Purpose and goals

  • Place the DDDEvent method in context of existing approaches
  • Address the current use of the method to compute return periods
  • Initiate a discussion on next steps before the project meeting and suggest ways forward

Framing the problem

Return periods for floods express the likelihood of an event occurring within a specific time frame. For example, the ‘100-year flood’ has a 1% chance of occurring in any given year, which is also known as the annual exceedance probability (AEP).

Generally, we have three categories of approaches available to us when we want to estimate the relationship between flood magnitude (return level) and flood frequency (return period or AEP): (1) statistical flood frequency analysis, (2) event-based simulation with rainfall-runoff models, and (3) continuous and hybrid rainfall-runoff simulation methods.

1. Statistical flood frequency analysis

Using observed streamflow data, we estimate design values using statistical models for extreme values. For example, the ECCO project flood frequency analysis for Sandsli.

2. Event-based simulation (rainfall-runoff modeling)

Using precipitation data as input, we simulate the design event(s) with a hydrologic model. These models focus on simulating flood events rather than continuous streamflow and require explicit specification of antecedent conditions.

While well-suited for simulating flood hydrographs, assigning exceedance probabilities to the results requires great care, as individual design event simulations are not inherently linked to event frequency over time.

We present three subcategories of event-based simulation as defined in the Australian Rainfall and Runoff guidebook, focusing on how they assign exceedance probabilities to simulated events.

Figure 1: Figure inspired by Ball et al (2019)
Figure 1: Figure inspired by Ball et al (2019)

(i) Simple design event

The simplest event-based approach involves selecting ‘typical’ values for key inputs like rainfall depth, temporal patterns, and antecedent conditions. These inputs are used in a hydrologic model to simulate a single design hydrograph. The return period of the simulated flood is assumed to match the return period of the input precipitation.

The simple design event approach is highly practical and widely used, but a key challenge is that if the fixed input values are not properly chosen, the simulated flood can be misestimated.

(ii) Ensemble event

The ensemble event approach evolved from the simple design event method. Instead of fixing all key inputs, we allow one influential input to vary, generating a set (an ‘ensemble’) of hydrographs. The design flood magnitude is estimated from the weighted average of these hydrographs, with weights reflecting the relative likelihood of the varying input value. The idea is that allowing one input value to vary along a range of reasonable values minimizes bias and avoids misestimation caused by selecting a single value for each input.

The ensemble event method still assumes the return period of the simulated floods matches the return period of the input precipitation.

It is hopefully more robust than the simple event method because we’re catering to the random variability of a key input, but this basic introduction of variability is very different from a full Monte Carlo approach, which requires joint modeling of precipitation and relevant antecedent conditions.

(iii) Monte Carlo event

A key difference between the Monte Carlo event approach and the simple and ensemble event approaches is how the input precipitation is treated.

In Monte Carlo event approaches, the distribution of precipitation across a range of exceedance probabilities, durations and temporal patterns is jointly modeled with distributions of key antecedent conditions.

Theoretically, if we know the probability of observing particular co-occurring precipitation characteristics and their occurrence with particular antecedent conditions over a specific time frame, we can propagate that probability through to individual design event simulations. In practice, it is challenging to treat precipitation this way.

There are several approaches.

Some practitioners assume independence between precipitation characteristics, like rainfall intensity and duration. Others use complex dependency modeling structures, such as the bivariate Archimedean copulas of Zegpi and Fernandez (2010) or the trivariate copulas of Balistrocchi and Bacchi (2011). Alternatively, rainfall inputs can be transformed to make duration and intensity independent (Svensson et al., 2013) or rainfall events of different duration can be randomly sampled and then ‘enveloped’ to find a relevant range of durations (Nathan et. al., 2013).

These approaches differ significantly from current design rainfall practices, which assume a critical rainfall duration for each catchment that produces the design flood.

Another challenge is that key antecedent conditions (e.g., soil moisture, snow water equivalent) often can’t be sampled directly from climatological data, requiring some sort of continuous simulation to generate their distributions.

In a practical setting, it is therefore more common to set up a continuous or semi-continuous simulation approach, despite the computational burdens of these methods.

3. Continuous and hybrid simulation methods (rainfall-runoff modeling)

We differentiate between continuous simulation and hybrid or semi-continuous simulation:

  • Continuous simulation constructs the entire time series using a modeling setup that realistically produces low, moderate, and extreme streamflow, as well as dry periods. Frequency estimates can be directly taken from the simulated time series by identifying specific time frames.

  • Hybrid or semi-continuous simulation focuses only on representing the larger rainfall events and their resulting flood events, and implicitly considers the influence of dry periods when sampling catchment-rainfall interactions (antecedent conditions, temporal patterns, storm durations) from externally derived initial condition distributions.

The latter approach is more efficient, but not inherently linked to event frequency over time. Various attempts to establish this link exist. For example:

SCHADEX (Paquet et al., 2013) chooses a single critical rainfall duration, randomly samples the depth from an appropriately defined uniform distribution, and then calculates a relative probability for the synthetic rainfall event (this is the link to event frequency over time). The relative probability is given in part by the likelihood of observing different weather patterns (MEWP, Multi-exponential weather patterns, Garavaglia et. al., 2010). The synthetic rainfall event is then inserted into the historical record and used to generate the possible hydrological states of the catchment.

The intent of hybrid simulation approaches is similar to the Monte Carlo approach, but in this markdown we differentiate between approaches that define a full probabilistic model for precipitation vs approaches that make simplifying assumptions (for example, assuming a critical rainfall duration) and rely on sufficiently lengthy simulation period to capture variation in catchment dynamics (for example SCHADEX).

There are also a number of approaches that take elements of the above methods but don’t fit neatly into a single category.

For example, stochastic PQRUT (Filipova et. al. 2019) uses a short period of continuous simulation to generate distributions of relevant antecedent conditions and their dependencies. Then, initial conditions for a hydrologic model (PQRUT) are sampled from the joint distribution of antecedent conditions. These initial conditions, along with a y% AEP rainfall depth, are used in the PQRUT model to generate a design flood.

The method then assumes the frequency of simulated floods matches the frequency of peak over threshold events in the rainfall record. Since we have an assumed frequency per year, the simulated events can be plotted against the flood frequency curve using plotting positions.

Stochastic PQRUT is like the ensemble event approach in that it considers variability in key antecedent conditions, while simultaneously adopting a simple approach to precipitation (each design flood generated from single depth and duration associated with the y% AEP).

However, instead of summarizing hydrographs across an ensemble, stochastic PQRUT treats each simulated flood event independently. The method then makes an assumption about the frequency over time of the simulated flood events (that the number of simulated flood events matches the number of POT events in the rainfall record). This assumption is different than the simplistic assumption of precip AEP = flood AEP that we would make in an ensemble event method, but stops short of assigning relative probabilities to specific rainfall characteristics, such as in SCHADEX.

The current approach with DDDEvent

The current method relies on a short period of continuous simulation from the DDD hydrologic model to select relevant input values for an event-based model (DDDEvent).

The precipitation input to the event-based model is the depth [mm] associated with the y% AEP at a single duration.

For each precipitation depth, two key model inputs are allowed to vary when we run DDDEvent: soil moisture and the temporal pattern of precipitation. Soil moisture is drawn from a gamma distribution. The temporal pattern of the precipitation event is drawn from a beta distribution. Dependencies between the two are not modeled.

The parameters of the gamma distribution \(Gamma(\alpha,\theta)\) are chosen such that the mean (that is, the product \(\alpha\theta\)) is equal to the subsurface capacity for the catchment. Subsurface capacity is an internal physical state computed from the continuous simulation run of the DDD hydrologic model. The practitioner is free to choose any values of \(\alpha\) and \(\theta\) so long as their product is equal to the computed subsurface capacity value.

All other inputs are fixed. The variation in soil moisture and precipitation pattern generates an ensemble of hydrographs at each precipitation depth.

This process is then repeated at each precipitation depth from 1 to 100 mm. The figure below shows the maximum value for every simulated event for precipitation depths from 1-100:

Figure 2: Example DDDEvent output for Sandsli, run 2500 times for each precipitation value. Second panel is a cut out of the first. The red points are values from the DDDv2 model; the green points are values from DDDEvent.
Figure 2: Example DDDEvent output for Sandsli, run 2500 times for each precipitation value. Second panel is a cut out of the first. The red points are values from the DDDv2 model; the green points are values from DDDEvent.


DDDEvent is like stochastic PQRUT in that it accounts for variation in key inputs (soil moisture and temporal pattern of precipitation for DDDEvent, initial discharge, soil moisture and swe for stochastic PQRUT). However, unlike stochastic PQRUT, DDDEvent does not jointly model the key inputs (that is, the method does not account for co-variation between soil moisture and temporal pattern of precipitation).

Additionally, the modeling of soil moisture variation in DDDEvent differs from stochastic PQRUT. In stochastic PQRUT, the DDD hydrologic model generates soil moisture values, and a distribution is directly fitted to these values for random sampling. In DDDEvent, the DDD hydrologic model also generates soil moisture values, but the sampling distribution is not directly fitted to the generated values. Instead, the practitioner generates a separate sampling distribution by choosing, rather than statistically estimating, the parameters of a gamma distribution.

Contextualizing DDDEvent within existing methods

DDDEvent resembles an ensemble event approach in that it accounts for the random variability of key model inputs. It generates an ensemble of hydrographs for each fixed precipitation depth associated with the y% AEP.

Key differences between DDDEvent, stochastic PQRUT, and a true ensemble event approach lie in how this ensemble of hydrographs is treated.

In an ensemble event approach, the ensemble of hydrographs at a single design precipitation value would be summarized with a weighted average, where the weights would correspond to the relative likelihood of the varying input value. As DDDEvent allows both soil moisture and the temporal pattern of precipitation to vary, this weight would have to correspond to the joint probability of the two inputs.

In stochastic PQRUT, each hydrograph in the ensemble would be treated separately. The hydrographs would then be assigned a relative frequency, where the frequency of simulated floods matches the frequency of peak over threshold events in the rainfall record. That is, if we observed on average four POT rainfall events per year, we would assume a group of four simulated floods all came from the same year.

In DDDEvent as it is currently used, we treat the ensemble of hydrographs and their generating precipitation value as a two dimensional density field from which it is possible to infer the joint probability of precipitation depth and flood peak. That is, we use the method as though it were a Monte Carlo method.

archived: numerical example

Next steps?

If we treat DDDEvent as an ensemble approach and assume the return period of the simulated floods matches the return period of the input precipitation…

Ignoring, for the time being, the weighting of hydrographs by the relative likelihood of input values, we can average the \(Q_{max}\) values for each y% AEP precipitation depth (15-minute duration) and compare them to return levels estimated through statistical flood frequency analysis (FFA) on observed data (example from Sandsli).

It is promising that the DDDEvent averages are in the same range as the FFA-estimated return levels. For the quantiles outside the range of observed data (0.05-0.005 quantiles, 20-200 year return periods), the DDDEvent estimated levels are well within the 90% credible interval from the FFA. However, at the lower 2-year return period, within the range of observed data, the DDDEvent estimated level falls outside the credible interval. This could be due to using a simple, rather than weighted, average in this example.

Useful items to consider prior to the ECCO meeting in Tartu in April:

  • Better place the DDDEvent approach in context with, for example, stochastic PQRUT and SCHADEX.

  • Decide whether to model dependencies in the antecedent conditions, and if so, how. If not, explain why.

  • Determine how to best summarize the ensemble of events at each precipitation depth (simple average? weighted average? probabilistic description? (histogram? kernel density estimate?))

Other ideas?

Looking forward to the project meeting and further discussions!

References

Ball J, Babister M, Nathan R, Weeks W, Weinmann E, Retallick M, Testoni I, (Editors) Australian Rainfall and Runoff: A Guide to Flood Estimation, Commonwealth of Australia (Geoscience Australia), Version 4.2, 2019.

Balistrocchi, M., and B. Bacchi. “Modelling the statistical dependence of rainfall event variables through copula functions.” Hydrology and Earth System Sciences 15.6 (2011): 1959-1977.

Filipova, Valeriya, Deborah Lawrence, and Thomas Skaugen. “A stochastic event-based approach for flood estimation in catchments with mixed rainfall and snowmelt flood regimes.” Natural Hazards and Earth System Sciences 19.1 (2019): 1-18.

Garavaglia, F., et al. “Introducing a rainfall compound distribution model based on weather patterns sub-sampling.” Hydrology and Earth System Sciences 14.6 (2010): 951-964.

Nathan, Rory, and Erwin Weinmann. “Australian rainfall and runoff discussion paper: monte carlo simulation techniques.” Australia: Engineering House (2013).

Paquet, Emmanuel, et al. “The SCHADEX method: A semi-continuous rainfall–runoff simulation for extreme flood estimation.” Journal of Hydrology 495 (2013): 23-37.

Svensson, Cecilia, Thomas R. Kjeldsen, and David A. Jones. “Flood frequency estimation using a joint probability approach within a Monte Carlo framework.” Hydrological sciences journal 58.1 (2013): 8-27.

Zegpi, Macarena, and Bonifacio Fernandez. “Hydrological model for urban catchments–analytical development using copulas and numerical solution.” Hydrological Sciences Journal–Journal des Sciences Hydrologiques 55.7 (2010): 1123-1136.