Spatial weights are used when fitting statistical models
with 'SSN2' to split the tail-up covariance function upstream of
network confluences, which allows for the disproportionate
influence of one upstream edge over another (e.g., a large stream
channel converges with a smaller one) on downstream
values. Calculating the spatial weights is a four-step process:
calculating the segment proportional influence (PI) values for
the edges,
calculating the additive function values (AFVs) for
the edges,
calculating the AFVs for the
observed and prediction sites, and
calculating the spatial
weights for observed and prediction sites.
Steps 1) and 2) are undertaken in afv_edges, Step 3) is
calculated in afv_sites(), and Step 4) is calculated in the
package 'SSN2' when spatial stream-network models that include the
tail-up covariance function are fit using ssn_lm
or ssn_glm.
The segment PI for each edge, \(\omega_j\), is defined as the
relative influence of the j-th edge feature on the edge directly
downstream. \(\omega_j\) is often based on
cumulative watershed area for the downstream node of each edge,
which is used as a surrogate for flow volume. However,
simpler measures could be used, such as Shreve's stream order
(Shreve 1966) or equal weighting, as long as a value exists for
every line feature in edges (i.e., missing data are not
allowed). It is also preferable to use a column that does not
contain values equal to zero, which is explained below.
The segment PI values produced in afv_edges() are
ratios. Therefore, the sum of the PI values for edges directly
upstream of a single node always sum to one. Also note that
\(\omega_j=0\) when \(A_j=0\).
The AFVs for the j-th edge, \(AFV_j\), is equal to the product of
the segment PIs found in the path between the edge and the network
outlet (i.e., most downstream edge in the network), including edge
j itself. Therefore, \(0 \le AFV \le 1\). If \(\omega_j=0\),
the AFV values for edges upstream of the j-th edge will also be
equal to zero. This may not be problematic if the j-th edge is a
headwater segment without an observed site. However, it can have a
significant impact on the covariance structure of the tail-up model
when the j-th edge is found lower in the stream network.
A more detailed description of the segment PIs and the steps used to
calculate AFVs are provided in Peterson and Ver Hoef (2010;
Appendix A).