## 2 Materials and Methods
### Data Processing and Statistical Analysis  
The assessment involved 11 trials per participant: 3 for the short-high ball, 3 for the mid ball, and 5 for the more challenging long-low ball condition, given its comparatively lower success rate. After initial video assessment, only 26 trials out of 44 were deemed to have substantial impact and were retained for further 3D quantification. Of the 26 selected trials, 8 were from the short-high ball (two from each subject), 8 from the mid ball (two from each subject), and 10 from the long-low ball (two subjects contributed 2 trials each, and the remaining 2 contributed 3 trials each). The acquired raw data underwent processing using a five-point smoothing filter. This processed dataset furnished critical three-dimensional coordinates of the 42 markers. Subsequently, the temporal change in the ball center's speed was calculated frame by frame by removing by employing the 3D coordinates of the three reflective tapes on the ball surface. Simultaneously, the temporal change in the head's speed was also determined frame -by frame based on the 3D coordinates of the four head markers. Following the quantification of speed changes over time, specific parameters were identified and determined, including maximal ball speed (Max V\({}_{\text{ball}}\)), maximal head speed (Max V\({}_{\text{head}}\)), head speed at impact (V\({}_{\text{head-impact}}\)), coordinates of ball center at impact (x\({}_{\text{b}}\), y\({}_{\text{b}}\), z\({}_{\text{b}}\)), coordinates of jumping foot's big toe at take-off (x\({}_{\text{t}}\), y\({}_{\text{t}}\), z\({}_{\text{t}}\)) and airborne/flying time from take-off to impact (t\({}_{\text{flying}}\)). Subsequently, the quantification of BSE and EOR was executed using the following equation:  
\[\text{BSE}=\text{Max V}_{\text{ball}}-0\text{ (i.e., ball speed before impact)}=\text{Max V}_{\text{ball}} \tag{1}\]  
Figure 4: An exemplary video frame **(left)** and 3D capture data **(right)** of a diving header. The ball position is indicated by the three yellow markers, and the trajectory of the head is illustrated by the four blue lines tracking the four head markers. \[\text{EOR}=\sur\left[(x_{\text{b}}-x_{\text{t}})^{2}+(y_{\text{b}}-y_{\text{t}})^{2}\right] \tag{2}\]  
All calculations were conducted using Microsoft Excel 2013. Owing to the limited sample size (four subjects), this pilot investigation fundamentally operates as a case study, characterized by a collection of observations rather than a comprehensive statistical study. One common practice in case studies involves the utilization of individual data to reveal potential trends and influences, employing descriptive statistics--specifically means (\(\upmu\)) and standard deviations (SD)--to portray the attributes of the designated parameters [24,25]. Consequently, descriptive statistics (\(\upmu\pm\text{SD}\)) were employed to numerically illustrate the ball speed enhancement (BSE) achievable through the application of diving headers and the distance limit (i.e., EOR) within which the diving header could be effectively employed. Previous studies [26,27,28,29,23] have underscored the significance of ball release speed (i.e., BSE in this study) as a crucial parameter for shot quality. Within the EOR, potential determinants influencing BSE encompass flying distance, ball height, and head speed at impact. To scrutinize these potential influences, Bravais-Pearson (parametric) correlations were conducted between BSE and the identified influencing factors. This analytical approach establishes the connection between two interval-scaled variables, yielding a correlation coefficient (r-value) with a corresponding significance level. The commonly accepted r-value interpretation in sports biomechanics and human kinetics [30] was applied, setting the threshold at a significance level of \(p=0.05\). This correlation analysis offers practitioners an in-depth comprehension of diving header quality and provides recommendations for strategic decisions in its effective application.