Probable Maximum Precipitation Estimation Using the Revised $K_m$-Value Method in Hong Kong

Probable maximum precipitation (PMP) has been defined as “the greatest depth of precipitation for a given duration meteorologically possible for a design watershed or a given storm area at a particular location at a particular time of year, with no allowance made for long term climatic trends” (WMO 2009). Hydrologists use PMP to estimate the probable maximum flood (PMF) in the design of a particular project in a given watershed, such as the height and reservoir storage capacity of a dam or dimension of flood-carrying structures (spillway and flood carrying tunnel). In this way, the risks of loss of life and damage could be reduced.

The available methods for PMP estimation are mainly categorized in two ways: hydrometeorological estimation methods and statistical methods in practice. Hydrometeorological estimation methods can be generally classified as follows (WMO 2009; Lin and AECOM 2014): a storm model approach (Collier and Hardaker 1996), moisture maximization (Micovic et al. 2015), a generalized method, storm transposition, and a depth-area-duration (D-A-D) method. The statistical estimation is a possible approach, a kind of modified frequency analysis based on historical precipitation records. Among several statistical procedures, Hershfield’s $K_m$-value method is the best known (National Research Council 1985; WMO 1986, 2009). It was initially developed by Hershfield (1963), who defined a frequency factor $K_m$ omitting the observed maximum value from a sample series for the estimation of PMP. Hershfield combined samples by either including or excluding the maximum value and produced some nomographs with adjustments. Koutsoyiannis and Papalexiou (2016) set up mathematical expressions of Hershfield’s adjustment factors through these nomographs. Subsequently, Hershfield (1965) improved the method by constructing an empirical graph with $K_m$ varying between 5 and 20 depending on the mean value of annual rainfall maxima and rainfall duration. Preliminary appraisal of this technique in Canada (Bruce and Clark 1966) and in the U.S. (Myers 1967) showed that the PMP estimates obtained by this approach are closely comparable to those obtained by the conventional moisture maximization and storm transposition methods. According to Wiesner (1970), this method has the advantage of taking into account the entire rainfall data set, expressing it in terms of statistical parameters. Using the same data set as Hershfield used, Koutsoyiannis (1999) fitted a generalized extreme value (GEV) to the frequency factors and found that the highest value, 15, of $K_m$ corresponds to a 60,000 year return period event. Papalexiou and Koutsoyiannis (2006) studied the maximum precipitation depths derived by moisture maximization at a few stations in the Netherlands. They concluded that a probabilistic approach to estimating extreme precipitation values is more consistent with natural behavior and provides a better basis for estimation. The World Meteorological Organization (WMO) in its various manuals and technical publications (WMO 1969, 1986, 2009) has also recommended this method for estimation of point PMP for those river basins whose daily rainfall data are available for a long period of time.

However, a standardized variable, ${\mathrm{\varphi }}_m$, the maximum deviation from the mean of a sample scaled by the standard deviation of the sample, derived directly from the frequency equation, was used to replace $K_m$ in China. Its envelope value was used by some researchers to calculate the PMP (Wang 1999; Hua et al. 2007) in China. They recommended that ${\mathrm{\varphi }}_m$ was more reasonable and consistent with the frequency equation. In contrast, Lin (1981) and Lin and Vogel (1993) recommended that $K_m$ replace ${\mathrm{\varphi }}_m$ under some circumstances, and Lin used this revised method to estimate the PMP of the Zhelin Reservoir on the Xiushui River in Jiangxi Province, China, in 1981 and in Austin, Texas, U.S., in 1993.

Regarding the fact that Hershfield’s $K_m$-value method has been used in China, the purpose of this study was to determine the differences and relationships between the frequency factor $K_m$ defined by Hershfeild and the standardized variable ${\mathrm{\varphi }}_m$. In this paper, Hershfield’s $K_m$-value method will be reviewed with some criteria developed for using the method in a reasonable way. The advantages and disadvantages of this revised $K_m$-value method will also be discussed in connection with a case study on the estimation of 24 h PMP in Hong Kong.

Hershfield’s procedure of estimating PMP was originally developed based on the general frequency equation (Chow 1951)where $X_T$ = rainfall for return period $T$; ${\overline{X}}_n$ and $S_n$ = mean and standard deviation of a series of $n$ annual maximum rainfall values for a given duration; and $K$ = frequency factor associated with $T$ and the probability distribution selected to fit the data. If the maximum observed rainfall, $X_m$, is substituted for $X_T$, $K_m$ for $K$, Eq. (1) could be as follows:

Then $K_m$ is the number of standard deviations to be added to the mean to obtain $X_m$. $K_m$ is computed using the equationwhere $X_m$ = highest value for this series; and ${\overline{X}}_{n-1}$ and $S_{n-1}$ are respectively the mean and standard deviation excluding the highest value from the series. By enveloping $K_m$ from a large number of computed $K_m$, Hershfield estimated PMP using Eq. (4):

The statistical expressions for PMP are rederived to obtain a better understanding of the method. The standardized variable $\varphi$ is introduced to replace $K$ from Eq. (1), soandwhere ${\varphi }_m$ = standardized variable or maximum deviation from mean scaled by its standard deviation $S_n$. It is computed directly from the equation

For the distribution of a series of $n$ annual rainfall maxima we havewhere $EX$ and $\sigma$ = expected value and population standard deviation, respectively. ${\varphi }_M$ is the maximum value of ${\varphi }_m$. To estimate $X_{\text{PMP}}$, the population parameters in Eq. (8) must be substituted, i.e., $EX$ by ${\overline{X}}_n$ and $\sigma$ by $S_n$. The true value of the population parameter ${\varphi }_M$ would be unknown before $X_{\text{PMP}}$ happened. If the estimate for $K_m$ is the same as that presented by Hershfield, then $K_m$ is defined as the number of standard deviations to be added to the mean to obtain $X_m$ of the incomplete sample series ($X_1,X_2,\dots ,X_{n-1}$), andwhere ${\overline{X}}_{n-1}$ and $S_{n-1}$ = mean and standard deviation of the incomplete sample series, respectively. Thus, the equation for calculating $K_m$ isThe relationship between ${\overline{X}}_{n-1}$ and ${\overline{X}}_n$ is ${\overline{X}}_n=[(n-1){\overline{X}}_{n-1}+X_m]/n$. The definition of $S_n$ is $S_n^2=1/n-1\sum _{i=1}^n{(X_i-{\overline{X}}_n)}^2$. Combining the foregoing two equations with Eqs. (7) and (10), Eq. (11) can be derived. The detailed derivations of Eq. (11) are included in the Appendix:

Since both $K_m$ and ${\mathit{\varphi }}_m$ are always greater than 0, making $C_1={(n-1)}^3/n^2(n-2)$, $C_2=n-1/n(n-2)$, Eq. (11) can be rewritten asor

Using Eq. (12) [or Eq. (13)], along with the obvious inequalities $0<1/K_m\le 1$ and $0<1/{\varphi }_m\le 1$, the following inequalities, which restrict the range of $K_m$ and ${\varphi }_m$, can be derived:

The lower limit of $K_m$ is approximately 1, while the upper limit of ${\varphi }_m$ is approximately $\sqrt{n}$. Obviously, $K_m$ is always greater than ${\mathrm{\varphi }}_m$. However, the difference in value between $K_m$ and ${\mathrm{\varphi }}_m$ gets smaller as $n$ increases. This means that the frequency factor $K_m$ approaches the upper limit ${\mathrm{\varphi }}_M$ when the sample size $n$ increases. For example, in the case of ${\mathrm{\varphi }}_m=5$ and ${\mathrm{\varphi }}_m=10$, Tables 1 and 2 show the variations of $K_m$. As $n$ increases, $K_m$ approaches ${\mathrm{\varphi }}_m$. Therefore, statistically, $K_m$ has a consistent relationship with ${\mathrm{\varphi }}_m$ and can be used in the estimation of PMP. For practical use, Eq. (4) is applied to estimate PMP. Because the standard deviation $S_n$ is defined as the coefficient of variation $C_{vn}$ multiplied by the mean ${\overline{X}}_n$, Eq. (4) can be written

Since Eq. (12) requires $(C_1-C_2{\mathrm{\varphi }}_m^2)>0$, $n$ should be greater than or equal to ${\mathrm{\varphi }}_m^2+2$. Therefore, the minimum data size $N_m$ is deduced a criterion to use the $K_m$-value eligibly

From Eq. (12) [or Eq. (13)] the following relation can also be derived:where $N_s$ is defined as the stable data size needed to give statistically reliable results for a particular ${\mathrm{\varphi }}_m$. If the ratio of $K_m$ to ${\mathrm{\varphi }}_m$ has a value of 1.1, or a relative error of 10%, the relation in Eq. (18) can be further approximated as

Eq. (19) can be used to perform a quick PMP estimate with respect to the length of data required to develop a stable estimate. For instance, it requires 104 years of data records as stable data size if ${\mathrm{\varphi }}_m=4$, referring to Fig. 1. Furthermore, when $N_s>3.5n$, it could raise the uncertainties of $K_m$, usually up to 50%. Hence, these stations should be discarded to reduce the computation burden as such stations will contribute nothing to the statistical results except errors.

The distribution of the sample mean, ${\overline{X}}_n$, approaches normal distribution as $n$ increases (Lin 1978). Let the correction factor of the population mean $EX$ to the sample mean ${\overline{X}}_n$ be $R_n$, which can be expressed as

If there was no sampling error, the $R_n$ would be exactly equal to 1. However, due to sampling error $R_n$ may be greater or less than 1 with equal chance. By taking a confidence level of 99.7%, i.e., the $X_n$ is varying within a confidence interval of $\pm 3{\sigma }_{\overline{X}}$ or $\pm 3{\overline{X}}_n{C}_{vn}/\sqrt{n}$. Then $R_n$ could be expressed as $(1+3C_{vn}/\sqrt{n})$ for maximum adjustment, and the adjustment ${\overline{X}}_n$ iswhere ${\overline{X}}_n^{\prime }$ = adjusted sample mean of $EX$. The relations of $R_n$ with $C_{vn}$ and $n$ are given in Table 3. It shows that $R_n$ decreases toward 1 while $C_{vn}$ decreases when $n$ is fixed; and $R_n$ decreases toward 1 while $n$ increases when $C_{vn}$ is certain.

Precipitation data are usually observed at fixed time intervals. For example, 8 a.m. to 8 a.m. means daily, and such data rarely yield the true maximum rainfall amounts for the indicated duration 24 h. As we know, the annual maximum observational day rainfall amounts are very likely to be appreciably less than the annual maximum 24 h rainfall amounts determined by 1,440 consecutive minutes with unrestricted beginning and ending time. Studies of thousands of station-years of rainfall data indicate that a conversion factor of 1.13 to convert 1-day rainfall data to 24 h is reasonable to be used in practice (Weiss 1964).

Now, if we use this revised $K_m$-value method to estimate PMP at a single station or over a small watershed, it should be taken by the following steps:

It is noted that the essence of the reviewed $K_m$-value method is storm transposition, but instead of transposing the specific rainfall amount of one storm, an abstracted statistic $K_m$ is transposed (WMO 2009). In the study, other parameters such as ${\overline{X}}_n^{\prime }$ and $C_{vn}$ are also regionalized within the study area for the sake of maximization.

In Hong Kong, PMP was first derived by Bell and Chin (1968). They used depth-area-duration (DAD) method together with moisture maximization to estimate the 24 h PMP from 21 major storms between 1955 and 1965. They further updated their PMP estimates through the introducing a seasonal adjustment factor. Hong kong Observatory (1999); Chang and Hui (2001) used the same method but without seasonal adjustment to analyze 53 storms occurring between 1966 and 1999, and got a result which was smaller than the former one. The statistical method was not used in the PMP estimates in Hong Kong before. Hence, the revised $K_m$-value method is applied to estimate 24 h PMP in Hong Kong for the first time. The 5 min data or hourly data of 64 rain-gauge stations in Hong Kong were selected. The annual maximum precipitation series of 24 h for each rain-gauge station was obtained by using a 24 h sliding window over the 5 min or hourly data. Since Shenzhen is very close to Hong Kong, the annual maximum precipitation series of Shenzhen acquired from the Hydrologic Bureau of Guangdong Province was also used in the study. Table 4 gives us the detailed information of all the rain-gauge stations. Fig. 2 shows the locations of the rain-gauge stations.

As described above, firstly, each rain-gauge station was displayed in descending order of maximum value, $X_m$. Secondly, the minimum data size, $N_m$ was calculated for each. And all the rain-gauge stations meet the criterion. Thirdly, by applying the criterion of a stable data size $N_s$, N09 and N15 were discarded. Following, the values of $K_m$, $C_{vn}$, and ${\overline{X}}_n$ for each eligible rain-gauge station are calculated. Since the annual maximum precipitation series for 24 h is directly obtained by a 24 h sliding window, which is true maxima for 24 h. There was no need to apply a conversion factor of 1.13 to the mean value of stations in Hong Kong. However, the mean daily value series of Shenzhen was multiplied by 1.13 to convert 1-day data to 24 h. Furthermore, each eligible rain-gauge station need a maximized adjustment of mean in terms of sampling error at a confidence level of 99.7%. Finally, the highest values of ${\overline{X}}_n$, $K_m$, and $C_{vn}$ were obtained. They were 426.30 (of N14), 5.46 (of N14), 0.57 (of R11), respectively. Therefore, according to the Eq. (16), the 24 h PMP estimate for Hong Kong is approximately 1,753 mm.

Table 5 shows the results of parameters of all those rain-gauge stations in descending order of $X_m$. It is shown that the highest values of ${\overline{X}}_n$ and $K_m$ come from the same station N14. As we know, rainfall is quite affected by topography, and rainfall in mountain areas is mostly larger than in plain areas. Observation value at automatic station may be a little higher than the ordinary observation station. N14 is an automatic station located in Tai Mo Shan, the highest elevation in Hong Kong. Therefore, the rainfall amount at station N14 is always larger than the other rain-gauge stations.

Although the frequency factor $K_m$ is always greater than the standardized variable, ${\mathrm{\varphi }}_m$, it usually rapidly approaches ${\mathrm{\varphi }}_M$ when $n$ increases. In this study, it has already proved that $K_m$ had an obvious statistical relationship with ${\mathrm{\varphi }}_m$. Therefore, some stations are discarded by applying the criteria of the minimum sample size $N_m$ and the stable sample size $N_s$ to reduce the computation burden. And the highest values of ${\overline{X}}_n$, $K_m$, and $C_{vn}$ are calculated and selected in parameter regionalization. Finally, the PMP at a single station or over a small watershed could be estimated by applying the Eq. (16).

If the selected data length $n$ is less than $N_m$, the revised $K_m$-value method could not be applied. For example, if ${\mathrm{\varphi }}_m=5$, the minimum data size of $n$ is at least 27 years. Although $N_s$ representing a stable data length, it cannot guarantee a true estimation of PMP. The criterion just presents a statistically reliable estimation of how long the data series should be. Obviously, the longer data series is better for the method. The estimate of the revised $K_m$-value method greatly depends on the data availability, including data quality and quantity. The PMP estimate will be more precise with a longer historical data series. In fact, there are few stations in the real world that are long enough to work out reliable PMP estimates. In general, a station having longer data with outlier occurring in history is very welcome for the statistical approach. So, caution must be taken when using this method to estimate PMP in practice.

Although the data used in the revised $K_m$-value method comes from the whole study area, the PMP result is still a point PMP and mostly to be representative to the potential storm center without temporal and spatial distribution. The major advantage of this approach over others is to provide a quick and preliminary estimate of PMP at a single station or over a small watershed. The suggested PMP estimates of the revised $K_m$-value method could only be considered as a reference value. It could not be recommended as final estimate for engineering design study. And it will be comparable with values of storm transposition or DAD method.

The 24 h PMP estimate by the revised $K_m$-value method in Hong Kong is 1,753 mm. It is substantially greater than 1,250 mm given by Chang and Hui (2001). The difference of the two is about 503 mm. The reason for the difference is that they are estimated by totally different method based on different data. The revised $K_m$-value method is a statistical way with the series of annual maxima rainfall till 2010. While the DAD method and moisture maximization utilized by Chang is a hydrometeorological method with the storm data and dew point data from 1955 to 1999 applied. Per the criteria presented above, if researchers want to get a stable estimate with 10% error in $K_m$, the data length $n$ should be longer than 89 years and 64 years for N14 and R11, respectively. But in this study, historical data are shorter than 30 years for a great majority of Hong Kong rain-gauge stations except for HKO station (119 years available). Therefore, this method cannot guarantee to yield a stable PMP estimate. Furthermore, the storm transposition and DAD method are recommended to apply to estimate the 24 h PMP in Hong Kong in the future if possible to get comparable results.

Appendix S1 is available online in the ASCE Library (www.ascelibrary.org).

The Hershfield’s procedure of estimating PMP was originally developed based on the general frequency equation (Chow 1951):where $X_T$ = rainfall for return period $T$; ${\overline{X}}_n$ and $S_n$ = mean and standard deviation of a series of $n$ annual maximum rainfall values for a given duration, respectively; $K$ is defined as a frequency factor, which is associated with $T$ and the probability distribution selected to fit the data. If the maximum observed rainfall $X_m$, is substituted for $X_T$, $K_m$ for $K$, Eq. (22) could be written as

Then $K_m$ is the number of standard deviation to be added to the mean to obtain $X_m$. $K_m$ is computed by the following equationwhere $X_m$ = highest value for this series; ${\overline{X}}_{n-1}$ and $S_{n-1}$ are, respectively, the mean and the standard deviation excluding the highest value from the series. By enveloping $K_m$ from a large number of computed $K_m$, Hershfield estimated Probable Maximum Precipitation (PMP) though Eq. (25)

The statistical expressions for PMP are re-derived to obtain a better understanding of the method. The standardized variable $\varphi$ is introduced to replace the $K$ of Eq. (22), soandwhere ${\varphi }_m$ = standardized variable or the maximum deviation from the mean, scaled by its standard deviation $S_n$. It is computed directly from the following equation

As known, the relation between ${\overline{X}}_{n-1}$ and ${\overline{X}}_n$ is

And according to the definition of $S_n$

Because $\sum _{i=1}^{n-1}(X_i-{\overline{X}}_{n-1})=0$, so Eq. (31) could be

Combining with Eqs. (30) and (32), there is

Both sides of the Eq. (35) were divided by $S_n^2$:

So, based on Eqs. (24) and (28), Eq. (36) could be

Finally, according to Eqs. (34) and (37) could be written as

Because both $K_m$ and ${\mathrm{\varphi }}_m$ are always greater than 0. Making $C_1={(n-1)}^3/n^2(n-2)$, $C_2=n-1/n(n-2)$, Eq. (38) could be rewritten as follows:

This work was supported by the 24 h PMP Updating Study Project of the Civil Engineering and Development Department of the Hong Kong Special Administrative Region (HKSAR). The data used in the paper were provided by Hong Kong Observatory (HKO).