# Performance Assessment of NAMI DANCE in Tsunami Evolution and Currents Using a Benchmark Problem
## 3 Results
### Performance Assessment of the Model  
The credibility of a numerical model is directly related to its performance. Using numerical models, it is impossible to obtain the same outcomes as those in natural systems since natural systems are never closed and model results are not always unique. The expected outcome is an acceptable level of agreement between analytical/experimental/field data and model prediction, as well as sufficient accuracy of the model. The performance of numerical models can be assessed by demonstrating the agreement between observation and prediction. Models can only be evaluated in relative terms, and their predictive value is always debatable. There is a variety of different methods to assess the performance of numerical models. Error statistics is one of these tools.  
Different types of errors are introduced to determine the correlation between the analytical/experimental/field data and model results. In National Oceanic and Atmospheric Administration (NOAA), two common errors are emphasized; namely, normalized root mean square error, NRMSE, and the error of the maximum value, MAX [20]. The percent NRMSE is applied within  
Figure 4: Comparison of results: (**a**) horizontal velocity in x-direction, U, recorded at \(X=13.0\) m, \(Y=0.0\) m and \(Z=0.75\) m at Gage 2; (**b**) horizontal velocity in y-direction, V, recorded at \(X=13.0\) m, \(Y=0.0\) m and \(Z=0.75\) m at Gage 2; (**c**) horizontal velocity in x-direction, U, recorded at \(X=21.0\) m, \(Y=-5.0\) m and \(Z=0.77\) m at Gage 9; (**d**) horizontal velocity in y-direction, V, recorded at \(X=21.0\) m, \(Y=-5.0\) m and \(Z=0.77\) m at Gage 9. Black line represents benchmark data, red line represents numerical results.  
a space segment or time period to all observed data points. The main use of NRMSE is to assess the accuracy of a model in predicting the entire set of observed data; in other words, it is used to define the overall model performance. NRMSE deals with an error of absolute values and does not show the bias of the code's prediction (i.e., underprediction or overprediction). MAX is used to quantify each model's predictive accuracy for the maximum data point regardless of the location and time.  
The formulas to evaluate percent NRMSE and MAX error values are given in Equations (6) and (7):  
\[\text{NRMSE}=100\ \times\ \frac{1}{f\left(x_{i}\right)_{max}-f\left(x_{i} \right)_{min}}\sqrt{\frac{\Sigma\left(f\left(x_{i}\right)-y_{i}\right)^{2}}{n}} \tag{6}\]  
\[\text{MAX}=100\times\frac{\left|f\left(x_{i}\right)_{max}-y_{i}\ max\right|}{ f\left(x_{i}\right)_{max}} \tag{7}\]  
where \(f(x_{i})\) represents the observed data and \(y_{i}\) represents the predicted data.  
The benchmark problem discussed in this study features a series of experiments collecting free surface elevation and velocity data. The velocity computations in numerical models differ at shallow water depths because of the nature and limit of two-dimensional depth-averaged shallow water equations. Therefore, larger error limits are accepted for velocity comparisons in shallow water depths. The NRMSE and MAX error limits for velocity values for experimental benchmarks can be selected as 15% and 25%, respectively [20]. For free surface elevation time series, NRMSE and MAX error limits may be reduced to 15% and 10%, respectively [20]. Statistical error analysis is applied to the results given in Sections 3.1 and 3.2. The results are listed separately for the free surface elevation time series, and velocity time series and are given in Tables 1 and 2.