realTimeloads provides code for computing loads from estimated timeseries of concentration (C, mass per unit volume) and discharge (Q, volume per unit time). Load (L) is the total mass of an analyte transported over a time period, while flux (F) is mass transported per unit time. That is, F = C(t)Q(t) and L = sum(C(t)Q(t)dt), where t is time.

Code for processing of acoustic backscatter from acoustic Doppler current profilers (ADCP) and acoustic Doppler veclocimeters (ADVM) are provided. Processed acoustic backscatter can be used to estimate suspended-sediment concentration reported as TSS or SSC.

Synthetic data are provided in realTimeloads::ExampleData. ExampleData is a list with the following data frames: “Site”, “ADCP”, “Echo_Intensity”, “Sonde”, “Height”, “Discharge”, and “Sediment_Samples”. Detailed context for these data are given in the cited guideline. Briefly, the data were generated from a 1D sediment transport model implemented for the Burdekin River of Queensland, Australia. The sediment transport model utilizes measured channel geometry, measured discharge, and sediment characteristics measured from bed samples.

“Site” contains site information such as site name, unique site number, and relevant elevation datum. “ADCP” and “Echo_Intensity” contain instrument measurement information and acoustic backscatter computed from acoustic scattering theory and suspended-sediment concentration (SSC) predicted from the sediment transport modal. “Sonde” contains water quality parameters needed to process acoustic backscatter. “Height” is water level height computed from Manning’s equation and measured discharge in “Discharge”. “Sediment_Samples” contains the SSC from the 1D sediment transport model at the elevation of the ADCP and the depth-averaged,velocity-weighted SSC. Acoustic backscatter from the ADCP was computed using the SSC at the elevation of the ADCP; however, for unbiased estimation of suspended-sediment load, users should estimate the depth-averaged,velocity-weighted suspended-sediment concentration.

### Load library and example data ----
#> Registered S3 method overwritten by 'quantmod':
#>   method            from
#> zoo
InputData <- realTimeloads::ExampleData

Process acoustic backscatter from the ADCP, regress processed acoustic backscatter to suspended-sediment concentration timeseries measured at the elevation of the ADCP, and predict estimated timeseries of SSC with uncertainty

### Assign list items to variables used in functions ------------------------ 
Site <- InputData$Site
ADCP <- InputData$ADCP
Echo_Intensity_Beam_1 <- InputData$Echo_Intensity
Echo_Intensity_Beam_2 <- InputData$Echo_Intensity # example code assumes backscatter is equal across beams; however, backscatter between beams will differ in practice 
Sonde <- InputData$Sonde
Height <- InputData$Height
Discharge <- InputData$Discharge
Sediment_Samples <- InputData$Sediment_Samples

### Process acoustic backscatter data  --------------------------------------
ADCPOutput <- realTimeloads::acoustic_backscatter_processing(Site,ADCP,Height,

### Compute estimate of analyte timeseries using --------

# threshold (minutes) for interpolation of Surrogate to Analyte timeseries
threshold <- 30

# Surrogate "X" (e.g, acoustic backscatter and/or turbidity)
tx <- ADCPOutput$time
x <-ADCPOutput$mean_sediment_corrected_backscatter_beam_1_and_beam_2_dB

# Analyte "y" (e.g., TSS, NOx, etc)
ty <- Sediment_Samples$time
y <- Sediment_Samples$SSCpt_mg_per_liter

# interpolate surrogate onto analyte timeseries
# Randomly sample timeseries n times to simulate sampling that would occur in the field 
n <- 50
ind_calibration <-sample(1:length(ty),n,replace=FALSE)
calibration <- realTimeloads::surrogate_to_analyte_interpolation(tx,x,

# Specify calibration variable names for later storage in regression data
names(calibration)[grepl('surrogate',names(calibration))] <- 'SNR_dB'
names(calibration)[grepl('analyte',names(calibration))] <- 'SSCpt_mg_per_liter'

# compute bootstrap regression parameters per
fun_eq <- "log10(SSCpt_mg_per_liter) ~ SNR_dB"

Regression <- realTimeloads::bootstrap_regression(calibration,fun_eq)

# predict concentration timeseries with uncertainty
Surrogate <- data.frame(ADCPOutput$time,
colnames(Surrogate) <- c('time','SNR_dB')

Estimated_Concentration <- realTimeloads::estimate_timeseries(Surrogate,Regression)

With timeseries of estimated concentration multiply by discharge to compute load

### Interpolate discharge onto surrogate time series (e.g., Backscatter t.s) ----
# All data in ExampleData are on the same time-step, however in practice this may not be the case
Q <- realTimeloads::linear_interpolation_with_time_limit(Discharge$time,
                                                         Discharge$Discharge_m_cubed_per_s,tx,threshold = 1)$x_interpolated

# compute dt (second) for load integration
dt =  c()
dt[2:length(tx)] <- as.numeric(difftime(tx[2:length(tx)],tx[1:length(tx)-1],units = "secs"))
dt[1] = median(dt,na.rm=TRUE) # assume time step 1 using median dt

# compute load (L) (kt) at each time "t"
# Estimated_Concentration$estimated_timeseries is a matrix with simulated timeseries drawn from Monte Carlo simulations on the bootstrapped regression parameters. Rows are simulations, columns are time. Methods are from 
iters <- nrow(Regression$regression_parameters) # number of Monte Carlo simulations
Qsi =  matrix(NA, nrow = iters, ncol = length(x))
for (i in 1:iters) {
  Qsi[i,] <- Estimated_Concentration$estimated_timeseries[i,]*Q*dt*1e-9 # assumes concentration is mg/l and discharge is cubic meters per sec, dt is in seconds

Compute uncertainty on flux (kt/s) and load (kt)

### Compute uncertainty on concentration and load timeseries ---------------
cQs = rowSums(Qsi,na.rm=TRUE) # total load for each iteration
quants <- c(0.0527, 0.1587, 0.5, 0.8414, 0.9473) # +/- 1 and 2 sigma and median (i.e., reported) estimate

Total_load_kt <- data.frame(t(quantile(cQs,quants,na.rm=TRUE)))
colnames(Total_load_kt) = c('minus_two_sigma_confidence','minus_one_sigma_confidence','median_confidence','plus_one_sigma_confidence','plus_two_sigma_confidence')

Reported_real_time_load_estimate <- Total_load_kt$median_confidence

# explicitly name timeseries with uncertainty for plotting below 
Analyte_concentration_timeseries_mg_per_liter <- Estimated_Concentration$estimated_timeseries_quantiles

Analyte_flux_timeseries_kt <- data.frame(t(apply(Qsi, 2 , quantile , probs = quants , 
                                                 na.rm = TRUE )))

colnames(Analyte_flux_timeseries_kt) = c('minus_two_sigma_confidence','minus_one_sigma_confidence','median_confidence',

Plot estimated versus actual timeseries

### Compare estimate of concentration to actual ----

# actual loads (kt)
SSCxs <- Sediment_Samples$SSCxs_mg_per_liter
SSCpt <- Sediment_Samples$SSCpt_mg_per_liter
Qspt <- SSCpt*Q*dt*1e-9 # load from SSC measured at a point
Qsxs <- SSCxs*Q*dt*1e-9 # load from depth-averaged, velocity-weighted SSC

indp <- seq(from = 1, to = nrow(ADCP), by = 1) # vector for plotting subset of timeseries if desired 

     col='red',type = "l",lwd= 2,xlab = "time (AEST)",ylab="Analyte concentration (mg/l)",
     main = "Estimated versus actual concentration",ylim = c(0,1500))
      col='grey',lty = c(5))
      col='grey',lty = c(5))
       col = 'black',pch = c(21))
      col='black',lwd= 1.5)

legend("topright",legend = c("Estimated concentration","Estimation uncertainty","Actual concentration used in regression","Actual concentration timeseries"),
       lty = c(1, 5, 0,1),col = c('red', 'grey', 'black', 'black'),pch = c(-1,-1,21,-1))

###  Compare estimate of flux to actual ----
     col='red',type = "l",lwd= 2,xlab = "time (AEST)",ylab="Analyte flux (ton per second)",
     main = "Estimated versus actual flux",ylim = c(0,20))
points(tx[ind_calibration],Qspt[ind_calibration]/dt[ind_calibration]*1e3,col = 'black',
       pch = c(21))
lines(tx,Qspt/dt*1e3,col = 'black',lwd= 1.5)
legend("topright",legend = c("Estimated flux","Estimation uncertainty","Actual flux of data used in regression","Actual flux timeseries"),
       lty = c(1, 5, 0,1),col = c('red', 'grey', 'black', 'black'),pch = c(-1,-1,21,-1))

# check estimated load / modeled load
#mean(cQs)/sum(Qspt,na.rm=TRUE) # relative to Cpt (biased) load
#mean(cQs)/sum(Qsxs,na.rm=TRUE) # relative to Cxs (actual) load

Output <- list("time" = tx,"surrogate_timeseries_used_for_prediction_of_analyte"= df,"regression_data" = calibration,"regression_parameters_estimated_from_bootstrap_resampling" = Regression,"Analyte_concentration_timeseries_mg_per_liter"= Analyte_concentration_timeseries_mg_per_liter,"Dicharge" = Discharge,"Analyte_flux_timeseries_kt" =Analyte_flux_timeseries_kt)

Compare estimate to actual load over period of record. Note that ratio of the estimated load computed from SSC measured at-a point (SSCpt) is ~ 1, while the ratio of the estimated load computed from SSC measured at-a point to the actual load from SSCxs is ~ 1.2. This illustrates that the load estimated from backscatter is a non-biased estimate of load from SSCpt, while loads from SSCpt overestimate the actual load by 20%.

### Compare total loads in barplot ----
# check ratios of estimated load to modeled load
#Total_load_kt$median_confidence/sum(Qspt,na.rm=TRUE) # estimated load relative to load at Cpt
#Total_load_kt$median_confidence/sum(Qsxs,na.rm=TRUE) # estimate total load relative to actual load from Cxs

loads <- c('Estimated load from SSCpt(dB)'= Total_load_kt$median_confidence,'Load from SSCpt' = sum(Qspt,na.rm=TRUE),'Load from SSCxs' = sum(Qsxs,na.rm=TRUE))

y <- c(Total_load_kt$median_confidence)
y0 <- c(Total_load_kt$plus_two_sigma_confidence)
y1 <- c(Total_load_kt$minus_two_sigma_confidence)
hb <- barplot(loads,main="Total load comparison",
        ylab="Total load (kt)",border=NA,ylim = c(0,8000))

For data collected in the field, missing data may occur for numerous reasons. To infill missing data the function impute_data() is provided. impute_data() uses ARIMA to infill gaps up to 3 hrs in duration and then uses regression trees to infill larger gaps. impute_data() has been found to be relatively quick to implement (when MC = 1, see ?impute_data) for large data sets (20,000 data points) and has produced reasonable results on numerous field data tested by the author. impute_data() can infill data that exhibit diurnal and semidiurnal variability using the argument “harmonic” set to “TRUE”

Below, 50% of the processed backsactter is imputed and uncertainty on the imputation is estimated using Monte Carlo simulations. Imputation uncertainty is provided to allow propagation of uncertainty into final load estimates if desired.

### Impute data ----
xo <- ADCPOutput$mean_sediment_corrected_backscatter_beam_1_and_beam_2_dB
# simulate 50% missing data
idata <- sample(which(is.finite(xo)),round(sum(is.finite(xo))*0.50),replace=FALSE)
x <- rep(NA,length(xo))
x[idata] <- xo[idata]
flow_ratio <- imputeTS::na_interpolation(Q/x)
Xreg <- cbind(Q,flow_ratio)
# random sampling of training set in impute_data() can generate warnings in decision tree algorithm 
# In impute_data(), ptrain = 1 can be set to preclude uncertainty estimation and decrease code run time. 
# ptrain=0.8 indicates that 80% of the finite data in x will be used in training regression trees and 20% of the data will be held-out for computing validation accuracy
suppressWarnings(Imputation <- impute_data(tx,x,Xreg,MC=10,ptrain=0.8))

ximputed <- Imputation$Imputed_data$x
ximputed[is.finite(x)] <- NA

plot(xo,ximputed,xlab = 'Backscatter (dB)', ylab = 'Imputed backscatter (dB)',ylim = c(min(xo,na.rm = T),max(xo,na.rm = T)),xlim = c(min(xo,na.rm = T),max(xo,na.rm = T)))
legend("topleft",legend = c("Median imputed value","Plus 2-sigma uncertainty on imputed value","Minus 2-sigma uncertainty on imputed value"), lty = c(0, 0,0),col = c('black', 'red', 'blue'),pch = c(21,21,21))


Livsey, D. N., Downing-Kunz, M. A., Schoellhamer, D. H., & Manning, A. J. (2020). Suspended sediment flux in the San Francisco Estuary: Part I—Changes in the vertical distribution of suspended sediment and bias in estuarine sediment flux measurements. Estuaries and Coasts, 43, 1956-1972.

Livsey, D. N., Turner, R. D. R., & Grace, P. R. (2023). Combining Optical and Acoustic Backscatter Measurements for Monitoring of Fine Suspended‐Sediment Concentration Under Changes in Particle Size and Density. Water Resources Research, 59(8), e2022WR033982.

Livsey, D.N. (in review). National Industry Guidelines for hydrometric monitoring–Part 12: Application of acoustic Doppler velocity meters to measure suspended-sediment load. Bureau of Meteorology. Melbourne, Australia