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# Statistical analysis for flood mapping estimation

Francesca Raffaele, Rita Nogherotto, Adriano Fantini

ICTP, Trieste, Italy

afantini@ictp.it

## flood hazard maps

• Different Return Periods (probabilities): 100, 200, 500 years...
• Italian territory
• Future climate projections
• Using data from hydrological model

## Methodology

Precipitation:

• Observations
• RCM output

Gridded netCDF:

• River network
• Discharges

hydrological model

For each RP, cell:

• Gumbel distr.
• Hydrographs
• Extreme Q

Statistical analysis

For each RP, cell:

• Flood extent
• Flood depth

(multiple simulations)

• RCM output
• Discharges
• Floods

Validation and change for

CA2D hydraulic model

Based on Maione et al., 2003

(over nine domains)

HOW?

HOW?

## Return Period

The RP is a common measure of probability used for extreme events: it represents the probability of the event happening any given year. For example:

• If an event has RP=100 yr, its probability to happen any given year is 1%
• RP=200 yr => p=0.5%
• Only a statistical measure!

## Synthetic Design Hydrographs

The SDH is the curve giving the "typical" flood event discharge (Q) as a function of time (t), for any given Return Period (RP):

SDH = Q_{RP}(t)

There are two components to the SDH (at a given RP):

• peak discharge
• shape

Time (h)

## Flow Duration Frequency curve

The FDF is the curve maximising the flood event discharge (Q) averaged over a duration (D) around the peak, so that for a given event:

Notice that, by definition, for an idealised event with Return Period RP, the peak flood discharge is:​

Q_{D=0}(RP) = Q_{RP}(t=0)
\text{FDF} = Q_D = \frac{1}{D} max \int_{t}^{t+D} Q \left(\tau \right) d\tau

## Example time!

FLOOD PEAK

DURATION

FDF(60)

From Maione, 2003

Volume = Q_D \times D
r_D

The shape of the SDH is dictated by the peak-duration ratio        , which is the ratio of the time before the peak and the total duration (D) of the averaging window.

The smaller the       , the more skewed the hydrograph will be towards steeper (flatter) rising (falling) limbs of the hydrograph.

Also notice that:

r_D

## Two assumptions

Following Maione et al. (2003), we assume that the reduction ratio (    ), which is the ratio of the FDF and the peak flood discharge                 is constant for any Return Period (RP), so that:

Q_0\left( RP \right)
\varepsilon_D
\varepsilon_D = \varepsilon_D\left( RP \right) = \frac{Q_D\left( RP \right)}{Q_0\left( RP \right)}

Which is a reasonable assumption also according to NERC (1975): the shape of the hydrograph, given by this ratio, does NOT depend on the RP!

Moreover, following Alfieri et al. (2014), we assume that the hydrograph is symmetric, that is to say that:

r_D = \frac{1}{2}

## Step 1: split the hydrograph

Set the flood peak as t=0, and split the left and right limb of the SDH as follows:

\int^{t=0}_{-r_DD} Q\left(\tau \right) =
\int^{\left(1-r_D\right)D}_{t=0} Q\left(\tau \right) =
\left(1-r_D\right) D Q_D\left(RP \right)
r_D D Q_D\left(RP \right)

t=0

FDF

## Step 2: differentiate for D

\text{SDH} = Q_{RP} \left( t \right) = \frac{d/dD \left[ \left( 1-r_D\right) D Q_D\left(RP\right)\right]|_{D=D\left(t\right)}}{d/dD \left[ \left( 1-r_D\right) D \right]|_{D=D\left(t\right)}}

Only for the falling limb, differentiate the previous equation with respect to D:

Where:

t=\left( 1-r_D\right)D

Once we know the reduction ratio and the FDF, we can then calculate the SDH!

But remember that we set the reduction ratio to one half, so that:

t = \tfrac{1}{2} D

## Step 3: obtain the reduction ratio

\varepsilon_D = \varepsilon_D\left( RP \right) = \frac{Q_D\left( RP \right)}{Q_0\left( RP \right)}

Remember:

We assume (Maione, 2003):

\varepsilon_D \simeq \sqrt{p_2 \left[ 2 + p_1 - \frac{3}{2}p_2 \left(1 - p_1 \right) \right]}

Where:

p_1 = e^{-4D/\theta} \quad ; \quad p_2 = \frac{\theta}{2D}

Where θ only depends on the (known!) drained area; a function for it is conveniently obtained by Maione based on observed data. We are only missing                   !

Q_0\left( RP \right)

## Step 4: obtain the peak discharge

Q_{D=0}(RP) = Q_{RP}(t=0)

Possible approach: fitting an extreme value distribution, such as a Gumbel distribution (Maione, 2003; Alfieri, 2015) to the distribution of yearly maxima:

we use the available years of data (up to 30) to estimate the peak discharge for any RP, thus extrapolating data for higher Return Periods!

Now:

• Do this procedure for each 5-10 km along rivers
• Run 5528 hydraulic simulations
• Aggregate data...

By odineidolon

# Statistical analysis for flood mapping estimation

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