You might consider using a kernel especially suitable for the sphere, such as a von Mises-Fisher density
$$f(\mathbf{x};\kappa,\mu) \propto \exp(\kappa \mu^\prime \mathbf{x})$$
where $\mu$ and $\mathbf{x}$ are locations on the unit sphere expressed in 3D Cartesian coordinates.
The analog of the bandwidth is the parameter $\kappa$. The contribution to a location $x$ from an input point at location $\mu$ on the sphere, having weight $\omega(\mu)$, therefore is
$$\omega(\mu) f(\mathbf{x};\kappa,\mu).$$
For each $\mathbf{x}$, sum these contributions over all input points $\mu_i$.
To illustrate, here is R
code to compute the von Mises-Fisher density, generate some random locations $\mu_i$ and weights $\omega(\mu_i)$ (12 of them in the code), and display a map of the resulting kernel density for a specified value of $\kappa$ (equal to $6$ in the code).
![[Figure]](https://i.stack.imgur.com/4Z2fg.png)
The points $\mu_i$ are shown as black dots sized to have areas proportional to their weights $\omega(\mu_i)$. The contribution of the large dot near $(100,60)$ is evident throughout the northern latitudes. The bright yellow-white patch around it would be approximately circular when shown in a suitable projection, such as an Orthographic (earth from space).
#
# von Mises-Fisher density.
# mu is the location and x the point of evaluation, *each in lon-lat* coordinates.
# Optionally, x is a two-column array.
#
dvonMises <- function(x, mu, kappa, inDegrees=TRUE) {
lambda <- ifelse(inDegrees, pi/180, 1)
SphereToCartesian <- function(x) {
x <- matrix(x, ncol=2)
t(apply(x, 1, function(y) c(cos(y[2])*c(cos(y[1]), sin(y[1])), sin(y[2]))))
}
x <- SphereToCartesian(x * lambda)
mu <- matrix(SphereToCartesian(mu * lambda), ncol=1)
c.kappa <- kappa / (2*pi*(exp(kappa) - exp(-kappa)))
c.kappa * exp(kappa * x %*% mu)
}
#
# Define a grid on which to compute the kernel density estimate.
#
x.coord <- seq(-180, 180, by=2)
y.coord <- seq(-90, 90, by=1)
x <- as.matrix(expand.grid(lon=x.coord, lat=y.coord))
#
# Give the locations.
#
n <- 12
set.seed(17)
mu <- cbind(runif(n, -180, 180), asin(runif(n, -1, 1))*180/pi)
#
# Weight them.
#
weights <- rexp(n)
#
# Compute the kernel density.
#
kappa <- 6
z <- numeric(nrow(x))
for (i in 1:nrow(mu)) {
z <- z + weights[i] * dvonMises(x, mu[i, ], kappa)
}
z <- matrix(z, nrow=length(x.coord))
#
# Plot the result.
#
image(x.coord, y.coord, z, xlab="Longitude", ylab="Latitude")
points(mu[, 1], mu[, 2], pch=16, cex=sqrt(weights))