Precipitation over the Murray-Darling Basin

This assignment uses a century of gridded rainfall to ask three questions that hydrologists actually argue about: how much water falls on the Murray-Darling Basin, whether that amount is changing, and how the droughts and wet periods of the last twenty-five years compare with one another. Along the way you will meet the one geometric correction that separates a defensible basin water balance from one that is wrong by nearly a fifth.

Total: 10 marks.

For EMSC3025, Questions 1–4 total 10 marks. Question 5 is an optional one-mark bonus.

For EMSC6025, Questions 1–3 total 7 marks. Question 4 is worth 2 marks, and required Question 5 is worth 1 mark.

Computer labs for this assignment

Complete these student notebooks before you begin the assignment. They are available in Colab and contain blank exercise cells.

• Lab 2: Shapefiles and basin masks explains the basin boundary and grid-cell mask.
• Lab 3: xarray gridded data explains OPeNDAP, coordinate selection, the descending-latitude trap, and complete years.
• Lab 4: Grid-cell areas derives the area formula that Question 2 uses.

Use the computer-lab page on the course website to read the executed solution notebooks after you attempt each exercise.

The rainfall data

We use the monthly rainfall product from the Bureau of Meteorology’s Australian Water Outlook (the AWRA-L v7 model archive), served by NCI over OPeNDAP. You do not need to download anything by hand: xarray can open the remote file directly and subset it on the server.

# Run this first in Google Colab.
%pip install --quiet xarray netCDF4

# Use xr as the short name for xarray.
import xarray as xr

# Store the remote data address.
url = (
    "https://thredds.nci.org.au/thredds/dodsC/"
    "iu04/australian-water-outlook/historical/v1/"
    "AWRALv7/processed/values/month/rain_day.nc"
)

# Use netcdf4 to open the OPeNDAP service.
ds = xr.open_dataset(url, engine="netcdf4")

What this code does

%pip install installs xarray and netCDF4 in the current Colab session. xarray reads and labels gridded data. netCDF4 lets xarray read the NCI OPeNDAP service.

import xarray as xr makes the xarray library available as xr. The url variable stores the data address. xr.open_dataset reads the file metadata and returns a dataset called ds.

At this stage, ds contains coordinates, variable names and attributes. It does not contain the full rainfall array in memory.

What you are getting:

• Variable rain_day, in millimetres. Despite the name and despite a long_name of “Daily Rainfall”, in this file each value is the monthly total rainfall. Ignore the long_name; trust the units and the time axis.
• A regular grid of 681 latitudes by 841 longitudes at 0.05 degrees (roughly 5 km), covering latitude -10 to -44 and longitude 112 to 154.
• Monthly steps beginning January 1911 and continuing to the present. The record is extended as new months are processed, so the last time step you receive depends on when you run your notebook. State in your notebook the first and last time step of the data you actually pulled, and exclude any incomplete final calendar year from annual statistics.

Everything in this assignment that refers to a climatology, a long-term mean or an anomaly uses the full record from 1911 up to the last complete calendar year as its baseline. Say so in your notebook, and use the same baseline throughout so your numbers are comparable.

The basin

The Murray-Darling Basin boundary is supplied as a pair of shapefiles, north and south, in

Unzip it and take the union of MDB_north_boundary.shp and MDB_south_boundary.shp. This assignment treats the basin as a single region; the north/south split is the subject of Assignment II. Lab 2 (lab02_shapefiles_and_masks) shows how to read the shapefiles and test whether a point falls inside a polygon. A grid cell belongs to the basin if its centre does; that is accurate enough here, and you can check it by comparing your masked area against the published basin area of about 1,060,000 km².

# Wrong: this order does not match the axis.
ds.rain_day.sel(latitude=slice(-38, -24))
# -> latitude: 0 cells. EMPTY, and no error.

# Right: north to south, following the axis.
ds.rain_day.sel(latitude=slice(-24, -38))
# -> latitude: 281 cells. Correct.

Questions

1. (2 marks) Plot monthly rainfall over the basin for four months, as a single figure with four panels arranged two rows by two columns:

   	dry	wet
   October	October 2006	October 2022
   December	December 2019	December 2010

   The layout is the point of the question. Each row holds the calendar month fixed, so the difference between the two panels in a row cannot be the seasonal cycle — it is purely interannual. Use a common colour scale across all four panels (otherwise the comparison is meaningless), show the basin outline, and label each panel with its date. Comment briefly on what the figure shows, including where in the basin the wet months put their rain.

2. (3 marks) Compute the total volume of rainfall falling on the basin each month, in gigalitres (1 GL = 10⁶ m³). This requires the area of every grid cell, and grid-cell area on a sphere depends on latitude. Use the exact formula derived in Lab 4 (lab04_grid_cell_areas),

   A = R^2 \, \big| \sin\phi_2 - \sin\phi_1 \big| \, \big| \Delta\lambda \big|

   with R = 6370 km and the cell edges at \phi \pm 0.025°, all angles in radians.

   • (i) Report the mean monthly and mean annual rainfall volume over the basin in GL, and the mean annual rainfall as a depth in mm. State your basin area in km² and compare it with the published value.
   • (ii) Now repeat the calculation assuming every 0.05° cell has the same area — the area a 0.05° x 0.05° cell would have at the equator, with no \cos\phi factor. Report the resulting mean annual volume and the percentage error relative to (i). Explain, in terms of , why the error has the sign it does.
   • (iii) There is a third, lazier method: take the plain unweighted mean of the rainfall in all masked cells, then multiply by the true basin area. Compute it. You will find it is within about 0.1 % of the correct answer, while (ii) is out by nearly 18 %. Explain why. What quantity does the unweighted mean implicitly assume about cell areas, and what is the only part of the area effect that survives it?

3. (2 marks) Build a time series of annual basin rainfall over complete calendar years and fit a straight line to it by ordinary least squares. Report the slope together with its standard error and p-value, and then answer the question the fit was for: is the fitted trend distinguishable from the basin’s year-to-year variability? Quote the interannual standard deviation alongside the slope to support your answer.

   Repeat the fit over at least two different starting years (for example 1911, 1970 and 2000) and comment on what happens to the slope. A good answer explains the behaviour you see rather than reporting a single number.

4. (EMSC3025: 3 marks; EMSC6025: 2 marks) Compare four periods, using the full-record baseline defined above:

   • the Millennium Drought, 2001-2009;
   • the Tinderbox Drought, 2017-2019;
   • the La Niña wet period 2010-2012;
   • the La Niña wet period 2020-2023.

   For each, report the mean annual rainfall in GL and mm and its anomaly from the baseline, both in mm and as a percentage. Then discuss: what was similar and what was different between the two droughts, and between the two wet periods? Your discussion should distinguish clearly between how deep a rainfall deficit was and how long it lasted, and should say which of those two you would expect to matter more for storage in the basin’s reservoirs and soils, and why. Support the argument with the year-by-year numbers, not just the period means.

Extension — ENSO

5. Obtain a monthly ENSO index and quantify how much of the basin’s rainfall variability it explains. Use one of:
   • the Southern Oscillation Index from the Climatic Research Unit, University of East Anglia: https://crudata.uea.ac.uk/cru/data/soi/soi.dat (1866 to present; a fixed-width text file, year followed by twelve monthly values followed by the annual mean, with -99.9 as the missing-value flag); or
   • the Niño 3.4 sea-surface-temperature anomaly from NOAA PSL: https://psl.noaa.gov/data/correlation/nina34.anom.data (1948 to present; two header numbers, then year plus twelve monthly values, with -99.99 missing and a few lines of prose at the end).
   Do not attempt to download the index from bom.gov.au. The Bureau blocks automated access and will return an HTTP 403 error. Its background page on El Niño and La Niña is worth reading, but it is reading, not a data source.Then:
   • Correlate the index against monthly basin rainfall anomalies (rainfall minus the climatology of that calendar month — correlating raw totals mostly measures the seasonal cycle, not ENSO). Report the correlation coefficient and its p-value.
   • Repeat the correlation with the index lagged by one to six months ahead of the rainfall. Does the index have any useful predictive lead, or is the relationship essentially concurrent?
   • Repeat the concurrent correlation separately for each calendar month. The answer is not the same in January as it is in September, and explaining that difference is the most interesting part of this question.
   Note the sign convention: a positive SOI corresponds to La Niña, whereas a positive Niño 3.4 anomaly corresponds to El Niño. State which index you used and check that the sign of your correlation makes physical sense.