### 3.1. Relationships Between Shoreline Position and Wave Parameters

[9] The relationships between the shoreline position and wave parameters are not linear—not only the instantaneous parameter value but also the cumulative value influences the shoreline variation. This indicates that a correlation coefficient does not fully represent the relationship between the shoreline position and the wave parameter. Hence, the relationship was investigated by examining the performances of shoreline prediction models expressed by equations (1) and (2).

yi=y0+âˆ‘j=1i(dydt)jÎ”t, (dydt)j=(a0+a1Pj2+a2Pj+a3yj), where *y* is the shoreline position, *t* is the time, Δ*t* is the time interval, *P* is the value of the wave parameter, and *a*_{0}, *a*_{1}, *a*_{2} and *a*_{3}are coefficients. The second and third terms of the right-hand side ofequation (2) are based on *Sunamura* [1983], who assumed that the shoreline change rate was expressed by a quadratic function of wave height in which the shoreline advances under calm wave conditions but retreats under severe conditions. The fourth term represents the assumption that the shoreline change rate is negatively proportional to the shoreline position [e.g., *Katoh and Yanagishima*, 1988; *Miller and Dean*, 2004]. The seaward shoreline change rate is enhanced by a more retreated shoreline position but suppressed by a more advanced one.

[10] The wave parameters investigated were the deep water wave energy flux *E*_{f} (=*ρgH*_{s,0}^{2} *c*_{g,s,0}/16), *H*_{s,0}/*T*_{s,0}, and *H*_{s,0}^{2}/*T*_{s,0}, where *H*_{s} is the significant wave height, *ρ* is the sea water density, *g* is the gravitational acceleration, and *c*_{g,s} is the group velocity corresponding to the significant wave period *T*_{s}. The subscript 0 denotes the deep water value.

[11] The values of *H*_{s,0}/*T*_{s,0} and *H*_{s,0}^{2}/*T*_{s,0} were obtained from the Dean number *H*_{0}/(*ωT*) [*Dean*, 1973] and the Dalrymple parameter *gH*_{0}^{2}/(*ω*^{3}*T*) [*Dalrymple*, 1992], respectively, both of which were developed for predicting the occurrence of storm and normal beach profiles. *H*_{0} is the deep water wave height, *ω* is the sediment fall velocity, and *T*is the wave period. The fall velocity is a function of the sediment diameter and the water temperature. The sediment diameter is constant, and the water temperature is also assumed to be constant during a four-month period. Accordingly, the fall velocity and the gravitational acceleration were omitted from the original parameters, and*H*_{0} and *T* were replaced by *H*_{s,0} and *T*_{s,0}, respectively.

[12] The coefficients of equation (2)were determined so that the error between the measured and predicted shoreline positions was minimal using the SCE-UA algorithm (Shuffled Complex Evolution-University of Arizona) [*Duan et al.*, 1993].

[13] The model performance was investigated using the root-mean-square error, the coefficient of determination*r*^{2}, and the Brier skill score *BSS* [*Murphy and Epstein*, 1989], which is defined by equation (3).

BSS=1âˆ’âˆ‘j(yp,jâˆ’ym,j)2âˆ‘j(yrefâˆ’ym,j)2, where *m* and *p* denote the measured and predicted values, respectively. The value of *y*_{ref} is for the reference prediction, which is mostly the initial value and sometimes the mean measured value. The value of *BSS* is equal to 1 when the prediction perfectly matches the measurement and falls below 0 when the model performance is poorer than the reference model, which assumes no change from the initial or from the mean. *van Rijn et al.* [2003] qualified model predictions of beach profile change with *BSS* > 0.8, 0.6–0.8, 0.3–0.6, 0–0.3 and <0 as Excellent, Good, Reasonable, Poor and Bad.

### 3.2. Relationships Among Shoreline Position, Wave Parameters and Climate Indices

[14] To examine the causes of the interannual variation of a wave parameter, the relationships between the wave parameter and climate indices shown below during the periods from January to April, from May to August, and from September to December were investigated. The indices used in the investigations were the PDO index [*Mantua et al.*, 1997], the Southern Oscillation Index (SOI) [*Ropelewski and Jones*, 1987], the Nino-West Sea Surface Temperature (SST) anomaly, and the Arctic Oscillation (AO) index [*Thompson and Wallace*, 1998]. The Nino-West area, located 0°–15°N and 130°E–150°E, is set for El Niño monitoring by the Japan Meteorological Agency.