Rainfall erosivity calculation

Konstantinos Vantas

2019-08-31

Introduction

The R coefficient (MJ.mm/ha/h/yr) is defined as the long-term average of the product of the kinetic energy of a storm and the maximum 30 min intensity (Renard et al. 1991):

\[R = \frac{1}{n} \sum_{j=1}^{n} \sum_{k=1}^{m_j} (EI_{30})_{k}\]

where

• \(n\) is the number of years with rainfall records,
• \(m_j\) is the number of storms during year \(j\) and
• \(EI_{30}\) is the erosivity of storm \(k\).

The erosivity \(EI_{30}\) (MJ.mm/ha/h) is equal to:

\[EI_{30} = \left( \sum_{r=1}^{m} e_r \cdot v_{r} \right) \cdot I_{30}\]

where:

• \(e_r\) is the energy of rainfall (MJ/ha/mm),
• \(v_r\) is the rainfall depth (mm) for the time interval \(r\) of the hyetograph, which has been divided into \(r = 1, 2, ..., m\) sub-intervals, such that each one of these is characterized by constant rainfall intensity and
• \(I_{30}\) is the maximum rainfall intensity for a 30 minutes duration.

The quantity \(e_r\) can be calculated for each \(r\) using one of the kinetic energy equations:

1. Wischmeier and Smith equation, used in USLE: \(e_r = 0.119 + 0.0873log_{10}(i)\) with the upper limit of 0.283 MJ/ha/mm if \({i} > 76\) mm/h. (Wischmeier and Smith 1958).
2. Brown and Foster equation, used in RUSLE \(e_r = 0.29(1 - 0.72 e^{-0.05i})\) (Brown and Foster 1987).
3. McGregor et al. equation used in RUSLE2 \(e_r = 0.29(1 - 0.72 e^{-0.082i})\) (McGregor et al. 1995).

In the above equations \(i_r\) is the rainfall intensity (mm/hr) and \(e_r\) is the kinetic energy per unit of rainfall (MJ/ha/mm) for the interval \(r\).

The rules that apply in order to single out the storms causing erosion and to divide rainfalls of large duration are:

1. A rainfall event is divided into two parts, if its cumulative depth for duration of 6 hours at a certain location is less than 1.27 mm.
2. A rainfall is considered erosive:
   • if it has a cumulative value greater than 12.7 mm or
   • during a time period of 15 mins a cumulative value of precipitation of at least 6.4 mm is recorded.

Example

This is an example that uses the internal data set in order to compute the corresponding rainfall erosivity values.

library(hyetor)
library(tibble)
library(dplyr)
library(lubridate)

# view data
prec5min
#> # A tibble: 48,209 x 2
#>    date                 prec
#>    <dttm>              <dbl>
#>  1 1954-12-14 07:40:00     0
#>  2 1954-12-14 07:45:00     0
#>  3 1954-12-14 07:50:00     0
#>  4 1954-12-14 07:55:00     0
#>  5 1954-12-14 08:00:00     0
#>  6 1954-12-14 08:05:00     0
#>  7 1954-12-14 08:10:00     0
#>  8 1954-12-14 08:15:00     0
#>  9 1954-12-14 08:20:00     0
#> 10 1954-12-14 08:25:00     0
#> # … with 48,199 more rows

The following code can be used to:

1. Fill the time-series.
2. Compute the rainfall erosivity values per storm.
3. Filter the above values using cumulative precipitation height and maximum 15 minutes intensity rules.

ei_values <- prec5min %>%
  hyet_fill(time_step = 5, ts_unit = "mins") %>%
  hyet_erosivity(time_step = 5) %>%
  filter(cum_prec > 12.7 | max_i15 > 4 * 6.4)

ei_values
#> # A tibble: 29 x 9
#>    begin               end                 duration cum_prec max_i15
#>    <dttm>              <dttm>              <drtn>      <dbl>   <dbl>
#>  1 1955-04-14 14:10:00 1955-04-14 23:40:00 575 mins     37.8    24.4
#>  2 1955-04-15 15:55:00 1955-04-16 06:30:00 880 mins     22.5    10.8
#>  3 1955-05-11 13:30:00 1955-05-11 19:00:00 335 mins     26.8    25.6
#>  4 1955-07-15 14:50:00 1955-07-15 15:45:00  60 mins     18.9    52.4
#>  5 1955-08-30 14:30:00 1955-08-30 16:35:00 130 mins     23.1    30.4
#>  6 1955-09-02 12:05:00 1955-09-02 17:25:00 325 mins     24.1    38.4
#>  7 1955-09-04 13:10:00 1955-09-04 15:25:00 140 mins     19.4    17.6
#>  8 1955-09-28 18:05:00 1955-09-28 19:30:00  90 mins     35.3    60  
#>  9 1955-10-01 11:40:00 1955-10-01 19:50:00 495 mins     18.5    17.6
#> 10 1955-10-07 14:50:00 1955-10-08 06:10:00 925 mins     76.4    26  
#> # … with 19 more rows, and 4 more variables: max_i30 <dbl>,
#> #   total_energy <dbl>, erosivity <dbl>, eros_density <dbl>

After the calculation of \(EI30\) values the \(R\) coefficient can be computed with:

# add years and months variables
ei_values <- ei_values %>%
  mutate( year = year(begin))

# compute R
R_coeff <- ei_values %>%
  group_by(year) %>%
  summarise(R = sum(erosivity)) %>%
  ungroup() %>%
  summarise(R = mean(R)) %>%
  unlist()

R_coeff
#>        R 
#> 1217.456

References

Brown, LC, and GR Foster. 1987. “Storm Erosivity Using Idealized Intensity Distributions.” Transactions of the ASAE 30 (2). American Society of Agricultural; Biological Engineers: 379–0386.

McGregor, K.C., Ron Bingner, A.J. Bowie, and G.R. Foster. 1995. “Erosivity Index Values for Northern Mississippi” 38 (January): 1039–47.

Renard, Kenneth G, George R Foster, Glenn A Weesies, and Jeffrey P Porter. 1991. “RUSLE: Revised Universal Soil Loss Equation.” Journal of Soil and Water Conservation 46 (1). Soil; Water Conservation Society: 30–33.

Wischmeier, Walter H, and Dwight D Smith. 1958. “Rainfall Energy and Its Relationship to Soil Loss.” EOS, Transactions American Geophysical Union 39 (2). Wiley Online Library: 285–91.