Detection of mean-field bias of the radar rain rate using rain gauges available within a small portion of radar umbrella: a case study of the Donghae (East Sea) radar in Korea

Abstract

Observation of a storm approaching from the ocean to the in-land area is very important for the flood forecasting. Radar is generally used for this purpose. However, as rain gauges are mostly located within the in-land area, detection of the mean-field bias of radar rain rate cannot be easily made. This problem is obviously different from that with evenly-spaced rain gauges over the radar umbrella. This study investigated the detection problem of mean-field bias of radar rain rate when rain gauges are available within a small portion of radar umbrella. To exactly determine the mean-field bias, i.e. the difference between the radar rain rate and the rain gauge rain rate, the variance of the difference between two observations must be small; thus, a sufficient number of observations are indispensable. Therefore, the problem becomes determining the number of rain gauges that will satisfy the given accuracy, that being the variance of the difference between two observations. The dimensionless error variance derived by dividing the expected value of the error variance by the variance of the areal average rain rate was introduced as a criteria to effectively detect the mean field bias. Here, the variance of the areal average rain rate was assumed to be the climatological one and the expectation for the error variance could be changed depending one the sampling characteristics. As an example, this study evaluated the rainfall observation over the East Sea by the Donghae radar. About 6.8 % of the entire radar umbrella covered in-land areas, where the rain gauges were available. It was found that, to limit the dimensionless error variance to 2 %, a total of 26 rain gauges are required for the entire radar umbrella; whereas, a total of 24 rain gauges would be required within the in-land area with available for the rain gauge data.

Appendix A: Derivation of Eq. (8)

Derivation of Eq. (8) can be found in North and Nakamoto (1989) and North et al. (1994). Summarizing their work is as follows. First, the rain rate field $\psi (r,\,t)$ at a point r in a region, with an area of A is assumed to be random. Now the variance of the error for an arbitrary j-th bin between the areal average radar rain rate and the rain gauge rain is expressed as:

$$ \varepsilon_{j}^{2} = \left\langle {\left( {\Uppsi_{{G_{J} }} - \Uppsi_{{R_{J} }} } \right)^{2} } \right\rangle = \frac{1}{{A^{2} }}\int\limits_{R} {\int\limits_{R} {\left\langle {\psi (r,\,t)\psi (r^{\prime } ,\,t^{\prime } )} \right\rangle \left[ {1 - K(r)} \right]\left[ {1 - K(r^{\prime } )} \right]d^{2} rd^{2} r^{\prime } } } $$

(A 1)

where, $K(r) = A\delta (r_{{g_{n} }} )$ and $< \psi (r,\,t)\psi (r',\,t') >$ is the lag covariance. And the lag covariance is expressed by the variance of the rain rate field $\psi (r,\,t)$, $\sigma^{2}$, and the space–time lagged autocorrelation having space lag $\xi$ and time lag $\eta$, $\rho$. And the spectrum of the rain rate field, S, is expressed as Fourier transform of the autocorrelation.

$$ \left\langle {\psi (r,\,t)\psi (r^{\prime } ,\,t^{\prime } )} \right\rangle = \sigma^{2} \rho (\xi ,\eta ) $$

(A 2)

Eq. (A 1) can be transformed by Fourier transform like below.

$$ \varepsilon_{j}^{2} = \sigma^{2} \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {\left| {D_{\varepsilon } (v)} \right|^{2} S(v)d^{2} v} } $$

(A 3)

where, S is the spatial spectrum. $D_{\varepsilon } (v) = D_{R} (v) - e^{{2\pi ivr_{g} }}$ and $D_{R}$ is the function defined for the shape of the region satisfying $D_{R} (v) = \,\,\frac{1}{A}\int\limits_{R} {e^{2\pi ivr} d^{2} r}$. And Expectation of $D_{\varepsilon }^{2} (v)$ is like below.

$$ E\left[ {\left| {D_{\varepsilon } (v)} \right|^{2} } \right] = 1 + D_{R}^{2} (v) - D_{R} (v)E\left[ {\cos \left( {2\pi v \cdot r_{g} } \right)} \right] = 1 - D_{R}^{2} (v) $$

(A 4)

Expectation of Eq. (A 3) is like below.

$$ E\left[ {\varepsilon_{j}^{2} } \right] = \sigma^{2} \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {[1 - D_{R}^{2} (v)]S(v)d^{2} v} } = \sigma^{2} \int\limits_{ - \infty }^{\infty } {\int\limits_{ - \infty }^{\infty } {H(v)S(v)d^{2} v} } $$

(A 5)

where, $1 - D_{R}^{2} (v)$ equal to the design filter $H(v)$.