vertmean: Vertical Volume Weighted Mean of Matter Concentrations in Water Bodies

Description

Calculate vertical mean values which respect to depths of different layers or lake morphometry.

Usage

vertmean(depth, vari, level, top, bot, vol, total=FALSE)

Value

Volumetric average respectively total value (for total =TRUE) for a given quantity (concentration, energy, temperature) in the requested layer between depths top and bottom.

Arguments

depth: sorted vector of sampled depths,
vari: measurements corresponding to depth (concentration, temperature, ...),
level: surface water level (above ground or above sea level (m a.s.l.), depending on bathymetric function used,
top: top water level of requested layer over which to average or integrate,
bot: bottom water level of requested layer over which to average or intgrate,
vol: hypsographic function to be used (e.g. vol.depth),
total: if TRUE the total sum over the water body is returned (integrated value), instead of the volumetric mean.

Author

Thomas Petzoldt

Examples

Run this code


## define a bathymetric formula for a given lake or basin
## z:     water depth  (m below surface)
## zz:    water column (m above ground)
## level: total water depth (m above ground or above reference level)
weight.vol <- function(z, level) {
  zz  <- level - z
  if (any(zz < 0)) stop("depth > maximum depth")
  vol <- 175947 * zz^2 + 2686 * zz^3 # m^3
}

## area is first derivative
area <- function(z, level) {
  zz  <- level - z
  A   <-   0.5 * 175947 * zz + 1/3 * 2686 * zz^2 # m^2
}

## dummy formula for depth-weighted averaging
## (water column, instead of bathymetric curve)
weight.column <- function(z, level) {z}

## Plot of lake volume (bathymetric curve)
par(mfrow = c(1, 2))
z <- 0:12
V <- weight.vol(z, 12)
plot(V, z, type = "l", ylim = c(12, 0), xlab = "Volume (m3)",
  ylab = "Depth (m)")
polygon(c(V, 0), c(z, 0), col = "cyan")

## Test Data
level <- 12
depth <- c(0, 1, 3.5, 5, 7, 10, 10.5, 11.5)
pconc <- c(3.7, 4.2, 6.1, 8.9, 7.8, 9.7, 11.4, 11.4)

## Plot test data
plot(pconc, depth, xlim=range(c(0, pconc)), ylim=c(12,0), type="n",
  xlab="P concentration (mu g / L)", ylab="Depth (m)")
segments(rep(0, 13), depth, pconc, depth, lwd=3)

## simple means
m <- mean(pconc[depth <= 4])
lines(c(m, m), c(0, 4), col="blue", lwd=2)

m <- mean(pconc[depth >= 4])
lines(c(m, m), c(4, 12), col="blue", lwd=2)

## depth weighted
m <- vertmean(depth, pconc, level, top=0, bot=4, weight.column)
lines(c(m, m), c(0, 4), col="red", lwd=2)

m <- vertmean(depth, pconc, level, top=4, bot=12, weight.column)
lines(c(m, m), c(4, 12), col="red", lwd=2)

## volume weighted
m <- vertmean(depth, pconc, level, top=0, bot=4, weight.vol)
lines(c(m, m), c(0, 4), col="green", lwd=2)

m <- vertmean(depth, pconc, level, top=4, bot=12, weight.vol)
lines(c(m, m), c(4, 12), col="green", lwd=2)

m <- vertmean(depth, pconc, level, top=4, bot=12, weight.vol)
lines(c(m, m), c(4, 12), col="green", lwd=2)

legend("topright", col=c("blue", "red", "green"), lwd=2, cex=0.7,
  legend=c("non weighted", "depth weighted", "volume weighted"))

## total sum over the whole water column
vertmean(depth, pconc, level, top=0, bot=12, weight.vol, total=TRUE)