# Presentation: 404818

## Estimation of sampling error and retrieval error of monthly average rainfall rate observed by TRMM/PR

- *Yasuhisa Iida,
- Ken’ichi Okamoto, and Shoichi Shige
-   (Osaka Prefecture University)

- The 3rd TRMM Meeting in Las Vegas                7  Feb., 2008

- @ JAXA home page

**TRMM/PR space-time average rainfall-rate product includes ****retrieval error**** and ****sampling error****.**

**Notes:**
This is an example of TRMM/PR space-time average rainfall-rate product. TRMM/PR space-time average rainfall-rate product includes retrieval error and sampling error. 

This figure indicates TRMM/PR-based monthly average rainfall-rate product over 5-deg by 5-deg grids in Oct., 2007. This product includes retrieval error and sampling error. Let’s explain these errors in nest slide. 

## Purpose of this study is ....

- To estimate retrieval error and sampling error in TRMM/PR monthly averaged rainfall products using radar-AMeDAS data set around Japan as true values.
- In this presentation, we will report the preliminary results during 3 year [1998-2000].
- To estimate both errors for various area sizes.
- To estimate both errors every calender month.
- To investigate the bias due to the PR rainfall-retrieval algorithm (2A25, Version 6).

**Notes:**
The purpose of this study is to estimate retrieval error and sampling error in TRMM/PR monthly averaged rainfall products using radar-AMeDAS data set around Japan as true values. In this presentation, we will report the preliminary results during 3 year [ from 1998 to 2000]. The contents (of this presentation) is to estimate retrieval error and sampling error for various area sizes, and every calender month. Finally, we will investigate the bias due to the PR rainfall-retrieval algorithm (2A25, Version 6).

## Error of TRMM/PR space-time average rainfall-rate product

- Total Error = PR value – True value
-      =(_PR value – Simulated value_) + (_Simulated value – True value_)

- Retrieval Error

- Sampling Error

**Simulated value ****=****Sampled true value corresponding to space-time domain observed by TRMM/PR swath**

- Time

- PR value           (PR observations)

- Simulated value

- True value    (radar-AMeDAS)

- (due to Rainfall Retrieval Algorithm)

- (due to PR Sparse Sampling)

**Notes:**
We consider error of TRMM/PR space-time average rainfall product. As true value, we assume radar-AMeDAS around Japan. The true value is obtained every hour during one month. On the other hand, PR value is defined as PR-observed rainfall rate. TRMM/PR observes the rainfall discontinuously in space and time like this. Simulated value is defined as sampled true value corresponding to space-time domain observed by TRMM/PR swath. Total error is defined as “PR value minus True value”. Total error is divided into retrieval error and sampling error. Retrieval error is defined as “PR value minus simulated value”, which is due to Rainfall Retrieval Algorithm. Sampling error is defined as “Simulated value minus True value”, which is due to TRMM/PR sparse sampling.
In the next slides, we will define these errors in detail. 

## Notations

- M: Total number of **month**s during 3 years (=36)

- Nt(m): Total number** **of **hour**s in the m-th month

- tm(i): the i-th **time** in the m-th month

- (i=1,2, ... , Nt(m))

_**True value is sampled every hour.**_

- (m=1, 2, ...., M)

**Notes:**

This is Notations of symbols that I use (below) in below slide. (This shows notations of symbols that I use below.) 
True value is sampled every hour.
M is defined as total number of all months during 3 years, which value is equal to 36. (its value takes 36).
Nt(m) is defined as total number of hours in the m-th month. Small m takes these values (m=1, 2, ...., M).
tm(i) represents the i-th time in the m-th month. Small i takes these values (i=1,2, ... , Nt(m)).
Nm is defined as total number of α-degree by α-degree areas around Japan in the m-th month, alpha α is variable. 
Ng represents total number of grids over the g-th area, small g takes these values (g=1,2, ... , Nm).
Small k is grid number over the g-th area, small g takes these values (k=1,2, ... , Ng).

## Notations

- Nm: Total number of    -degree by   -degree **area**s around Japan in the m-th month

- Ng : Total number of **grid**s over the g-th area

- k : **Grid** number over the g-th area

- : variable

- (g=1,2, ... , Nm)

**The g-th area**

**The k-th grid**

-  (degree)

-  (degree)

- (k=1,2, ... , Ng)

**Notes:**

This is Notations of symbols that I use (below) in below slide. (This shows notations of symbols that I use below.) 
True value is sampled every hour.
M is defined as total number of all months during 3 years, which value is equal to 36. (its value takes 36).
Nt(m) is defined as total number of hours in the m-th month. Small m takes these values (m=1, 2, ...., M).
tm(i) represents the i-th time in the m-th month. Small i takes these values (i=1,2, ... , Nt(m)).
Nm is defined as total number of α-degree by α-degree areas around Japan in the m-th month, alpha α is variable. 
Ng represents total number of grids over the g-th area, small g takes these values (g=1,2, ... , Nm).
Small k is grid number over the g-th area, small g takes these values (k=1,2, ... , Ng).

## Monthly averages of true values, PR-observed values, and simulated values

- Ng : Total number of **grid**s over the g-th area

- Nt(m): Total number** **of **hour**s in the m-th month

- rt(k, tm(i)): **True value** at the k-th grid of the g-th area and at the i-th time in the m-th month

**Notes:**
We define monthly averages of true values, simulated values, and PR-observed values.
r_hat,t(g,m) is .... 

Devided by all the observed (sampled) grids
Note that these denominators are different from true-value case and PR-value case. 

## Monthly averages of true values, PR-observed values, and simulated values

- Ng : Total number of **grid**s over the g-th area

- Nt(m): Total number** **of **hour**s in the m-th month

- rPR(k, tm(i)): **PR-observed value** at the k-th grid of the g-th area and at the i-th time in the m-th month

- NPR(k, tm(i)):**Information of PR observation** at the k-th grid of the g-th area and at the i-th time in the m-th month

- (NPR(k, tm(i)) = 1 for PR obs.; NPR(k, tm(i)) = 0 for no PR obs.)

**Notes:**
We define monthly averages of true values, simulated values, and PR-observed values.
r_hat,t(g,m) is .... 

Devided by all the observed (sampled) grids
Note that these denominators are different from true-value case and PR-value case. 

## Monthly averages of true values, PR-observed values, and simulated values

- Ng : Total number of **grid**s over the g-th area

- Nt(m): Total number** **of **hour**s in the m-th month

- rsim(k, tm(i)): **Simulated value** at the k-th grid of the g-th area and at the i-th time in the m-th month

- NPR(k, tm(i)):**Information of PR observation** at the k-th grid of the g-th area and at the i-th time in the m-th month

- (NPR(k, tm(i)) = 1 for PR obs.; NPR(k, tm(i)) = 0 for no PR obs.)

**Notes:**
We define monthly averages of true values, simulated values, and PR-observed values.
r_hat,t(g,m) is .... 

Devided by all the observed (sampled) grids
Note that these denominators are different from true-value case and PR-value case. 

## Total, retrieval, and sampling errors over the g-th area in the m-th month

-                  : **Total error** over the g-th area in the m-th month

- : **Retrieval error** over the g-th area in the m-th month

- : **Sampling error** over the g-th area in the m-th month

**Notes:**
εtotal(g, m) represents Total error over the g-th area in the m-th month, which is defined as r_hat, PR(g,m) minus r_hat,true(g,m)

## Definition of total, retrieval, and sampling errors

**Total Error**: Etotal/R(%)

- Nm: Total number of      -degree by    -degree **area**s around Japan in the m-th month

- M: Total number of **month**s during 3 years (=36)

- R: **Average rainfall rate**

**Notes:**
Finary, we represent definition of total, retrieval, sampling errors, used in hereafter slides.
R is Average rainfall rate for 3 years around Japan, which is defined by this equation.
Total error is defined as root-mean-square total error Etotal for 3 years and all areas around Japan, as a percentage of average rainfall rate R.
(Mean-square total error E_total^2 for 3 years and Japanese areas is defined by this equation,)
Retrieval error is difined as the root-mean-square retrieval error for 3 year and all areas around Japan, which is divided by average rainfall rate R. 
Sampling error is difined as ...

## Definition of total, retrieval, and sampling errors

**Sampling Error**: Esamp/R(%)

**Retrieval Error**: Eretr/R(%)

- Nm: Total number of      -degree by    -degree **area**s around Japan in the m-th month

- M: Total number of **month**s during 3 years (=36)

- R: **Average rainfall rate**

**Notes:**
Finary, we represent definition of total, retrieval, sampling errors, used in hereafter slides.
R is Average rainfall rate for 3 years around Japan, which is defined by this equation.
Total error is defined as root-mean-square total error Etotal for 3 years and all areas around Japan, as a percentage of average rainfall rate R.
(Mean-square total error E_total^2 for 3 years and Japanese areas is defined by this equation,)
Retrieval error is difined as the root-mean-square retrieval error for 3 year and all areas around Japan, which is divided by average rainfall rate R. 
Sampling error is difined as ...

## Data

- radar-AMeDAS (RA) rainfall-rate data sets
- Hourly available data surrounding Japanese regions (In this study, we use the 3-year [1998-2000] rainfall rates.)
- Original spatial resolution of longitude 0.0625-deg x latitude 0.05-deg grids (about 5-km x 5-km grids)
- TRMM/PR-derived instantaneous rainfall rates
- 2A25 (TRMM/PR standard algorithm), Version 6
- Near surface rain

**Notes:**
In this study, we use radar-AMeDAS dataset and TRMM/PR-derived rainfall rates. 
We use these two data sets: radar-AMeDAS and TRMM/PR-derived rainfall rates. Radar-Amedas is hourly available data ..... Radar-amedas has original resolution of  .... TRMM/PR data sets are derived from TRMM/PR standard rainfall-retrieval algorithm 2A25, Version 6. We use near surface rainfall. 

## True value=radar-AMeDAS (RA)

**              **

- 18(UTC), 10/June 2000

- Monthly Average   (June, 2000; m=30)

**(c) True value****(****radar-AMeDAS****)**

**(a)PR observation**

**(b)Simulated value **

**Notes:**
This is the example of PR observation, simulated value, and true value at this time (18 UTC, June 10th, 2000). 
We average these values in this month (June, 2000), and display here. 

## Error Estimation (0.5-deg. by 0.5-deg. areas)

- 95%

- rt(g,m)

- rPR(g,m)

- Etotal/R(%):

- Retrieval Error (%)

- rsim(g,m)

- rsim(g,m)

- rt(g,m)

- 86%

- 88%

- Eretr/R(%):

- Esamp/R(%):

- 16473

- R=0.29(mm/h)

- Total Error (%)

- Number of averaging data:

- Average rainfall rate

- rt(g,m), rPR(g,m), rsim(g,m): **Monthly average** of true value, PR value, simulated value **(mm/h)**

- Sampling  Error (%)

- rPR(g,m)

**Notes:**
The results of error estimation for this area size (0.5-degree by 0.5-degree) is shown in these figures (of this slide). 
This is Retrieval error, (this is) sampling error, and (this is) total error. Total number of averaging data, average rainfall rate, and the estimated (value of) error are displayed here. Retrieval error is a bit smaller than sampling error. 

## Error Estimation (1.0-deg. by 1.0-deg. areas)

- Total Error (%)

- 79%

- rt(g,m)

- rPR(g,m)

- Etotal/R(%):

- Retrieval Error (%)

- Sampling  Error (%)

- rsim(g,m)

- rPR(g,m)

- rsim(g,m)

- rt(g,m)

- 69%

- 67%

- Eretr/R(%):

- Esamp/R(%):

- 3409

- R=0.29(mm/h)

- Number of averaging data:

- Average rainfall rate

- rt(g,m), rPR(g,m), rsim(g,m): **Monthly average** of true value, PR value, simulated value **(mm/h)**

**Notes:**
Next we display the results (of error estimation) for this area size (1.0-degree by 1.0-degree) (in these figures).
The outline of this slide (figure) is as same as the previous one. Retrieval error is a bit larger than sampling error. 

## Error Estimation (2.0-deg. by 2.0-deg. areas)

- Total Error (%)

- 64%

- rt(g,m)

- rPR(g,m)

- Etotal/R(%):

- Retrieval Error (%)

- Sampling  Error (%)

- rsim(g,m)

- rPR(g,m)

- rsim(g,m)

- rt(g,m)

- 58%

- 47%

- Eretr/R(%):

- Esamp/R(%):

- 556

- R=0.32(mm/h)

- Number of averaging data:

- Average rainfall rate

- rt(g,m), rPR(g,m), rsim(g,m): **Monthly average** of true value, PR value, simulated value **(mm/h)**

**Notes:**
Next we display (This is) the results (of error estimation) for this area size (2.0-degree by 2.0-degree) (in these figures).
The outline of this slide (figure) is as same as the previous one. In this case, Retrieval error is larger than sampling error.

## Error Estimation (4.0-deg. by 4.0-deg. areas)

- Total Error (%)

- 48%

- rt(g,m)

- rPR(g,m)

- Etotal/R(%):

- Retrieval Error (%)

- Sampling  Error (%)

- rsim(g,m)

- rPR(g,m)

- rsim(g,m)

- rt(g,m)

- 41%

- 33%

- Eretr/R(%):

- Esamp/R(%):

- 148

- R=0.29(mm/h)

- Number of averaging data:

- Average rainfall rate

- rt(g,m), rPR(g,m), rsim(g,m): **Monthly average** of true value, PR value, simulated value **(mm/h)**

**Notes:**
This is the results for this area size. In this case, Retrieval error is larger than sampling error.

Next we display the results (of error estimation) for this area size (4.0-degree by 4.0-degree) (in these figures).
The outline of this slide (figure) is as same as the previous one. Retrieval error is larger than sampling error.

## Error Estimation (5.0-deg. by 5.0-deg. areas)

- Total Error (%)

- 42%

- rt(g,m)

- rPR(g,m)

- Etotal/R(%):

- Retrieval Error (%)

- Sampling  Error (%)

- rsim(g,m)

- rPR(g,m)

- rsim(g,m)

- rt(g,m)

- 35%

- 22%

- Eretr/R(%):

- Esamp/R(%):

- 54

- R=0.25(mm/h)

- Number of averaging data:

- Average rainfall rate

- rt(g,m), rPR(g,m), rsim(g,m): **Monthly average** of true value, PR value, simulated value **(mm/h)**

**Notes:**
Next we display the results (of error estimation) for this area size (5.0-degree by 5.0-degree) (in these figures).
The outline of this slide (figure) is as same as the previous one. Retrieval error is larger than sampling error.

## Area-Size Dependence of Errors

- Smaller error as larger area size
- Magnitude relation between sampling error and retrieval error changes with area size.

| Area Size (α-deg x α-deg) | Sampling Error        Esamp /R  | Retrieval Error   Eretr /R  | Total Error   Etotal /R | R(mm/h) | Number of data |
| --- | --- | --- | --- | --- | --- |
| 0.5x0.5 | 88% | 86% | 95% | 0.29 | 16473 |
| 1.0x1.0 | 67% | 69% | 79% | 0.29 | 3409 |
| 2.0x2.0 | 47% | 58% | 64% | 0.32 | 556 |
| 4.0x4.0 | 33% | 41% | 48% | 0.29 | 148 |
| 5.0x5.0 | 22% | 35% | 42% | 0.26 | 54 |

**Notes:**
We summerize the previous results (shown in those previous slides) in this table. Area size, sampling error, ....., average rainfall rate R, total number of averaging data. 
We can find smaller error as larger grid size. Magnitude relation between sampling error and retrieval error changes with area size. 

## Monthly Errors (1998-2000, 1.0-deg. areas)

**Sampling Error****(****%****)**

**Retrieval Error****(****%****)**

**Total Error****(****%****)**

**Smaller errors in rainy summer seasons (larger rainfall amounts),              Larger errors in dry winter seasons (smaller rainfall amounts)**

**E****total**** (I)/R(I)**

**E****retr**** ****(****I****)**** ****/R ****(****I****)**

**E****samp ****(****I****)**** ****/R ****(****I****)**

- Seasonal Dependence

- 40

- 115

- 100

- 80

- 60

- 1     2     3     4      5     6     7     8      9    10   11   12

- Estimate errors every calendar month

- Calendar Month  I

**Notes:**
We estimate total, sampling, retrieval errors every calendar month.  For example, In Jan, from 1998 to 2000, we average the (total) errors for three months and Japanese areas. The root-mean-square total error in the I-th calendar month is devided by R(1), which is average of true values in the same calendar month. Sampling, and retrieval errors are obtained (defined) similarly. The error estimates every calendar month (12 calendar months) are shown in this figure. We can find seasonal dependence, that is, smaller error in rainy summer seasons, larger error in dry winter seasons.    

## Retrieval Bias = PR obs. – Simulated Value

- PR algorithm overestimate or underestimate the rainfall  compared with radar-AMeDAS?
**Evaluate the bias due to PR-retrieval algorithm:**

- _Definition of Bias:_

**Retrieval Bias**: Bretr/R(%)

- R:**Average rainfall rate**

- : Monthly average of **true values** over the g-th area in the m-th month

- : Monthly average of **PR-observed values** for the same area and month

- : Monthly average of **simulated values** for the same area and month

- M: Total number of **month**s for 3 years (=36)

- Nm: Total number of α-degree by α-degree **area**s around Japan in the m-th month

**Notes:**
Does PR algorithm overestimate or underestimate the rainfall compared with radar-AMeDAS?
We evaluate the bias due to PR-retrieval algorithm, which is defined as this equation. Notations are as same as the previous slides. Retrieval Bias is 3-year and all-areas average of monthly average of PR-observed values minus simulated value, as a percentage of average rainfall rate R. 

## Retrieval Bias estimation for area sizes

- PR algorithm underestimates the rainfall compared with radar-AMeDAS.
- Retrieval Bias is almost independent of area size.

| Area Size (α-deg x α-deg) | Retrieval Bias     Bretr /R  | R(mm/h) | Number of data |
| --- | --- | --- | --- |
| 0.5x0.5 | -25% | 0.29 | 16473 |
| 1.0x1.0 | -26% | 0.29 | 3409 |
| 2.0x2.0 | -29% | 0.32 | 556 |
| 4.0x4.0 | -26% | 0.29 | 148 |
| 5.0x5.0 | -26% | 0.26 | 54 |

**Notes:**
This is the eatimated retrieval bias for various area sizes. PR algorithm underestimates the rainfall compared with radar-AMeDAS. Retrieval bias do not change with area size.  

## Monthly Bias (1998-2000, 1.0-deg. areas)

- Monthly Bias (1998-2000, 1.0-deg. areas)

**Retrieval Bias****(****%****)**

- PR algorithm largerly underestimates in the rainy summer months

**B****retr**** ****(****I****)**** ****/ R ****(****I****)**

- Seasonal Dependence

- Sampling bias (%)

- -40

- 10

- 0

- -10

- -30

- -20

- Estimate retrieval bias every calendar month

- 1     2     3     4      5     6     7     8      9    10   11   12

- Calendar Month I

**Notes:**
We estimate retrieval bias every calendar month. The estimation method is as same as the total, retreival, sampling errors. We display the retrieval bias every calendar month. Seasonal dependence was found, PR algorithm largerly underestimates in the rainy summer months. 

## Concluding Remarks

- Retrieval error as well as sampling error are smaller as larger area size.
- Magnitude relation between sampling error and retrieval error changes with area size.
- Retrieval error is larger than sampling error for 2.0-5.0 degree areas.
- Retrieval error as well as sampling error depend on season: larger errors in dry winter seasons, smaller errors in rainy summer seasons.
- The PR algorithm underestimates rainfall compared with radar-AMeDAS annually for any area size.
- Seasonal dependence was found: it largely underestimates in rainy summer months (June-Sep.).

## Appendix

## Relation among total, retrieval, and sampling errors

- : Total error over the g-th area in the m-th month

- : Retrieval error over the g-th area in the m-th month

- : Sampling error over the g-th area in the m-th month

- (if                       and                           are uncorrelated )