Qmulti - Composite drought index

Overview

Qmulti is a composite drought index created by RAM. Its objective is to convey the performance of the growing season in an easy to understand numerical scale, where performance means the quality of crop and pasture development.

The index is built by default around commonly available gridded variables used for early warning activities. It integrates key dimensions of rainfall (amount, distribution and timing) as well as vegetation and dry spells. Rainfall being the key driver for crop growth; while vegetation and temperature are response variables to rainfall dynamics. Additional variables can be added if desired.

The index accounts for the variable importance of different phases of the season by giving more weight to anomalous behavior in the wetter stages of the season.

Formulation

Point in time Quantiles

For any location and time $xyt$, we represent the index by:

$$Qmulti_{xyt} = f(\theta rainfall_{xyt}, \theta dryspell_{xyt}, \theta NDVI_{xyt})$$

where $\theta$ is some form of anomaly, i.e. a measure of where a given value falls in the historical distribution of values.

The anomaly $\theta$ is constructed as the logit of an observation's percentile value given the historical distribution for that time of year. In other words, an observation in October would be converted to a percentile based on all historical observations for October only.

Formally, at time $t=T$ and variable $i$:

$$\theta i_{xyT} = logit(percentile(i_{xyT} \: | \: i_{xyt} \; {t \in 1...T}) )$$

where:

$$logit(x) = ln(x/(1-x))$$

The logit transform preserves the proper distance between percentiles towards the extremes of the scale, by ensuring, for example, that difference in the anomaly value the 97th percentile compared to the 96th is significantly larger than, say, the 67th percentile compared to the 66th.

It follows that $\theta$ will, for most real-world values, fall approximately within -5 and 5.

The combined anomaly is constructed from weighted average of the indicator anomalies. Which weights ($w_{i}$) should apply to each variable will remain a somewhat subjective choice. Our initial choice is as follows- rainfall: 0.33, dryspells: 0.47, NDVI: 0.2.

Therefore:

$$\theta multi_{xyt} = \sum_{i} w_{i} *\theta i_{xyt}$$

For reporting purposes, we would like to convey seasonal performance using an intuitive 0-100 bracket. To do so, we apply an inverse logit transformation to recover an index as percentiles

$$Qmulti_{xyt} = 100 * invlogit(\theta multi_{xyt}) = 100 * e^{\theta multi_{xyt}}/ (1 + e^{\theta multi_{xyt}})$$

Seasonal Quantiles

When monitoring the performance of the growing season, we would like to take into account for the influence of the variables of interest across the whole season to date, and not just at a specific point in time. In other words, we need an aggregation function that delivers seasonally-integrated (or cumulative) percentiles:

$$\theta i_{xy} = f(\theta i_{xyt} \; {\text{for all} \: t \in season} )$$

Our analysis is carried out at monthly time steps. We use a weighted sum of monthly percentiles with the weights reflecting the contribution of each month to season.

These are derived from long term average (LTA) rainfall and are the proportion of the total rainfall that (on average) falls in each month. For example, in the context of Southern Africa, October contributes a much smaller amount to the index compared to February, since February is a much rainier month. “Total” here is the whole season or whichever part of the season the user is considering. The weights always add to 1.

These weights ($w_{xyt}$) apply to all variables that are part of the index, whether they are rainfall related or not, i.e. there aren’t variable-specific weights.

Therefore:

$$w_{xyt} = LTArainfall_{xyt} / \sum_{t \in season} LTArainfall_{xyt}$$

and:

$$\theta i_{xy} = \sum_{t \in season} w_{xyt} *\theta i_{xyt}$$

All calculations are pixel based, so this scheme accounts for the variability in rainfall seasonality that may exist across the region.

Finally, the seasonally aggregated indicator anomalies $\theta i$ can be combined using the importance weights as above into a seasonally aggregated, composite anomaly $\theta multi$. Finally, an inverse logit transformation is applied to report results using the 0 - 100 range.