Dependence of estimated precipitation frequency and intensity on data resolution

Abstract

Precipitation frequency (F) and intensity (I) are important characteristics that climate models often fail to simulate realistically. Their estimates are highly sensitive to the spatial and temporal resolutions of the input data and this further complicates the comparison between models and observations. Here, we analyze 3-hourly precipitation data on a 0.25° grid from two satellite-derived datasets, namely TRMM 3B42 and CMORPH_V1.0, to quantify this dependence of the estimated precipitation F and I on data resolution. We then develop a simple probability-based relationship to explain this dependence, and examine the spatial and seasonal variations in the estimated F and I fields. As expected, precipitation F (I) increases (decreases) with the size of the grid box or time interval over which the data are averaged, but the magnitude of this change varies with location, and is strongest in the tropics and weakest in the subtropics. Our simple relationship can quantitatively explain this dependence of the estimated F and I on the spatial or temporal resolution of the input data. This demonstrates that large differences in the estimated F and I can arise purely from the differences in the spatial or temporal resolution of the input data. The results highlight the need to have similar resolution in comparing two datasets or between observations and models. Our estimates show that extremely low frequencies (<1%) are seen over the subtropics while the highest frequencies (20–40%) are located mostly over the tropics, and that the high frequency results from both longer and more frequent precipitation events. Precipitation intensity is more uniformly distributed than frequency. Strong correlations between the amount and frequency confirm the notion that the frequency plays a bigger role than intensity in determining precipitation variations.

Appendix: A simple statistical relationship for the scale-dependence of precipitation frequency

What is the reason for the dependence of estimated precipitation frequency (F) and intensity (I) on the spatial and temporal resolution of the data used? Intuitively, larger grid boxes are more likely to capture some precipitation than smaller ones over a given time period (Kedem and Chiu 1987). For instance, a 0.5° grid box is four times larger than a 0.25° gird box, but the possibility of precipitation (i.e., precipitation frequency) is not four times larger for a 0.5° grid box than for a 0.25° grid box because precipitation events over different grid boxes are not independent (e.g., they could occur at the same time over two or more 0.25° grid boxes within the 0.5° grid box), and they are often correlated spatially (Bell et al. 1990). Here, we aim to derive the relationship between the precipitation frequencies for a 0.25° grid box and a 0.5° grid box based on a pure probability consideration. The same kind of relationship applies for the dependence on temporal resolutions as well. In the following, we propose a simple statistical relationship based on basic probability concepts to explain the dependence of estimated precipitation frequency on the spatial and temporal resolutions of the data. The purpose of the this exercise is two-folds: first to demonstrate that the frequency on two different grids are analytically linked and second to test whether the simple relationship we develop actually works as expected.

We first consider the dependence on the spatial resolution of the precipitation data. As illustrated in Fig. 18, the frequency of precipitation events for a coarse 0.5° grid box is related to the frequency for the finer 0.25° grid boxes inside it. Assuming the frequency (or probability) of precipitation events for the small boxes are P₁, P₂, P₃ and P₄ (which can be calculated using the TRMM or CMORPH data), then the frequency for the lower two boxes (i.e., box 1 and 2 combined) is

$${P_{12}}={P_1}\mathop \cup \nolimits {P_2}=~{P_1}+{P_2} - {P_1}\mathop \cap \nolimits {P_2}$$

(1)

where P₁∩P₂ is the overlapping frequency accounting for the events that occur in both box 1 and 2 at the same time. This frequency is included in both P₁ and P₂ and thus needs to be subtracted once from the sum. The same reasoning applies for the upper boxes (i.e., box 3 and 4 combined):

$${P_{34}}={P_3}\mathop \cup \nolimits {P_4}={P_3}+{P_4} - {P_3}\mathop \cap \nolimits {P_4}$$

(2)

Fig. 18

A schematic diagram illustrating the overlapping (i.e., concurring) part of the precipitaiton events over two small grid boxes (denoted as P1–P4) shown in the top-left panel. P12 and P34 represent the occurrence frequency for the lower and upper part of the larger box. The circles represent the occurence frequency of precipitation events for each of the indicated boxes, and the overlapped area of the circles represent the concurring part of the frequency (i.e., the frational times when precipitaiton happens over both boxes)

Similarly, the frequency for the big 0.5° grid box can be calculated as

$$P={P_{12}}+{P_{34}} - {P_{12}}\mathop \cap \nolimits {P_{34}}$$

(3)

Substitute Eqs. (1, 2) into Eq. (3), but use $P_{{12}} \mathop \cap \nolimits^{} P_{{34}} = \left( {P_{1} \mathop \cup \nolimits^{} P_{2} } \right)\mathop \cap \nolimits^{} \left( {P_{3} \mathop \cup \nolimits^{} P_{4} } \right)$ and $\left( {a\mathop \cup \nolimits b} \right)\mathop \cap \nolimits c=a\mathop \cap \nolimits c+b\mathop \cap \nolimits c - a\mathop \cap \nolimits b\mathop \cap \nolimits c$ to expand this term, we have

$$\begin{aligned}P &= \left( {~{P_1}+{P_2}+{P_3}+{P_4}} \right)~~ \hfill \\ & \quad - \left({{P_1}\mathop \cap \nolimits {P_2}+{P_1}\mathop \cap \nolimits {P_3}+{P_1}\mathop \cap \nolimits {P_4}+{P_2}\mathop \cap \nolimits {P_3}+{P_2}\mathop \cap \nolimits {P_4}+{P_3}\mathop \cap \nolimits {P_4}} \right) \hfill \\ & \quad +\left( {{P_1}\mathop \cap \nolimits {P_2}\mathop \cap \nolimits {P_3}+{P_1}\mathop \cap \nolimits {P_2}\mathop \cap \nolimits {P_4}+{P_1}\mathop \cap \nolimits {P_3}\mathop \cap \nolimits {P_4}+{P_2}\mathop \cap \nolimits {P_3}\mathop \cap \nolimits {P_4}} \right) \hfill \\ & \quad - \left( {{P_1}\mathop \cap \nolimits {P_2}\mathop \cap \nolimits {P_3}\mathop \cap \nolimits {P_4}} \right). \hfill \\ \end{aligned}$$

(4)

In Eq. (4), P₁ ∩ P₂ ∩ P₃ is for the frequency when precipitation occurs over all the 3 boxes at the same time (i.e., the frequency of joint occurrences), and similarly for the other terms with multiple “∩”. These terms can be calculated directly using the data on the 0.25° grid. Thus, the precipitation frequency for a 0.5° box is analytically linked to the precipitation frequencies (P1 to P4 in Eq. 4) for the 0.25° boxes within the 0.5° box. We can use Eq. (4) and the frequencies (including the frequencies of joint occurrences) calculated using the TRMM or CMOPRH data on a 0.25° grid to estimate the frequency on a 0.5° grid, and compare this estimate with that calculated directly using precipitation rates averaged onto the 0.5° grid. Strictly speaking, the threshold used to define the precipitation events should also be slightly different for the 0.25° and 0.5° cases if we want to use Eq. (4), except when it is zero. Thus, we will simply use a threshold of 0 mm/h in calculating the frequency on both the 0.25° and 0.5° grids using the TRMM 3B42 3-hourly data. The results are shown in Fig. 9 and discussed in Sect. 4.3.

Please note that Eq. (4) applies only to a special case where each grid box on the coarse grid contains exactly four grid boxes on the fine grid. The occurrence frequency terms on the fine grid boxes (P₁, P₂, P₃ and P₄) and the other terms of frequencies of joint occurrences in Eq. (4) will need to be calculated by counting the number of such events using the data on the fine grid (e.g., TRMM data on 0.25° grid). Thus these terms will depend on the actual data used.

We will apply similar reasoning to derive the equation linking the frequencies for two different temporal resolutions, namely, for 3-hourly and daily data. Replacing each small box in Fig. 18 with a 3-h time period, and considering the big box as the 12-h period consisting of the four 3-h periods, then Eq. (4) can be used to calculate the frequency over the 12-h period using the frequency calculated using 3-hourly data. The same method can be applied to the other 12-h period of the day. Following the same reasoning, the frequency or probability for the whole day (P) can be estimated as

$$P={P_{1234}}+{P_{5678}} - {P_{1234}}\mathop \cap \nolimits {P_{5678}}$$

(5)

where P₁₂₃₄ and P₅₆₇₈ are, respectively, the frequency for the first (i.e., 00–03, 03–06, 06–09 and 09–12Z) and second (i.e., 12–15, 15–18, 18–21 and 21–24Z) 12-h periods calculated using Eq. (4) and the frequencies estimated using 3-hourly data. The last term in Eq. (5) (i.e., ${P_{1234}}\mathop \cap \nolimits {P_{5678}}$) represents the overlapped frequency for the first and second 12-h periods, and we will estimate it directly using 12-hourly averaged precipitation rates by computing their concurring frequency. Thus, Eq. (5) links the daily precipitation frequency to the precipitation frequency estimated using 3-hourly (and 12-hourly) precipitation rates. This estimate from Eq. (5) can be compared with that calculated directly from daily mean precipitation rates (both on the same grid). For this comparison, we will again use zero as the threshold to simplify the definition of precipitation events. The results are shown in Fig. 10 and discussed in Sect. 4.3.