KR20220048205A - Method for interpolating rain gauge data considering rainfall characteristics - Google Patents

Abstract

The present invention relates to a method for interpolating rain gauge data reflecting rainfall characteristics, comprising the steps of: registering a variogram; registering a Kriging matrix equation; calculating a covariance constituting the Kriging matrix equation by applying rain gauge data to the variogram; calculating a weight by applying the covariance to the Kriging matrix equation; and interpolating the rain gauge data based on the weight, wherein the intermittence and irregularity of the rain gauge data are reflected on a correlation coefficient constituting the variogram. Accordingly, rainfall can be forecasted more accurately by applying a simple Kriging method having the irregularity and intermittence of rain gauge data reflected thereon.

Description

Interpolation method of rain gauge rainfall with rainfall characteristics reflected {METHOD FOR INTERPOLATING RAIN GAUGE DATA CONSIDERING RAINFALL CHARACTERISTICS}

The present invention relates to an interpolation method of rain gauge rainfall in which rainfall characteristics are reflected, and more particularly, to a method of interpolation of rain gauge rainfall through a simple kriging secret method in which non-regularity and intermittent properties of rain gauge rainfall are reflected.

Kriging is one of the local interpolation techniques widely used in the field of geostatistics and is an empirical method for estimating the value of an arbitrary point by weighted linear combination of the values of nearby observation points.

In the 1950s, South African mining engineer Krige introduced an empirical method using the distribution characteristics of existing veins to find new veins, and it was systematically established by French geological mathematician Matheron.

After that, the kriging technique was established statistically and mathematically to be suitable for computational experiments by Sacks et al.

In recent years, the kriging technique is most often used in the field of environmental science, and is also used variously in the hydrology/meteorological field, such as soil maps, groundwater mapping, and temperature and precipitation estimation.

In the hydrology/meteorological field, the kriging technique was mainly used to identify or predict the spatial distribution of target factors by integrating various additional data such as observation data or remote sensing data.

Overseas, the kriging technique was used for precipitation distribution estimation using weather radar data or numerical elevation models, soil moisture distribution, soil organic matter distribution, chemical composition analysis, and groundwater hydraulic conductivity analysis.

In Korea, the kriging technique was mainly applied to studies related to estimating the spatial distribution of rainfall or estimating area rainfall. In addition, Kriging was used in studies such as estimating the surface temperature, understanding the spatial distribution of the drought index, analyzing the runoff according to the spatial distribution of rainfall, creating a groundwater level distribution map using a numerical elevation model, creating a wind speed prediction map, and creating a new appropriate depth map.

Regarding rainfall intensity measurement using the kriging technique, 'rainfall intensity measuring apparatus and method' disclosed in Korean Patent No. 10-1352568 calculates the radar rainfall intensity error distribution by joint kriging and calculates the radar rainfall error distribution. A technique for adjusting radar rainfall intensity data by using is disclosed.

In general, the following two assumptions are applied in such a kriging technique. First, the target data to which the kriging technique is applied should follow a normal distribution. If the target data do not follow a normal distribution, variable transformation is performed so that the predictor and dependent variables are linear. In this case, the value estimated by the kriging technique can be easily inversely transformed, but there is a problem in that the estimate of the variance cannot.

Next, the data to which the kriging technique is applied must be continuous and stationar. Most spatial data exhibit a specific trend or have a problem that the average changes as the location changes. If the kriging technique is applied to data with strong trends, mis-specification may occur in spatial dependence and the predicted values may be biased. Therefore, the predicted value reflecting the spatial characteristics of the target data can be calculated only when the kriging technique is applied by removing the tendency of the data.

The biggest problem in the case of applying the kriging technique to rainfall data is that the rainfall data do not satisfy the above assumptions of the kriging technique. Rainfall data generally show a strong positive distortion, and a strong intermittent. In some cases, the direction is also significant. For this reason, when the kriging technique is performed using rainfall data, the predicted values are highly likely to appear biased or distorted.

Therefore, in order to estimate the systematic rainfall field using the kriging technique, it is necessary to reflect the characteristics of the rainfall data following a distorted distribution rather than the normal distribution and the temporal and spatial distribution characteristics of no rainfall, that is, intermittent. The method did not reflect this, making it difficult to accurately predict rainfall.

Accordingly, the present invention has been devised to solve the above problems, and an object of the present invention is to provide a method for interpolation of rain gauge rainfall through a simple kriging secret method in which non-regularity and intermittent properties of rain gauge rainfall are reflected.

The above object is, according to the present invention, in the interpolation method of rain gauge rainfall in which rainfall characteristics are reflected, the steps of registering a variogram, registering a kriging matrix equation, and applying the variogram to the rain gauge rainfall data to obtain the kriging matrix equation calculating the covariance constituting the , calculating the weight by applying the covariance to the kriging matrix equation, and interpolating the rain storm based on the weight; It is achieved by an interpolation method of rainfall characteristics in which rainfall characteristics are reflected, characterized in that the intermittent and non-normality of the rainfall data is reflected in the correlation coefficient constituting the variogram.

Here, the correlation coefficient may be calculated by applying a dimeric mixed lognormal distribution.

In addition, the correlation coefficient is calculated between the rain gauge rainfall data measured at two rainfall stations; The correlation is a first case having a positive observation value at both rainfall stations, a second case in which at least one of the two rainfall stations has a positive observation value, and both rainfall stations are '0' ' It can be calculated by being divided into a third case having more than one observation value.

And, the correlation coefficient of the first case is

는 상기 제1 케이스의 상관계수이고,

는 이변량 대수정규분포의 상관계수이고,

및

는 두 곳의 강우 관측소 각각의 이변량 대수정규분포의 분산이다)에 의해 산출되고; 상기 제2 케이스의 상관계수는 수학식(

is the correlation coefficient of the first case,

is the correlation coefficient of the bivariate lognormal distribution,

and

is the variance of the bivariate lognormal distribution of each of the two rainfall stations); The correlation coefficient of the second case is

및

는 단변량 대수정규 분포함수이고,

는 이변량 대수정규 결합분포함수이고,

는 대수정규분포의 평균(

)과 분산(

)이고,

는 이변량 대수정규분포의 평균(

,

)과 분산(

,

)이고,

는 상관계수이다)에 의해 산출되며; 상기 제3 케이스의 상관계수는 수학식(

and

is a univariate lognormal distribution function,

is the bivariate lognormal joint distribution function,

is the mean of the lognormal distribution (

) and variance (

)ego,

is the mean of the bivariate lognormal distribution (

,

) and variance (

,

)ego,

is the correlation coefficient); The correlation coefficient of the third case is

and the moment of

It can be calculated through the relationship between the moments of

According to the configuration as described above, according to the present invention, a simple kriging method that reflects the non-regularity and intermittent nature of the rainfall gauge is applied, thereby enabling more accurate rainfall prediction.

1 is a view for explaining an interpolation method of rain gauge rainfall in which rainfall characteristics are reflected according to an embodiment of the present invention;
2 to 8 are diagrams for explaining simulation results for verifying the effect of the rain gauge rainfall interpolation method according to the present invention.

Advantages and features of the present invention and methods of achieving them will become apparent with reference to the embodiments described below in detail in conjunction with the accompanying drawings. However, the present invention is not limited to the embodiments disclosed below, but may be implemented in various different forms, and only these embodiments allow the disclosure of the present invention to be complete, and common knowledge in the art to which the present invention pertains It is provided to fully inform those who have the scope of the invention, and the present invention is only defined by the scope of the claims. Like reference numerals refer to like elements throughout.

Hereinafter, embodiments according to the present invention will be described in detail with reference to the accompanying drawings.

1 is a view for explaining a method of interpolating rainfall in a rain gauge in which rainfall characteristics are reflected according to an embodiment of the present invention.

Referring to FIG. 1 , first, a variogram for interpolation of rainfall in the rain gauge is registered ( S10 ).

The variogram is the expected value of the square of the difference between data separated by a certain distance, and is a measure of the similarity of data. A variogram is defined as in [Equation 1], and a variogram derived for a given data using [Equation 1] is called an empirical variogram.

[Equation 1]

는 베리오그램,

은 관측 지점의 수, 즉 강우 관측소의 수이고,

는 관측 지점에서의 관측 강우량이다. 그리고,

는 자료 간 떨어진 거리, 즉 관측 지점 간의 거리로 분리거리(Separation distance)라고 한다. 관측 자료간 거리가 가까우면 관측 자료의 유사성이 커져 베리오그램은 작아지고, 거리가 멀수록 크게 나타나는 특성을 갖는다.In [Equation 1],

is the variogram,

is the number of observation points, i.e. the number of rainfall stations,

is the observed rainfall at the observation point. And,

is the distance between data points, that is, the distance between observation points. When the distance between observation data is close, the similarity of the observation data increases, and the variogram becomes smaller, and the greater the distance, the larger the characteristic.

In order to model the empirical variogram, it is necessary to determine the threshold value (Sill height), correlation length (Correlation length), and nugget components. Here, the correlation distance means the maximum separation distance at which the observation data show correlation, and the threshold is a constant value of the variogram at the correlation distance and represents the variance of the observation data. A nugget is a case where the variogram does not become 0 when the separation distance is 0, and it represents the uncertainty of observation data for a short separation distance.

If the economic variogram is derived as described above, a theoretical variogram that can represent it should be determined. Since it directly affects the prediction result of the kriging technique, which will be described later, it should be determined as an optimal model for the rain gauge rainfall interpolation method according to the present invention.

In the present invention, a Gaussian model suitable for observation data following a normal distribution among a linear model, a spherical model, an exponential model, a Gaussian model, and a nugget model is applied take as an example

The Gaussian model is used to analyze data with strong correlation or continuity between observation data at small separation distances. This is suitable for observation data that follow a normal distribution. The Gaussian model determines the separation distance corresponding to 95% of the threshold as the actual correlation distance. The Gaussian model can be expressed as [Equation 2].

[Equation 2]

는 문턱값,

는 상관거리,

는 분리거리다.In [Equation 2],

is the threshold,

is the correlation distance,

is a separation distance

As described above, when the theoretical variogram is determined and registered, a krigging matrix equation for applying the kriging technique is registered. Hereinafter, a krigging technique applied to the rain gauge rainfall interpolation method according to an embodiment of the present invention will be described.

In general, the main purpose of the krigging technique is to determine the weight to be applied to predict the value of an unknown point. In determining the weight, the error between the predicted value and the true value is minimized, and the condition that the estimate is not biased is assumed. In addition, the sum of weights must be 1 in order not to always be biased in the estimation equation in the kriging technique.

Here, there are various methods for determining the weight, and in the present invention, a simple kriging technique for determining the weight by minimizing the error variance is applied as an example.

개의 관측 자료를 이용하여 단순 크리깅 기법으로 관측 지점

에서 미지값을 예측하는 방법은 [수학식 3]과 같이 나타낼 수 있으며, 오차 분산식은 [수학식 4]와 같이 나타낼 수 있다.More specifically, the known

Observation point using simple kriging technique using observation data

A method of predicting an unknown value in [Equation 3] can be expressed as [Equation 3], and the error dispersion equation can be expressed as [Equation 4].

[Equation 3]

[Equation 4]

는 예측하고자 하는 참값이고,

는 단순 크리깅 기법에 의한 예측값이고,

은 관측 자료의 총 개수이고,

는 오차 분산이다. 여기서,

는

지점에서의 특성값으로

를 나타낸다.

는 위치벡터로 기하학적 차원에 따라 위치성분을 모두 포함한다.here,

is the true value to predict,

is the predicted value by the simple kriging technique,

is the total number of observations,

is the error variance. here,

Is

as a characteristic value at a point

indicates

is a position vector and includes all position components according to the geometric dimension.

)과 예측값(

)의 차의 제곱에 대한 기대값으로 정의되며, 가중치의 함수이다. 이는 참값은 알려져 있지 않지만 상수이고 예측값은 가중치의 함수이기 때문이다.Looking at [Equation 4], the error variance is the true value (

) and the predicted value (

) is defined as the expected value of the square of the difference, and is a function of the weight. This is because the true value is unknown, but a constant, and the predicted value is a function of the weight.

If [Equation 4] is expanded and expressed as a relational expression between variance and covariance, it is the same as [Equation 5], and if [Equation 4] is substituted for [Equation 5], it can be expressed as [Equation 6] there is.

[Equation 5]

[Equation 6]

는

와

사이의 분리거리에 따라 결정되는 공분산이다.here,

Is

Wow

The covariance is determined by the separation distance between them.

In [Equation 6], it can be confirmed that the error variance is a function of the weight. Partially differentiating the error variance with respect to each weight to obtain an extreme value that becomes 0, and partial differentiating it twice to obtain a weight that minimizes the error variance if the value is greater than 0. Expressing this as an equation, it is as in [Equation 7], and when it is partially differentiated and rearranged, it becomes [Equation 8].

[Equation 7]

[Equation 8]

에 대하여 편미분하면

이 되고, 이는 분산을 나타내므로 양의 값이 된다. 따라서 [수학식 8]을 만족시키는 가중치들이 오차 분산을 최소로 한다. [수학식 8]을 정리하면, [수학식 9]로 나타낼 수 있으며, [수학식 9]가 크리깅 방정식으로 정의된다.[Equation 8] again

If partial differentiation with respect to

, which is a positive value because it represents the variance. Therefore, the weights satisfying [Equation 8] minimize the error variance. If [Equation 8] is arranged, it can be expressed as [Equation 9], and [Equation 9] is defined as the Kriging equation.

[Equation 9]

In order to more easily solve the Krigging equation [Equation 9], it is preferable to represent it as a matrix equation as in [Equation 10]. More specifically, when there are n known observation data, the weight is obtained by solving the Krigging equation consisting of the covariance between each observation point, the prediction point, and the covariance of each data. The diagonal component of the matrix given in [Equation 10] is the variance because it is the covariance of the observation data existing at the same point.

[Equation 10]

In [Equation 10], the matrix component consists of covariance, and the covariance, which is a component of the Krigging matrix equation, can be calculated as in [Equation 11] using a theoretical variogram.

[Equation 11]

는 관측 자료간 거리

에 따른 공분산이고,

는 분산이고,

는 앞서 설명한 이론적 베리오그램이다.In [Equation 11],

is the distance between observations

is the covariance according to

is the variance,

is the theoretical variogram described above.

As described above, when the Krigging matrix equation is registered, the covariance of the Krigging matrix equation can be calculated through [Equation 11], and when the Krigging matrix equation is solved by substituting the components of the Krigging matrix equation, that is, the covariance, the weight to be applied to the Krigging method is calculated (S13). Then, the calculated weight is applied to the kriging technique to interpolate rainfall in the rain gauge (S14).

Here, in the rain gauge rainfall interpolation method according to the present invention, the intermittent and non-normality of the rain gauge rainfall data are reflected in the above-described correlation coefficient constituting the variogram. Prior to the description of the process of reflecting intermittent and non-normality in the correlation coefficient of the variogram in the present invention, the relationship between non-normality and intermittency and the correlation coefficient will be first described.

Correlation coefficient when non-normality is taken into account

In the process of applying the kriging technique, in order to construct the krigging matrix equation of [Equation 10], it is necessary to calculate the covariance between the stations, which is the basis of the matrix component. In general, the direction and extent of the increase or decrease of the random variable Y according to the increase or decrease of the random variable X is expressed without normalization as covariance, and the normalization is the correlation coefficient. Therefore, the relationship between the covariance and the correlation coefficient can be expressed as [Equation 12].

[Equation 12]

는 관측 자료

와

에 대한 공분산이고,

는 두 관측 자료의 상관계수이고,

,

는 각각의 관측 자료의 표준편차이다.In [Equation 12]

is the observation data

Wow

is the covariance for

is the correlation coefficient between the two observations,

,

is the standard deviation of each observation data.

Through [Equation 12], it can be confirmed that the covariance is directly affected by the correlation coefficient between observation data. The correlation coefficient appears differently depending on which distribution the observed data follows. For example, the correlation coefficient is determined differently when the observation data follows a normal distribution and when the observation data follows a lognormal distribution. This is explained based on the theory of lognormal distribution as follows.

의 평균과 분산은 [수학식 13] 및 [수학식 14]와 같이 나타낼 수 있다.First, a variate following a lognormal distribution

The mean and variance of can be expressed as [Equation 13] and [Equation 14].

[Equation 13]

[Equation 14]

와

는 대수정규분포의 모수로

의 평균과 표준편차를 나타낸다. 평균과 분산에 대한 상기와 같은 관계를 이용하면, [수학식 12]와 같은 공분산은 기대값 정리에 의해 결정될 수 있다. 공분산에 대한 기대값 정리는 [수학식 15]와 같이 나타낼 수 있고, 이를 평균과 분산에 대한 [수학식 13] 및 [수학식 14]로 나타내면 [수학식 16]과 같이 표현될 수 있다.here,

Wow

is the lognormal distribution parameter

represents the mean and standard deviation of Using the above relationship for the mean and variance, the covariance as in [Equation 12] can be determined by the expected value theorem. The expected value theorem for covariance can be expressed as [Equation 15], and if it is expressed by [Equation 13] and [Equation 14] for the mean and variance, it can be expressed as [Equation 16].

[Equation 15]

[Equation 16]

,

가 대수정규분포를 따른다고 가정하고, 대수를 취한 변량

,

가 정규분포를 다른다고 가정한다. 또한,

,

의 평균과 분산을 각각

,

및

,

라 하고, 두 관측 자료 사이의 상관계수를

라고 하면,

과

사이의 상관계수는 [수학식 17]과 같이 유도될 수 있다.By applying the above relation to [Equation 12], it is possible to determine the correlation coefficient of observation data following a lognormal distribution. First, two variables

,

A variate taking logarithm, assuming that is lognormal

,

Assume that is not normally distributed. In addition,

,

mean and variance of

,

and

,

, and the correlation coefficient between the two observations is

If you say

class

The correlation coefficient between them can be derived as in [Equation 17].

[Equation 17]

The expression on the left in [Equation 17] is a summary of [Equation 12] about the correlation coefficient, and generally represents the correlation coefficient of data following a normal distribution. On the other hand, when the data follow a lognormal distribution, the correlation coefficient is different from the case of the data following the normal distribution, as shown in the equation on the right of [Equation 17].

Therefore, it can be seen that the form of the correlation coefficient is different when the data follow a lognormal distribution and when the data follow a normal distribution. As a result, the shape of the variogram is determined, and it affects the covariance calculation, which affects the kriging weight. Therefore, more accurate interpolation is possible only when the non-normality of the rain gauge rainfall data is reflected.

Correlation coefficient when intermittent is considered

,

,

그리고

로 분류될 수 있다. 여기서

,

,

, 및

는 양의 관측 자료를 나타낸다. 따라서, 3가지 형태의 관측 자료의 구조에 대해 두 강우 자료 사이의 상관관계를 정의하는 것이 가능하다.The correlation coefficient varies greatly depending on the type of '0' included in the observation data. According to the characteristics of the rainfall, the observation data measured at the two rainfall stations are

,

,

And

can be classified as here

,

,

, and

represents positive observation data. Therefore, it is possible to define the correlation between the two rainfall data for the structure of the three types of observation data.

로, 관측 자료

를 이용하는 경우이고, 제2 케이스는 두 곳의 강우 관측소 중 적어도 하나가 양의 관측값을 갖는 경우, 즉

로, 관측 자료

,

및

를 이용하는 경우이다. 마지막으로 제3 케이스는 두 곳의 강우 관측소 모두 '0' 이상의 관측값을 갖는 경우, 즉

로, 모든 관측 자료를 이용하는 경우이다.Case 1 has positive observations at both rainfall stations, i.e.

furnace, observation data

, and the second case is when at least one of the two rainfall stations has a positive observation value, i.e.

furnace, observation data

,

and

in case of using Finally, the third case is when both rainfall stations have an observation value of '0' or higher, i.e.

This is the case when all observational data are used.

로 나타낼 수 있다. 이 때 상관계수

와

는 조건

와

에 대한 조건부(Conditional) 상관계수가 되고,

의 경우는 전체 관측 자료를 사용하므로 비조건부(Unconditional) 상관계수가 된다.As above, the correlation coefficients between the rainfall stations for the three cases are

can be expressed as In this case, the correlation coefficient

Wow

is the condition

Wow

becomes a conditional correlation coefficient for

In the case of , since the entire observation data is used, it becomes an unconditional correlation coefficient.

와

하에서의 조건부 확률분포함수를 알고 있다면, 평균, 분산 또는 강우 관측소 사이의 상관계수와 같은 통계치들을 산출하는 것이 가능하다. 이와 같은 통계치들의 유도를 위해서는 [수학식 18]과 같은 전확률공식(Total probability theorem)의 적용이 필요하다.If, condition

Wow

If the conditional probability distribution function under In order to derive such statistics, it is necessary to apply the Total Probability Theorem as in [Equation 18].

[Equation 18]

로 나타나는 조건

하에서의 모멘트를 유도할 수 있으며, 아울러

또는

로 표시된 조건

와

하에서의 모멘트를 구하는 것도 가능하다. 각각의 경우 중 먼저

와

의 관계를 유도하면 다음과 같다.first

condition represented by

It is possible to induce a moment under

or

condition marked with

Wow

It is also possible to find the moment under in each case first

Wow

If we derive the relationship of

이며, 이에 대한 여집합은

이다.

의 확률은

이다. 따라서 [수학식 18]을 이용하면 [수학식 19]와 같은 관계식을 유도할 수 있다.

, and the complement of this is

am.

the probability of

am. Therefore, using [Equation 18], a relational expression such as [Equation 19] can be derived.

[Equation 19]

와

의 관계 또한 [수학식 20]과 같이 유도될 수 있다.And,

Wow

The relationship of can also be derived as in [Equation 20].

[Equation 20]

In the same way, it is possible to derive relational expressions such as [Equation 21] and [Equation 22].

[Equation 21]

[Equation 22]

,

,

의 산출이 가능하게 된다. [수학식 19] 내지 [수학식 22]의 모멘트는 서로 관계되어 있기 때문에 어떤 조건에서의 모멘트로부터 다른 조건에서의 모멘트를 산출하는 것도 가능하다.Therefore, using the moment given in [Equation 19] to [Equation 22], the correlation coefficient between rainfall stations is

,

,

can be calculated. Since the moments of [Equation 19] to [Equation 22] are related to each other, it is also possible to calculate a moment in another condition from a moment in one condition.

는 [수학식 23]과 같이 정의될 수 있다.First, the correlation coefficient

can be defined as in [Equation 23].

[Equation 23]

는 [수학식 24]와 같이 유도될 수 있다.Using the relational expressions given in Equation 19] to [Equation 22], the correlation coefficient

can be derived as in [Equation 24].

[Equation 24]

는 [수학식 25]와 같이 정의되고, [수학식 21] 내지 [수학식 24]을 이용하면

는 [수학식 26]과 같이 나타낼 수 있다.If one or both rainfall stations are positive,

is defined as in [Equation 25], and using [Equation 21] to [Equation 24],

can be expressed as [Equation 26].

[Equation 25]

[Equation 26]

일 때, 즉, 두 강우 관측소에 동시에 무강우가 기록되지 않는 경우에는

가 된다. 상기와 같은 수학식을 통해 관측 자료에 '0'이 포함된 경우와 그렇지 않은 경우에 대하여 상관계수가 다르게 나타날 수 있음을 확인할 수 있으며, 이는 우량계 강우 자료의 간헐성이 반영되어야 함을 나타낸다.In [Equation 26]

, that is, when no rainfall is recorded at the two rainfall stations at the same time

becomes Through the above equation, it can be confirmed that the correlation coefficient may be different for the case where '0' is included in the observation data and the case where '0' is not included, which indicates that the intermittency of the rain gauge rainfall data should be reflected.

As described above, it can be confirmed that the non-normality and intermittent nature of the observation data for rainfall affect the correlation coefficient, and in the interpolation method of the rainfall gauge according to the present invention, the correlation coefficient is will decide

More specifically, it is known that observation data, which are rainfall data, do not follow a normal distribution, as described above, and can generally be described as a lognormal distribution rather than a normal distribution. Therefore, the correlation coefficient of the observation data, which is the rainfall data, will appear different from the correlation coefficient when 0 is included in the data following the normal distribution.

In the present invention, as an example, applying a bivariate mixed log-normal distribution to the observation data corresponding to the case where '0' is included in the data following the lognormal distribution, and the correlation coefficient for this is can be explained as

A bivariate mixed lognormal distribution is defined as [Equation 27]. Here, the meaning of mixing indicates that both discontinuous and continuous distribution are considered.

[Equation 27]

,

이다. 일반적으로 단변량 대수정규(Log-normal) 분포함수는

나

로 나타낼 수 있고, 이변량 대수정규 결합분포함수는

로 나타낼 수 있으므로, [수학식 27]은 [수학식 28] 내지 [수학식 30]과 같이 나타낼 수 있다.[In Equation 27,

,

am. In general, the univariate log-normal distribution function is

me

It can be expressed as , and the bivariate lognormal joint distribution function is

Since it can be expressed as, [Equation 27] can be expressed as [Equation 28] to [Equation 30].

는 대수정규분포의 평균

와 분산

을 나타내고,

는 이변량 대수정규분포의 평균

,

, 분산

,

, 상관계수

를 나타낸다. 따라서 이변량 혼합 대수정규분포의 모수는 [수학식 31] 내지 [수학식 39]와 같이 나타낼 수 있다.here,

is the mean of the lognormal distribution

with dispersion

represents,

is the mean of the bivariate lognormal distribution

,

, Dispersion

,

, correlation coefficient

indicates Therefore, the parameters of the bivariate mixed lognormal distribution can be expressed as in [Equation 31] to [Equation 39].

[Equation 31]

[Equation 32]

[Equation 33]

[Equation 34]

[Equation 35]

[Equation 36]

[Equation 37]

[Equation 38]

[Equation 39]

또는

가 대수정규분포를 따를 때

인 경우에 대하여 모멘트는 [수학식 40] 내지 [수학식 43]과 같이 나타낼 수 있다.

or

When is lognormally distributed

For the case of , the moment can be expressed as in [Equation 40] to [Equation 43].

의 결합분포에서 제1 케이스의 조건

에 대한

의 기댓값은 [수학식 44]와 같이 나타낼 수 있다.given by a bivariate lognormal distribution

The condition of the first case in the joint distribution of

for

The expected value of can be expressed as [Equation 44].

[Equation 44]

를 [수학식 45]와 같이 산출할 수 있다.Using [Equation 40] to [Equation 44], the correlation coefficient of the first case according to [Equation 12]

can be calculated as in [Equation 45].

[Equation 45]

,

를 산출하기 위해서는

과

,

의 조건이 필요하다.

과

인 경우에

의 분포가 평균과 분산

와

를 갖는 대수정규분포를 따르므로 [수학식 46] 및 [수학식 47]의 관계를 도출할 수 있다.Correlation coefficients for case 2 and case 3

,

in order to calculate

class

,

condition is required.

class

in case of

distribution of mean and variance

Wow

Since it follows a lognormal distribution with

In addition, it is possible to derive relational expressions such as [Equation 48] and [Equation 49].

를 산출할 수 있다. 또한,

의 조건으로 주어지는 제2 케이스의 상관계수

는 [수학식 28] 내지 [수학식 30]을 통해 산출할 수 있다.Using the relationship between [Equation 21] and [Equation 22] and the moments of [Equation 46] and [Equation 49], the correlation coefficient of the third case

can be calculated. In addition,

Correlation coefficient of the second case given under the condition of

can be calculated through [Equation 28] to [Equation 30].

Through the above process, in interpolating rain gauge rainfall data with non-normality and intermittent nature by applying the kriging technique, correlation coefficients are calculated according to the first case, the second case, and the third case, and the non-normality and the intermittent The application of the kriging technique becomes possible.

Hereinafter, effects of the rain gauge rainfall interpolation method according to the present invention will be described with reference to FIGS. 2 to 8 .

As shown in FIG. 2 , a simulation was performed in which the interpolation technique of the rain gauge rainfall data according to the embodiment of the present invention was applied to the rain gauge rainfall data of 45 stations of the Korea Meteorological Administration distributed within the radar observation range of Mt. Gwanak. Then, the simulation results are compared with the distribution of the Gwanaksan radar rainfall. This is because, although radar rainfall is lower in degree than rain gauge rainfall, the spatial distribution of rainfall can be observed relatively accurately.

Among the heavy rain events that occurred on September 11, 2010, we used observation data at two points at 4:30 (Event 1) and 8:30 (Event 2) on September 11, 2010, when a large amount of rainfall was recorded. Simple kriging was performed. Fig. 3 (a) shows the radar rainfall distribution of Event 1, and Fig. 3 (b) shows the radar rainfall distribution of Event 2.

In order to verify the effect of the interpolation method of the rain gauge rainfall data according to the embodiment of the present invention, the simulation was performed by dividing the rainfall characteristics into four cases depending on whether or not the rainfall characteristics were considered. When the characteristics of rainfall data are not considered (Case 1), when only intermittency is considered (Case 2), when only non-normality of data is considered (Case 3), when both intermittent and non-normality are considered as in the embodiment of the present invention (Case 4). In Case 1 and Case 2, raw data of rainfall gauge were used, and in Case 3 and Case 4, rain gauge rainfall data using natural logarithms was applied. This approach is based on the assumption that the data to which the natural logarithm is applied will follow the normal distribution, considering the characteristics of the data that do not follow the normal distribution.

[Table 1] summarizes the average, variance, and maximum values of rainfall gauge and radar rainfall.

[Table 1]

4 and 5 show an empirical and theoretical variogram of radar rainfall, FIG. 4 is a variogram of Event 1, and FIG. 5 is a variogram of Event 2. 4 and 5 (a) is Case 1, FIGS. 4 and 5 (b) are Case 2, FIGS. 4 and 5 (c) are Case 3, FIGS. 4 and 5 (d) are Case 2 4 is And, [Table 2] summarizes the threshold values and correlation distances of variograms.

[Table 2]

As can be seen from FIGS. 4 and 5 and [Table 2], when the variogram is induced in consideration of the intermittent or non-normality of the rainfall data, it can be confirmed that the threshold value is small and the correlation distance is large compared to the case where it is not. can In particular, the intermittent rainfall data had a greater effect on the correlation distance than the non-normality. When both the intermittent and non-normality of the rainfall data were considered, the threshold value was smaller than in all other cases, and the correlation distance was very large. This is because, when intermittent rainfall data is considered, a large amount of '0' included in radar rainfall reduces variance and increases correlation between rainfall data. If the non-normality of the rainfall data is additionally taken into account, the effect of the outlier is reduced as data applied with the natural logarithm is used, and the correlation of the data becomes larger. This indicates that the correlation of rainfall data can be maximized when considering the characteristics of rainfall data.

On the other hand, (a) of Figure 6 is the location of the rain gauge rainfall for Event 1, Figure 6 (b) is the location of the rain gauge rainfall for Event 2, Figure 6 (c) is the radar rainfall for Event 1 location, and (d) of FIG. 6 is the location of radar rainfall for Event 2. The hollow squares in FIG. 6 represent rain gauge rainfall and radar rainfall recorded as no rainfall at the rain gauge point. As can be seen in FIG. 6 , there are parts where the radar rainfall and the rain gauge rainfall appear similarly, but in most cases, the radar rainfall was observed to be smaller than the rain gauge rainfall.

7 and 8 show simple kriging results using the rain gauge rainfall data and radar rainfall data shown in FIG. 6, respectively. 7 and 8 (a) to (d) are the results applied to Event 1, (e) to (h) are the results applied to Event 2, (a) and (e) are Case 1, (b) and (f) ) shows Case 2, (c) and (g) show Case 3, and (d) and (h) show Case 4.

As can be seen from FIGS. 7 and 8 , it can be seen that simple kriging results using rain gauge rainfall and radar rainfall are generally similar for each case. However, it can be seen that the size of the rainfall was larger in the result using the rain gauge rainfall than the result using the radar rainfall, and the rainfall area was also larger.

When simple kriging was performed without considering the characteristics of rainfall data, the rainfall area was small in both the rain gauge rainfall and radar rainfall simple kriging results. This is because the correlation distance was the smallest. When simple kriging was performed considering only the intermittent rainfall data, the correlation distance was large, and the rainfall area was larger compared to the case where the characteristics of the rainfall data were not considered. When simple kriging was performed considering only the non-normality of the rainfall data, the rainfall area was larger than the previous two results for both rain gauge rainfall and radar rainfall. When both intermittent and non-normality of rainfall data were considered, the spatial distribution of rainfall was larger. In particular, in Event 1, the rainfall area was the largest compared to the other results in both the rain gauge and radar rainfall. This is because the correlation was the largest.

[Table 3] and [Table 4] summarize the test statistics between the simple kriging results for rainfall gauge and radar rainfall and the observed radar rainfall. As a result of comparing the simple kriging results for rain gauge rainfall and radar rainfall with the observed radar rainfall, the RMSE difference was smaller when the characteristics of the rainfall data were considered than when the characteristics of the rainfall data were not considered. In particular, the RMSE was shown to be the minimum when the intermittency of the data was taken into consideration in the simple kriging results using rain gauge rainfall and radar rainfall. Through these results, it was confirmed that the case where simple kriging was performed in consideration of the characteristics of the rainfall data could improve the prediction result than the case where it was not.

[Table 3]

[Table 4]

Although several embodiments of the present invention have been shown and described, it will be apparent to those skilled in the art that changes may be made to these embodiments without departing from the spirit or spirit of the invention. . The scope of the invention will be defined by the appended claims and their equivalents.

Claims (4)

In the interpolation method of rain gauge rainfall in which rainfall characteristics are reflected,
A step in which the variogram is registered;
a step in which the kriging matrix equation is registered;
The step of calculating the covariance constituting the kriging matrix equation by applying the variogram to the rain gauge rainfall data;
calculating a weight by applying the covariance to the kriging matrix equation;
interpolating rain-based rainfall based on the weight;
An interpolation method of rainfall characteristics reflecting rainfall characteristics, characterized in that the intermittent and non-normality of the rain gauge rainfall data is reflected in the correlation coefficient constituting the variogram. According to claim 1,
The correlation coefficient is an interpolation method of rain gauge rainfall reflecting rainfall characteristics, characterized in that it is calculated by applying a bimodal mixed lognormal distribution. 3. The method of claim 2,
the correlation coefficient is calculated between the rain gauge rainfall data measured at two rainfall stations;
The correlation is a first case having a positive observation value at both rainfall stations, a second case in which at least one of the two rainfall stations has a positive observation value, and both rainfall stations are '0'' Rain gauge rainfall interpolation method with rainfall characteristics reflected, characterized in that it is calculated by being divided into a third case having more than one observation value. 3. The method of claim 2,
The correlation coefficient of the first case is

(

is the correlation coefficient of the first case,

is the correlation coefficient of the bivariate lognormal distribution,

and

is the variance of the bivariate lognormal distribution of each of the two rainfall stations); The correlation coefficient of the second case is

(

and

is a univariate lognormal distribution function,

is the bivariate lognormal joint distribution function,

is the mean of the lognormal distribution (

) and variance (

)ego,

is the mean of the bivariate lognormal distribution (

,

) and variance (

,

)ego,

is the correlation coefficient); The correlation coefficient of the third case is

and the moment of

An interpolation method of rainfall in the rain gauge reflecting rainfall characteristics, characterized in that it is calculated by the relationship between the moments of

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