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Temporal Data Sampling Methods
Temporal data sampling is intended to account for the dynamic nature of storms. Since storms are moving spatially, direct interpolation between values at the same time step may be overly simplistic. In an idealized example shown below, assume a storm that is one mile in diameter moving at one mile per hour, and the center of that storm passes over two gages that are three miles apart. In this case, the center of the storm will pass over the first gage two hours before it passes over the second.
If a target point is exactly in the center of the two gages above, direct interpolation will lead to a value of 0.5 for the first hour and a value of 0.5 for the third hour, as shown below:
Given that the storm is moving, it's highly unlikely that the interpolated pattern is correct. It is much more likely that the all of the precipitation would fall during the second hour at the target point. Given this trivial example, it is clear that interpolating data directly for each time step has potential to misrepresent the expected pattern of a storm. MetInterp provides methods for handling these storm dynamics, and are referred to as Temporal Data Sampling. Three methods are provided: None, McCormick and Modified McCormick. None is the default, and means there is no temporal sampling (direct interpolation for each time step). The McCormick method was proposed by Brian McCormick, HEC-MetVue's original developer, and the Modified McCormick method has an additional adjustment proposed by the HEC-RAS developers to constrain the shape if the storm in addition to the spatial movement of the storm.
McCormick Method
The McCormick method starts by computing the temporal Center of Mass (CoM) of the data from each gage. This is computed as follows:
T_{gage}^{CoM}=T_1+\frac{\sum_{t=1}^{N_T}T_t*z_t}{{\sum_{T=1}^{N_T}z_t}} |
In the above equation, T_{gage}^{CoM} is the time center of mass of the given gage, T_t is the time value at time t (seconds), z_t is the observed value at time t. Once all the gages have a temporal center of mass, then each grid cell is assigned a temporal center of mass by interpolating with the selected spatial interpolation method (i.e Inverse Distance Weighting). Prior to interpolating the z value for a given cell, the McCormick sampler selects the z value from each gage following the equation below:
T_{gage}^{cell_t}=T_t+(T_{gage}^{CoM}-T_{cell}^{CoM}) |
In the above equation, T_{gage}^{cell_t} is the time to sample data from the given gage when interpolating the given cell and T_t is the time at index t. The associated z_{gage}^{cell_t} is then sampled from the given gage at time T_{gage}^{cell_t}, and used for spatial interpolation.
Modified McCormick Method
Along with moving in space, storms shape changes over time. From one gage to another, a storm can appear shorter or longer. This means the storm length in between two gages should be interpolated as well. In order to account for this, the left and right standard deviation of the storm in each gage is computed as follows:
SD_{gage}^{L}=\sqrt{\frac{\sum_{t=1}^{T_{gage}^{CoM}}(T_t-T_{gage}^{CoM})^2*z_{t,gage}}{{\sum_{t=1}^{T_{gage}^{CoM}}z_{t,gage}}}} |
SD_{gage}^{R}=\sqrt{\frac{\sum_{T_{gage}^{CoM}}^{N_T}(T_t-T_{gage}^{CoM})^2*z_{t,gage}}{{\sum_{T_{gage}^{CoM}}^{N_T}z_{t,gage}}}} |
In the above equations, SD_{gage}^{L} and SD_{gage}^{R} are the left and right standard deviations for the given gage. These standard deviations are interpolated to each grid cell, using the selected spatial interpolation method.
\begin{matrix} T_{gage}^{cell_t}=(T_t-T_{cell}^{CoM})*\frac{SD_{gage}^{L}}{SD_{cell}^{L}}+T_{cell}^{CoM} & when & T_t\lt T_{cell}^{CoM} \\ T_{gage}^{cell_t}=(T_t-T_{cell}^{CoM})*\frac{SD_{gage}^{R}}{SD_{cell}^{R}}+T_{cell}^{CoM} & when & T_t\gt T_{cell}^{CoM} \end{matrix} |
The above equation leads to temporal sampling the both aligns the center of mass of storms observed between different gages, and scales the leading and trailing edges of the storm to be aligned properly.