Antecedent Precipitation Index (API)

Purpose

Construct a simple model of the relationship between antecedent precipitation and soil moisture.

Overview

Using self-collected data, data from the data archive or on-line data sets, develop a simple model between precipitation and soil moisture. This will be most meaningful for daily observations of each.

The model, a decaying exponential function, can be represented as a simple mechanical process, a table or spreadsheet computation, or as an exponent of time to make it accessible to a wide range of students. Besides working with integrated data sets, the students can look for characteristic differences between the drying rates in different climates.

Time Required

  • 1-3 class periods, depending upon how much data exploration and analysis is done.

Skill Level

  • Intermediate-Advanced

Key Concepts and Skills

  • Concepts: Decay rates, Correlation, Comparison
  • Skills: Analyzing, Graphing, Modeling, Fractions, Multiplication, Word Problems, Logarithms, Equation of a Line

Materials and Tools

  • Beginning: dry kidney beans
  • Intermediate; paper for tables and graphs, hand calculator
  • Advanced: spreadsheet program

System Requirements

  • Data Sets: precipitation, soil bulk density, soil water content
  • Period: Try to get a time period with several distinct wet-dry cycles. Daily precipitation and soil moisture (gypsum block) data sets are best.
  • Note: If bulk density is not available, try requesting via GLOBEmail, or assume a bulk density of 1.4 g/cm^3 for a typical loam soil.

Key Words

  • GLOBE 3; Soil Moisture; Precipitation; Bulk Density; Model; Analysis;

Background

The Antecedent Precipitation Index (API) is a simple number derived from rainfall depth, which can be compared with or used to estimate soil moisture. The goal of this activity is to develop quantitative skills that will allow you to analyze soil moisture based on a simple model of rainfall storage and evaporation. If you cannot answer these questions yet, read on. Below, we present several ways of thinking about data comparison. Following that, we develop a simple model of evaporation, starting with a game using beans. Finally, we present some ideas for analyzing GLOBE data from your site or from sites around the world.

Getting Started

Data analysis and making comaprisons are part of our everyday lives, so much so that we might not always be awear of it.

Common characteristics or properties - something that can be quantified - are the basis of most comparisons, but these things or variables must be identified and defined. In the Soil Characteristics activity "Water Race", one topic is to compare how much water passes through different kinds of soil. This is a fairly direct comparison - the volume of water that seeps through two soil samples of equal size is compared. Often times, scientists will look for consistent relationships between things in nature, and use observations of the more easily measured quantity to predict or estimate a quantity that is difficult to measure. This is the basis of the antecedent precipitation index. In this case, several complex processes, eg. infiltration and evaporation separate the two variables of interest, which are precipitation and soil water content.

Example 1: The Bean Game

One way to introduce students to data analysis and API is to have them play the following game with a pile of dry beans. Explain that the beans are being used to "model" soil water content and the beans that are taken away at each step are analogous to the process of evaporation.

Data Analysis

Graphs and plots help us "see" relationships between variables or quantifiable things of interest. One potential outcome to the above exercise is illustrated below. In this case, we consider what would happen if you started with the number 80 and repeatedly take half of it (see table below). This is equivalent to multiplying any number by what we will call the recession constant (k=0.50 in this case), to get the following number (the evaporation rate is 1-k). The graph also illustrates recession constants of 0.75 and 0.90.

Compare the Table of Numbers below with the same information presented in a simple graph:

Time Step Soil Water
Content
(k=0.50)
0
1
2
3
4
5
6
7
8 		80
40
20
10
5
2.5
1.25
0.625
0.3125 		Recession rates

Ask your students how they would determine the recession rate of the Soil Water Content curves shown in the first figure? Possible answers might be:

One characteristic of our Soil Water Content curves is that they appear to be decreasing such that they almost level off after a period of time. Mathematicians and Scientists refer to this as asymptotic behavior. Many physical phenomena are characterized by this kind of decay such as radioactive decay, battery/capacitor charging rate, dilution. It turns out that a single equation can be used to describe this behavior. It is called exponential decay and we can use logarithmic graphing paper to find recession constants using the equation for a line.

Time Step Log10 SWC
(k=0.50)
0
1
2
3
4
5
6
7
8 		1.903
1.602
1.301
1.000
0.699
0.398
0.097
-.204
-.505 		Log Decay rates
Ask your students:

Relationships between Precipitation and Soil Moisture

What we have just done might seem trivial since we started off knowing the answer. But what happens when we do not control the experiment - something that happens all the time when we are making observation in the real world? Under many conditions, rainfall stored as soil moisture will "evaporate" like the beans did in the exercise above. Below several ways of describing API in terms of words and equations are given. Which makes the most sense to you?

Following a rain, surface soils dry by the combined processes of evaporation and (if vegetation is present) transpiration. A simple approximation to the complex, interactive processes that control evaporation is to assume that only a fixed percentage (k) of the previous rainfall (stored in the surface soils) is retained every day.

"The rate at which moisture is depleted from a particular basin (or location) under specified meteorological conditions is roughly proportional to the amount in storage. In other words, the soil moisture should decrease logarithmically (or asymptotically) with time during periods of no precipitation" (Linsley, Kohler and Paulhus, 1982). It turns out that, given a location and season, this method works reasonably well.

Mathematically, this could be described in terms of the soil water content (SWC) and time (t) by the following equation: 

Current storage is proportional to Previous storage times Retention rate

               SWC(t) is proportional to SWC(t-1) * k 

to predict the SWC after two days of drying we could write,

               SWC(t) is proportional to SWC(t-2) * k * k

in general, we can write,

               SWC(t) is proportional to SWC(t=0) * k^t
or
               API(t) is proportional to API(t=0) * k^t
Now write the equation for what was just done above using logarithmic graph paper, starting with the rate equation and transforming it into the equation for a line: 
               SWC(t) is proportional to SWC(t=0) * k^t

                     SWC  = b * k^t 

               log10(SWC) = log10(b * k^t)

               log10(SWC) = log10(b) + log10(k) * t
 
        log10(SWC) = log10(zero intercept) + log10(recession constant) * time

                        y =  b + m * x
The important thing to remember if the mathematics is daunting, is that if you replot a decaying function on log paper and end up with something close to a straight line, then the logarithm of the recession constant is the slope of the strainght line. For the case of the 50% recession rate given above: 
              m = slope = (change in y)/(change in x)

               log10(k) = (1.301 - 1.602)/(2 - 1)

               log10(k) = -0.301

                     k  = 10^(-0.301) = 0.50

Basic Assumptions

Example 2: Artifical Field Data

Time Precipitation Soil Water
Content (X)		 Soil Water
Content (O)		 Time since
Precipitation		 Normallized
SWC (X)		 Normallized
SWC (O)
[days] [mm] [gm/gm] [gm/gm] [days] [-] [-]
1
2
3
4
5
6
7
8
9
10
11
12 		0
10
0
0
0
0
5
0
0
0
0
0 		1
16
12
9
7
5
8
6
4
3
2
2 		10
29
27
26
24
23
27
25
24
23
22
21 		?
0
1
2
3
4
0
1
2
3
4
5 		12
100
75
56
44
31
100
75
50
38
25
25 		71
100
95
89
87
82
100
94
91
85
82
76
soil moisture

A typical (somewhat contrived) relationship that could be observed between rainfall and soil water content is given above. The blue bars represent the amount of rainfall that fell on a given date.

This is a relatively complex relationship. But we now have the tools to analyze it. Every time it rains, soil moisture jumps up (increases). During the time between precipitation events, soil moisture decreases. In fact, soil moisture appears to be decreasing such that it almost levels off after a period of time. This should suggest to us to use a logarithmically transformed SWC scale. We can more readily work with all the data by normallizing SWC by the maximum value after a rain and just keep track of days since precipitation, as shown in the table above. I have normallized to 100 to avoid decimal numbers.

log soil moisture

The best fit lines with decay constants or log slopes of 0.725 and 0.955 are very close to the values of 0.75 and 0.95 used to simulate this data. The error reflected in the x's off the best-fit line is due to the fact that the numbers were rounded to the nearest integer. Lets see how a real data set would look ...

Example 3: GLOBE Data - site?

What to do with your data

What to do with other GLOBE data

Activities: Simple -> Complex

Further Investigations

It is possible to apply the API method to individual records of soil water content at a given depth but it makes more sense to compare the total amount of water that infiltrates into the ground (related to precipitation) with the amount of water lost through evaporation and transpiration (related to the average soil column SWC). Here is how to estimate near-surface soil moisture storage from a GLOBE soil moisture depth profile.
First, find the average soil water content (avg_SWC) from the surface to a depth of 100 cm.

avg_SWC/m = 0.20 * SWC(10) + 0.25 * SWC(30) + 0.30 * SWC(60) + 0.25 * SWC(90)

Now convert this to effective soil water depth by multiplying by the bulk density. Note the units below: 

 Avg_SWC/m  * Soil Bulk Density / Water Density = Soil Water Storage

   [g/g]          [g/cm^3]          [g/cm^3]           [mm]

(wet - dry) wt.    dry wt.           cm^3    10 mm
--------------- * ----------    *   ------ * -----  =   SWS 
   dry wt.        sample vol.        1 g     cm
Last updated: 5/5/98
Comments? globe@hwr.arizona.edu

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