# forecastverification.com

### Sidebar

References
References
Verification software

Miscellaneous
LateX users

metrics:rev

# Relative Economic Value

In the absence of a flood warning system (FWS), a user's flood losses will be determined by the climatological frequency of flooding and consist of unmitigated losses, which is the sum of the losses avoided through warning response $L_{\rm a}$, and the losses that cannot be avoided $L_{\rm u}$ for every flood event,

$$\label{eqn:benchmark_ead_nowarn} V_{\rm noFWS}~=~o~\left(~L_{\rm a}~+~L_{\rm u}~\right).$$

If the FWS generates perfect forecasts, a flood event is always preceded by a warning and flood damage can always be reduced by mitigating action. False alarms and missed events do not occur. The expected damage then consists of the sum of cost for warning response and unavoidable losses for every flood event:

$$\label{eqn:benchmark_ead_perfect} V_{\rm perfect}~=~o~\left(C~+~L_{\rm u}\right).$$

The FWS performance based on imperfect forecasts can be assessed using a contingency table. Missed events result in unmitigated flood losses, which equal the sum of avoidable and unavoidable losses $L_{\rm a}$\,+\,$L_{\rm u}$. Loss mitigation through warning response can only be achieved at a cost $C$. In case of false warnings, these are the only costs incurred by a user. A user's expected costs and losses consist of those associated with hits, misses and false alarms:

\begin{eqnarray} \label{eqn:benchmark_ead_ffwrs} V_{\rm FWS}&=&h~\left(C~+~L_{\rm u}\right)~+~f~C~+~m~\left(L_{\rm a}~+~L_{\rm u}\right) \nonumber \\ &=&o~L_{\rm u}~+~\left(h~+~f\right)~C~+~m~L_{\rm a} \textrm{.} \end{eqnarray}

The Relative Economic Value ($V~\left[-\right]$) of an imperfect warning system is defined as the value relative to the benchmark cases of No Warning ($V=0$) and Perfect Forecasts ($V=1$):

$$\label{eqn:rev_scaling} V~=~\frac{V_{\rm noFWS}~-~V_{\rm FWS}}{V_{\rm noFWS}~-~V_{\rm perfect}}.$$

Note that REV can be less than~0 if the cost of false alarms is higher than the benefits attained by the warning system.

Substituting first three equations into the fourth, subsequent division by $L_{\rm a}$ and substitution of $C/L_{\rm a}$ by the cost-loss ratio $r$ yields:

\begin{eqnarray} \label{eqn:rev} V&=&~\frac{o~L_{\rm a}~-~\left(h~+~f\right)~C~-~m~L_{\rm a}}{o~L_{\rm a}~-~o~C} \nonumber \\ &=&~\frac{o~-~\left(h~+~f\right)~r~-~m}{o~-~o~r} \nonumber \\ &=&~\frac{o~-~\left(h~+~f\right)~r~-~m}{o~\left(1~-~r\right)} \textrm{,} \end{eqnarray}

which allows for expressing $V$ as a function of $r$.