Introduction

In the past 10 years, climate change has become a major concern to the general public. One source of anxiety surrounding global warming is the impact it will have on river flows. As temperatures rise and precipitation patterns change due to a warming climate, water supplies will shrink and strain the existence of various ecosystems and wildlife species. One area particularly at risk is the US Southwest. Warmer climates are already reducing the snowfall in the mountains which in the past have been an adequate source of water for Western rivers during the drier, warmer seasons. Further, hotter temperatures are causing more evaporation in river reservoirs. A relatively new study suggests that a third of the loss of flow in the Colorado River can be attributed to global warming (Udall and Overpeck 2017).

While climate change is already a large issue, it could potentially become a larger one if communities lose access to water due to rivers shrinking. Although, it was caused by a different issue, the Flint, Michigan water crisis was a cause of a federal state of crisis. Flint is an example of what happens when a community loses access to its water source. Perhaps a more likely problem that would be caused by a potential water shortage is the effect it would have on the agriculture in the Southwest. Investigating how rising temperatures affect river discharge could provide vital information into how we can better prepare to meet freshwater needs around the Colorado River basin and perhaps beyond.

Data

Since rising temperature is the main catalyst for climate change, we need some sort of time series data measuring temperature or temperature changes. Given the motivation, we are not necessarily interested in how the average global temperature has increased over time, but more how the temperature has changed relative to a base period over time. For this reason, we chose our temperature data from the GISS Surface Temperature analysis which provides data in the form of monthly mean temperature anomalies in degrees Celsius from a base period of 1950-1980. The data can be downloaded from this link https://datahub.io/core/global-temp.

As noted earlier, climate change pertaining to rivers is particularly important to the Southwest region of the US. Because of this, it is particularly of interest to explore how temperature change is effecting rivers in this region. The Colorado river is perhaps the principal river of the Southwestern US. Studying how global temperature anomalies have affected the discharge of the Colorado river could perhaps lend us insight into how global warming effects rivers in the Southwest on a larger scale. The data used in this study is the discharge rate of the Colorado river in cubic feet per second measured below the Laguna Dam (AZ-CA).

## Warning: `panel.margin` is deprecated. Please use `panel.spacing` property
## instead

Model Fitting

## 
## Call:
## arima(x = discharge, order = c(0, 0, 2), xreg = anomaly)
## 
## Coefficients:
##          ma1     ma2  intercept  anomaly
##       0.6125  0.3556     6.4435  -0.1586
## s.e.  0.0614  0.0606     0.1274   0.1885
## 
## sigma^2 estimated as 0.1247:  log likelihood = -90.93,  aic = 191.85

Formally, the model we have fit takes the following form: \[Y_n = \beta_0 + \beta_1 t_n + \epsilon_n + \theta_1 \epsilon_{n-1} + \theta_2 \epsilon_{n-2}\] where \(Y_n\) represents the log(Discharge) at time \(n,\) \(t_n\) represents mean temperature anomaly from base period 1950-1980, \(\beta_0 = 6.4435,\) \(\beta_1 = -0.1586,\) \(\theta_1 = 0.6125,\) \(\theta_2 = 0.3556,\) and \(\epsilon_n\) is simply white noise with variance \(\sigma^2 = 0.1247.\) This indicates for an increase in 1 degree celcius of mean temperature anomaly from the mean, we expect that the log discharge rate of the Colorado River to decrease by -0.1586 log cubic meters per second.

nullarma <- arima(discharge,order=c(0,0,2))
dif <- arma02$loglik - nullarma$loglik 
p.val <- 1-pchisq(2*dif,df=1)
p.val

## [1] 0.4005494

Frequency Analysis

From a scientific standpoint, it makes sense that river discharge is effected by different seasons throughout the year. In hotter, drier seasons we would expect the discharge rate of a river to be lower. Additionally, the ACF plot of the residuals from the model fitted without arma errors showed high ACF values in later lags in addition (e.g., 10 and 11) perhaps indicating some seasonality. We can explore further via frequency analysis.

The spectrum plot shows that there is a peak at a frequency just under 1. This corresponds to a period of 12 months which confirms our intuition that there is yearly seasonality in the data. Let’s take a look at AIC values of various linear models with SARMA errors which may do a better job of capturing the data.

aic_table <- function(data,P,Q, xreg=NULL){
  table <- matrix(NA,(P+1),(Q+1))
  for(p in 0:P) {
    for(q in 0:Q) {
      table[p+1,q+1] <- arima(data,order=c(0,0,2),seasonal=list(order=c(p,1,q),period=12),xreg=xreg)$aic
    }
  }
  dimnames(table) <- list(paste("<b> AR",0:P, "</b>", sep=""),paste("MA",0:Q,sep=""))
  table
}
aic <- aic_table(discharge,3,3,anomaly)
kable(aic,digits=2)

According to the table above the AIC values of linear models with SARMA errors are all much higher than the simpler linear regression models with ARMA errors. This suggests that adding seasonality to our model does not add very much information.

Conclusion

Initially the side by side time series suggested that the decreasing trend underlying the log discharge rate of the Colorado River could be explained by the mean temperature anomaly from base period 1950-1980. However, when a linear regression model with ARMA(0,2) errors was fit, the likelihood ratio test showed that we did not have enough evidence to reject the null hypothesis that the coefficient of the mean temperature anomaly from base period 1950-1980 in the model was 0. This is not to say that our model is invalid, it simply means from an inferential standpoint with the data from this project, we do not have evidence to suggest that the discharge rate of the Colorado River and monthly temperature anomaly from a base period have the inverse relationship that we initially believed might have existed (i.e., monthly temperature anomaly cannot explain the trend in the discharge rate). Lastly, although the data displays seasonality, incorporating seasonality into our model does not improve the likelihood.