lower least). This produces a practical tradeoff: storage-limited designs (small $t_{\mathrm{GI}}$) tend to favor locations that directly affect the late-arriving peak-forming contributions (often upper-zone GI), whereas storage-sufficient designs (large $t_{\mathrm{GI}}$) can also benefit from targeting the sharpest, least-dispersed component of the composite peak (often lower-zone GI).

3.1.2. Uniform Rainfall with the Nash Cascade

We now quantify the uniform rainfall baseline using the dispersive routing model adopted throughout the paper, namely the Nash cascade as the reference dispersive routing kernel. This serves two purposes: (i) it tests whether the synchronization-based structure identified above persists when travel times are distributed, and (ii) it yields analytical transition curves that partition the $(T_r,T_n)$ plane into downstream-, mid-network-, and upstream-optimal regions under uniform rainfall.

For Nash cascade routing, the dimensionless peak reduction admits the following closed form:

$${\tilde{R}}_k(T_r,T_n)=G_k\left({\displaystyle \frac{2T_r}{T_n}}\right)-G_k\left({\displaystyle \frac{2(T_r-1)}{T_n}}\right),$$

(31)

where $G_m(·)$ is the unit-scale Gamma CDF.

Equation (31) makes the regime logic explicit. When $T_r\le 1$ (storage-sufficient storms), the second term vanishes and ${\tilde{R}}_k=G_k(2T_r/T_n)$; the ordering is fixed and the lower subcatchment is always optimal. When $T_r>1$ (storage-limited storms), the GI-active window has a fixed unit length in nondimensional time, and equating ${\tilde{R}}_1={\tilde{R}}_2$ and ${\tilde{R}}_2={\tilde{R}}_3$ yields the two analytical transition curves

$$T_r^{(1\leftrightarrow 2)}\left(T_n\right)=1+{\displaystyle \frac{1}{exp(2/T_n)-1}},T_r^{(2\leftrightarrow 3)}\left(T_n\right)=1+{\displaystyle \frac{1}{exp(1/T_n)-1}},$$

(32)

which partition the $(T_r,T_n)$ plane into lower-, middle-, and upper-optimal regions under constant rainfall (Figure 3(B-1)).

Importantly, Section 3.1.1 identified regime-specific candidate sets from synchronization alone: the boundaries $T_r=\frac{1}{2}{T}_n$ and $T_r=T_n$ determine which branch contributions can overlap to form the peak (and thus which locations can matter). Equations (31) and (32) provide the quantitative selection rule that resolves, within each synchronization sector, which of these candidates is actually optimal once travel times are distributed.

Figure 3(B-1) overlays these analytical boundaries and marks three representative points $(T_r,T_n)=(2,6)$, $(3,4)$, and $(3,2)$ selected from the lower-, middle-, and upper-optimal regions, respectively. Figure 3(B-2) shows the corresponding hydrographs and reductions, illustrating that the optimal location is the subcatchment whose routed response places the largest effective mass within the GI-active lag window at $t_p$. A mechanistic interpretation of these transition curves in terms of a sliding lag-time sampling window is developed further in Section 4.1.

It is important to note that the optimal-location map in Figure 3(B-1) is a categorical $argmax$ representation of the reduced peak-reduction metric ${\tilde{R}}_k(T_r,T_n)$. Accordingly, regime boundaries arise at points where competing ${\tilde{R}}_k$ values cross (i.e., ${\tilde{R}}_i={\tilde{R}}_j$ for $i\ne j$). The corresponding efficacy magnitude fields—showing the best attainable reduction (combined) and the reductions for each individual location—are reported in Appendix B (Figure A1), providing a quantitative view of how the categorical regimes are formed.

3.2. Within-Storm Variability

We next relax the constant-rainfall assumption and quantify how within-storm temporal structure reorganizes the optimal-location map in the $(T_r,T_n)$ plane. Using the two-parameter Beta hyetograph family, we vary (i) rainfall mode (peak position within the event) and (ii) rainfall concentration (sharpness of the peak), and compute the optimal location based on the resulting ${\tilde{R}}_k$.

Figure 4A shows that allowing within-storm variability can substantially alter the constant-rainfall baseline over wide portions of the plane. Two robust patterns emerge. First, departures from the constant-rainfall partition grow with increasing concentration: when a large fraction of rainfall is delivered over a short portion of the event, the effective timing of runoff generation strongly modifies which travel-time bands dominate the outlet peak, producing optimal-location maps that differ markedly from the uniform-rainfall case.

Second, early-peaking storms (small mode) exhibit particularly strong deviations from the uniform-rainfall predictions. Compared with the uniform-rainfall regime boundaries, front-loaded rainfall expands regions favoring more upslope placement and shrinks regions in which downstream (lower) placement is optimal. These shifts occur even when the bulk ratios $(T_r,T_n)$ are unchanged, demonstrating that within-storm timing can reorder the relative peak-reduction potential of the three subcatchments beyond what is implied by storm duration and network response time alone.

Figure 4A also highlights an “All similar” region (gray), where the relative difference among ${\tilde{R}}_k$ values is negligible (less than 5%), implying weak sensitivity of the optimal choice to GI placement. Appendix B (Figure A2) confirms this interpretation by showing representative efficacy magnitude maps for selected $(m,\kappa )$ combinations, in which competing locations indeed yield nearly indistinguishable peak-reduction percentages. These features anticipate event-wise results: within-storm variability matters most where competing locations have comparable interceptable contributions.

3.3. Analysis in the Virtual Catchments

We now test the framework using 2351 observed hourly storms evaluated across three virtual catchments spanning fast to slow network response. For each event–catchment pair, we compute $(T_r,T_n)$, determine the event-specific optimal location $k^{*}$, and quantify the decision risk of using the constant-rainfall optimum via the regret metric (Section 2.7.3).

Figure 5 (top row) maps the observed storm population in the $(T_r,T_n)$ plane for the small, medium, and large catchments. Point color indicates the event-specific optimal location, point size encodes rainfall concentration, and point shape encodes peak position within the event; the background colors show the constant-rainfall optimal regions as a baseline. The event cloud does not collapse onto this baseline: points of different colors are interspersed within the same constant-rainfall region, indicating that within-storm structure can overturn the uniform-rainfall ranking without changing the bulk ratios $(T_r,T_n)$. Notably, even in the storage-sufficient domain ($T_r<1$), where the constant-rainfall analysis predicts downstream (lower) placement, a visible subset of events is middle-optimal (and occasionally upper-optimal), showing that temporal structure can re-rank candidate locations well away from the constant-rainfall boundaries. Disagreement is most pronounced near $T_r\approx 1$–2 and around the analytical transition curves, where small event-to-event shifts in within-storm timing modify both the effective GI active duration and the relative weighting of travel-time bands that form the outlet peak.

The frequency of these re-rankings is substantial. For the small, medium, and large catchments, approximately 48.7%, 27.4%, and 15.8% of events achieve larger peak reductions when GI is placed at the event-specific optimal location rather than at the constant-rainfall optimal location. This confirms that sensitivity to within-storm variability is not confined to rare extremes but occurs systematically across the observed storm population.

Figure 5 (bottom row) shows the corresponding regret values. Regret is generally small in the fast-response (small) catchment, but becomes more variable and can be appreciable in the medium and large catchments, with the largest values concentrated near $T_r\approx 1$–2 and near the constant-rainfall regime boundaries. These patterns quantify the decision risk of adopting a constant-rainfall placement in the presence of realistic within-storm variability and motivate the mechanistic interpretation developed in Section 4.1.

4. Discussion

Below, we first interpret optimal placement as a finite-time sampling problem (Section 4.1) and then examine its event-wise consequences across scales (Section 4.2) and robustness under storm variability (Section 4.3). Section 4.4 reinterprets prior empirical findings through the $(T_r,T_n)$ framework, and Section 4.5 discusses limitations and future directions.

4.1. A Timing View of Optimal GI Placement: Peak Reduction as a Finite-Time Sampling Problem

Section 3.1, Section 3.2 and Section 3.3 collectively show that optimal GI placement is, at its core, a timing problem. Outlet peak attenuation does not depend simply on how much runoff GI can intercept, but on when interception occurs relative to the superposition that forms the outlet peak. In the present framework, this is expressed most cleanly by interpreting peak reduction as a finite-time sampling of routed travel-time responses through a storage-limited “GI active window” of duration ${\tau }_{\mathrm{cap}}$.

In nondimensional terms, peak reduction for each candidate subcatchment k can be interpreted as an overlap between the within-storm rainfall weighting $f(\tau /T_r;{\mathbf{p}}_r)$ and the corresponding dimensionless routing response ${\tilde{g}}_k$, evaluated only over the period during which GI storage remains available. Runoff generated while GI is active contributes to the outlet peak through a lag-time interval ending at the no-GI peak time ${\tau }_p(=t_p/t_{\mathrm{GI}})$; the subcatchment whose routed response places the largest weighted mass within this interval tends to yield the largest peak reduction.

It is useful to view this mechanism from the perspective of the outlet peak itself. From this peak-centered viewpoint, the routed response is most naturally expressed in terms of reverse lag time, since peak reduction is evaluated at ${\tau }_p$. Here, reverse lag time refers simply to measuring travel time backward from the outlet peak, i.e., $\ell ={\tau }_p-\tau$. Accordingly, the contribution from subcatchment k is represented by ${\tilde{g}}_k({\tau }_p-\tau )$. In this representation, earlier rainfall corresponds to larger values of ${\tau }_p-\tau$ (longer travel times), while rainfall occurring closer to the peak corresponds to smaller values (shorter travel times). This peak-centered view underlies the schematic representations in Figure 6 and Figure 7.

This perspective immediately explains the regime structure obtained under uniform rainfall. When rainfall intensity is constant ($f\equiv 1$), the weighting is flat and the dominant control reduces to the position of the GI active window relative to the reversed routing responses ${\tilde{g}}_k$. Along rays of fixed $T_r/T_n$, linear routing produces self-similar no-GI hydrographs (Appendix A), so varying the GI storage timescale simply shifts the active window across a response shape that is otherwise unchanged (Figure 6). The dashed black line crossing three circles in Figure 3(B-1) provides a particularly clear example of this mechanism. The resulting lower–middle–upper transitions in the uniform rainfall location map (Figure 3B) therefore arise not from changes in routing dynamics, but from changes in which travel time bands are sampled during peak formation, as illustrated by the sliding-window interpretation in Figure 6.

Allowing within-storm temporal variability modifies this mechanism in two coupled ways. First, rainfall structure reshapes the length and placement of the GI active window ${\tau }_{\mathrm{cap}}$ because filling depends on the temporal accumulation of rainfall: front-loaded or highly concentrated storms could reach the storage threshold early and shorten ${\tau }_{\mathrm{cap}}$, whereas back-loaded storms delay filling and extend interception later into the event. Second, rainfall timing shifts the peak time ${\tau }_p$ and redistributes weight across reversed lag times in ${\tilde{g}}_k({\tau }_p-\tau )$. Early rainfall preferentially emphasizes longer reversed lag times (typically associated with more upslope contributions), whereas late rainfall emphasizes shorter lag times (typically associated with more downslope contributions). These two effects jointly reassign which portions of each ${\tilde{g}}_k$ are sampled during peak formation, as illustrated schematically in Figure 7.

As a result, storms that share identical bulk ratios $(T_r,T_n)$ can nonetheless favor different GI locations once within-storm temporal structure is considered. Sensitivity to within-storm variability is therefore greatest when competing subcatchments contribute comparable interceptable masses, most notably near the constant-rainfall regime boundaries and when $T_r\approx 1$, where modest shifts in ${\tau }_p$ or ${\tau }_{\mathrm{cap}}$ determine whether GI remains active during the peak-forming period and which travel-time bands ultimately dominate outlet peak formation.

4.2. Event-Wise Consequences Across Scales: How Often the Baseline Fails, and When It Matters Most

Figure 8 condenses the practical consequence of the timing mechanism described above: within-storm variability not only changes which location is optimal, but does so in systematic, interpretable ways tied to storm concentration, peak position (mode), and magnitude relative to storage.

Effect of concentration. The frequency of event-specific location changes rises with rainfall concentration in all three catchments. In the small catchment, the change rate is already high under low-to-moderate concentration (44.9–51.8%) and increases further for the most concentrated storms (64.3%). The medium catchment ranges from 22.4 to 36.7%, and the large catchment from 10.6 to 25.6%. Regret—the loss in peak reduction incurred by adopting the constant-rainfall choice—shows a clearer scale dependence: it remains small in the small catchment (mean up to 0.4%, max up to 3.0%) but grows in the medium (mean up to 2.4%, max up to 8.0%) and large catchments (mean up to 3.6%, max up to 12.0%). This behavior is consistent with the timing interpretation: increasing concentration amplifies the influence of small shifts in filling and peak timing, and the penalty for “missing” the peak-forming travel-time band increases as routing spreads and branch separation becomes more consequential.

To translate these percentages into engineering terms, consider a catchment with a no-GI peak discharge of $Q_0=20$ m³/s. If the event-specific optimal placement achieves 20% peak reduction ($Q=16.0$ m³/s) but the constant-rainfall optimal location achieves only 16% ($Q=16.8$ m³/s), the resulting four percentage points of regret translate to 0.8 m³/s at the outlet—a margin that, in a capacity-limited system, can separate a safely conveyed event from urban flooding. At the upper tail of 12% regret observed in the large catchment, a 20% baseline reduction would fall to 8%, more than halving the GI benefit for that event.

Effect of peak position (mode). Location-change frequency decreases as storms peak later within the event. In the small catchment, it drops from 61.8% (mode < 0.2) to 38.6% (mode ≥ 0.8). In the medium catchment, it drops from 41.2% to 10.2%, and in the large catchment, from 21.3% to 10.2%. Mean regrets are less strictly monotone with mode, but the upper tail is clearly heavier for early-peaking storms, especially in the medium and large catchments (maxima of 7.6–8.0% in the medium catchment and up to 12.0% in the large catchment for mode < 0.4). Thus, early internal peaks more frequently invalidate the constant-rainfall ranking and can be costly when they do.

Effect of magnitude. Because $T_r=t_r/t_{\mathrm{GI}}$ and $t_{\mathrm{GI}}=S_{\mathrm{GI}}/\left(\alpha \overline{i}\right)$, $T_r$ scales with event depth $P=\overline{i}{t}_r$ as $T_r=\alpha P/S_{\mathrm{GI}}$. With the fixed design used here ($S_{\mathrm{GI}}=300$ mm; $\alpha =6$), this implies $T_r\approx P/50$ mm, so the depth bins in Figure 8 can be interpreted directly as regimes relative to GI storage. The largest location-change frequencies occur for intermediate depths where storm magnitude is comparable to storage (i.e., $T_r$ near unity): for the medium catchment the peak occurs at 50–75 mm (68.3%), and for the large catchment, at 50–75 mm (62.8%) and 75–100 mm (58.7%). Regret is likewise highest here: mean regret reaches 3.3% (max 8.0%) in the medium catchment and 3.9% (max 12.0%) in the large catchment at 50–75 mm. In design terms, $T_r\approx 1$ marks the event depth at which GI storage is just exhausted; the sensitive range $T_r\approx 1$–2 therefore corresponds to storms that moderately exceed the design capacity—precisely where flood risk emerges and placement decisions carry the greatest consequence. Mechanistically, this is the regime where filling and peak formation occur on similar timescales, so the exact temporal placement of rainfall within the event most strongly controls whether GI is still active at the decisive moment.

A useful cross-scale contrast emerges from these patterns. Fast catchments show frequent changes in the optimal location but typically small regrets, indicating that multiple locations can deliver similar peak reductions when routing is very rapid. This is consistent with the weak sensitivity of GI placement reported by Hung et al. [17] in the limit $T_n\ll 1$. In larger, more dispersive catchments, location changes are less frequent, but the cost of an incorrect placement can be substantially larger when it occurs.

4.3. From a Single Optimal Site to Robustness Under Storm Variability

These results highlight a practical design risk: selecting GI placement using a single temporal design pattern (e.g., a prescribed Huff-type hyetograph) can bias decisions toward locations that are not reliably effective across real storms [25]. The event classes most likely to overturn the constant-rainfall (or single-pattern) optimum are (i) highly concentrated storms and (ii) storms with early internal peaks, with the strongest penalties when storm magnitudes are comparable to GI storage ($T_r\sim 1$–2) and when routing is moderate to slow.

A direct implication is that optimal placement is better framed as a robustness problem rather than a single-storm optimum [38,39]. In that spirit, two practical strategies follow naturally: (1) evaluate candidate placements against an ensemble of plausible within-storm structures spanning peak position and concentration (e.g., Huff-type families), and/or (2) use continuous simulation with long records to sample realistic event-to-event variability [14,17]. The regret metric used here makes this robustness framing operational: candidate designs can be compared not only by expected peak reduction, but also by high-percentile regret (e.g., 90th percentile or maximum regret), explicitly managing worst-case underperformance.

More broadly, the $(T_r,T_n)$ diagram suggests a workflow for robust siting: estimate a characteristic network response time, characterize the effective GI filling timescale for candidate designs, map the local storm population into timing space, and then test placements under a representative spread of within-storm structures. Where uncertainty is large—especially near regime boundaries or when storms cluster around $T_r\approx 1$—a hedging strategy such as distributing GI across multiple zones may reduce sensitivity compared with concentrating capacity at a single nominal optimum, though potentially at the cost of peak reduction under any one assumed pattern. Because the framework is expressed in nondimensional terms, the same workflow extends under a changed climate: a designer can re-map projected future storm populations into the $(T_r,T_n)$ space to assess whether the fraction of events in the sensitive regime ($T_r\approx 1$–2) increases or decreases, and adjust the hedging strategy accordingly.

4.4. Reinterpreting Prior Findings Through the Timescale Framework

The nondimensional timescales $T_r$ and $T_n$ reinterpret prior findings and extend them into testable hypotheses. For example, Giacomoni and Joseph [40] optimized placement of green roofs and permeable pavements across five subcatchments of a 0.117 km² urban catchment and found that downstream placement reduced outlet peak flow most. They attributed this to travel-time attenuation—longer routing paths dampen upstream runoff so that it contributes less to the peak—the mechanism that $T_n$ formalizes. Yet, the compact drainage network (484 m total length) implies a low $T_n$, and all tested storms fell in the storage-sufficient regime ($T_r<1$). Their study used a 2-year design storm (24.8 mm) and ten recorded events of comparable or smaller depth. The study therefore explored only the low-$(T_r,T_n)$ corner of the regime map. Our framework predicts that in catchments with longer networks—larger $T_n$—the downstream advantage widens within the storage-sufficient regime (Appendix B, Figure A1). Once $T_r$ exceeds unity, however, the optimal position depends on both timescales (Figure 5).

Chen et al. [10] simulated small synthetic stormwater networks (∼0.81 km²) using fixed 2 h uniform hyetographs at return periods of 2–100 yr and found that downstream GI loses its advantage at high return periods. In the $(T_r,T_n)$ plane, their design traces a ray from the origin (Section 4.1). Because both $T_r$ and $T_n$ share the same denominator (GI filling time), increasing storm intensity raises them simultaneously, so the ray conflates storage exhaustion with increasing relative network delay. The regime map separates these two controls: a steep ray ($t_n\gg t_r$) stays in the region where downstream placement is favorable, whereas a shallow ray—typical of compact networks whose travel time is shorter than the storm duration—crosses into regions where mid-network or upstream placement becomes optimal, as Chen et al. [10] found. Had the network travel time been longer—producing a steeper ray—downstream placement would retain its advantage at higher return periods, though with diminishing efficiency as storage is exhausted.

Both studies explored restricted regions of the $(T_r,T_n)$ plane—Giacomoni and Joseph [40] the low-$(T_r,T_n)$ corner, Chen et al. [10] a single ray—and neither varied within-storm temporal structure. Yet, at any point in the plane, hyetograph shape alone can reorder optimal placement: concentrated or early-peaking storms shift when GI storage is active and alter which travel-time band dominates the outlet peak (Figure 5). As field and numerical evidence links temporal variability to GI performance, the same coordinate system can interpret these results and generate testable placement hypotheses that account for storm timing.

4.5. Limitations and Future Directions

Several limitations should be noted. Because the low-order analytical model developed here is designed to reveal the dimensionless groups that govern behaviour and the regime transitions rather than to reproduce site-specific complexity, the simplifications below are inherent to the approach. With the governing controls now identified, more complex models can progressively reintroduce the complexities listed below to test how the identified regime structures are modified under more realistic conditions.

First, GI is modeled as a storage-only interceptor with no within-event release to the network. In contrast, many real GI systems include underdrains that allow stored water to be released back to the drainage network during events, altering both the magnitude and timing of downstream contributions. Fiori and Volpi [16] demonstrated that within-event release can even increase outlet discharge when stored water re-enters the network during peak-forming periods, shifting the relative advantage of upslope versus downslope placement.

Second, infiltration and interactions with soil moisture and groundwater are neglected. These processes would modify effective storage recovery and filling behavior, would introduce dependence on antecedent conditions and seasonality, and could either dampen or amplify sensitivity to within-storm patterns depending on inter-event recovery dynamics.

Third, routing is treated as linear and time-invariant. Real stormwater conveyance networks exhibit nonlinearities (e.g., surcharge, backwater, and capacity limits), particularly during large rain events, and routing timescales vary with flow magnitude. Because these effects intensify with discharge, the linear assumption grows less secure as $T_r$ increases. For a large $T_r$, the practical consequence is limited because GI storage is overwhelmed regardless of placement. In the moderate-$T_r$ regime, however, the assumption is most consequential: the framework predicts the sharpest placement transitions there, and flow-dependent travel times could reshape the regime boundaries.

Fourth, the catchment is discretized into three equal-weight subcatchments, neglecting within-zone travel-time dispersion, detailed branching topology, and differences in area, impervious fraction, and GI loading ratio that would affect each zone’s relative influence on the outlet peak. These simplifications could redistribute the timing of routed contributions and shift regime boundaries; the continuum formulation (Section 2.3) provides a testable pathway toward higher-resolution networks. A complementary direction is to test whether the identified regime structures persist in fully distributed, dynamically routed models (e.g., SWMM), which can simultaneously relax several of the assumptions above—nonlinear and time-variant routing, finer spatial discretization, and spatially variable contributing areas and GI loading ratios.

Fifth, the rainfall representation is limited in both temporal resolution and spatial detail. The storm set is based on hourly data, whereas some urban systems respond on sub-hourly timescales; finer temporal resolution would resolve sub-hourly intensity structure that hourly aggregation smooths away, potentially amplifying within-storm variability effects. Additionally, rainfall is assumed spatially uniform—a deliberate simplification that isolates the temporal mechanism but excludes a distinct source of variability. Spatial gradients in rainfall intensity would cause each subcatchment’s GI storage to fill at a different rate, shifting which active window closes first and, consequently, which placement controls the routed peak. Because both $T_r$ and $T_n$ are scaled by a single characteristic filling time $t_{\mathrm{GI}}=S_{\mathrm{GI}}/\left(\alpha \overline{i}\right)$, uniform across subcatchments under this assumption, spatial variability would make $t_{\mathrm{GI}}$ location-specific—and whether a compact regime structure can still organize the placement problem under distributed rainfall remains to be tested.

Sixth, the Beta density is unimodal by construction and cannot represent multi-burst storms—those with multiple intensity peaks separated by lulls. In such storms, each burst generates its own peak-forming superposition at the outlet, and the synchronization among bursts introduces timing interactions that the current two-ratio framework cannot capture. Extending to multi-burst events would require either mixture-density parameterization or an event-segmentation approach, and this is left for future work.

Despite these simplifications, the timescale interpretation and the event-wise regret analysis provide a transferable way to diagnose when within-storm temporal structure can meaningfully alter GI siting decisions—and quantify the decision risk of relying on a single uniform or single-pattern design storm.

5. Conclusions

Optimal green infrastructure (GI) placement at the catchment scale is, at its core, a timing problem. Because GI provides finite storage, interception is possible only during a limited GI-active window, while the outlet peak emerges from a peak-forming superposition of routed contributions that arrive with heterogeneous travel times. Peak reduction therefore depends not simply on how much runoff is captured, but on whether interception occurs at the times that control peak formation.

Starting from a general convolution representation of rainfall forcing, interception, and network routing, this study developed a compact timing framework that organizes the placement problem under a fixed GI design. Nondimensionalization yields two ratios, $T_r$ and $T_n$, that scale storm duration and network response time, respectively, by a characteristic GI filling time. In this plane, uniform rainfall provides a transparent baseline: as storms transition from storage-sufficient to storage-limited, peak reduction can be interpreted as a finite-time sampling of the network travel-time distribution, producing systematic shifts between downstream-, mid-network-, and upstream-favorable locations. The formal derivation of these ratios from the governing equations—rather than from physical reasoning alone—establishes them as the complete independent controls of the placement problem, providing an analytical foundation on which within-storm variability can be systematically incorporated.

The central result is that within-storm temporal variability can reorganize this baseline significantly. Rainfall structure modifies both the duration of the GI-active window through storage filling and the weighting of travel-time bands that dominate outlet peak formation. Consequently, storms that share identical bulk timing ratios $(T_r,T_n)$ can nonetheless favor different GI locations. Highly concentrated storms and storms with early internal peaks are particularly influential, as they tighten the coupling between filling dynamics and the timing of routed overlap at the outlet.

An event-wise evaluation using 2351 observed storms demonstrates that this mechanism has practical consequences. A substantial fraction of events achieved greater peak reduction at an event-specific optimal location than at the uniform-rainfall optimum, indicating that reliance on a single design hyetograph can misrank candidate sites. By making the relevant temporal alignment explicit, the proposed timing framework clarifies when uniform storm assumptions are informative and when they are likely to be misleading, providing a clear basis for timing-aware and more robust GI siting under realistic storm variability.