METHOD FOR QUANTIFYING STRUCTURAL FEATURE FORECAST ERROR OF METEOROLOGICAL ELEMENT BASED ON GRAPHICAL SIMILARITY

Abstract

A method for quantifying a structural feature forecast error of a meteorological element based on a graphical similarity is based on the concept of graphical similarity to propose a normalized evaluation technique for a forecast error of a scalar meteorological element such as rainfall, radar reflectivity, temperature, visibility, or wind speed. The method can objectively and truly reflect the true capability of forecasting the meteorological element such as precipitation.

Claims

1. A method for quantifying a structural feature forecast error of a meteorological element based on a graphical similarity, comprising the following steps: S1: defining a structural feature of a meteorological element field; defining an overall structural feature d of the meteorological element field as an area S of a spatial range covered by the meteorological element, a total numerical size R of the meteorological element in a target region, and a structure H of the meteorological element; $\begin{array}{cc}\stackrel{.fwdarw.}{}=\left[S,R,H\right]& \left(1\right)\end{array}$ wherein, {right arrow over ()} denotes the overall structural feature of the meteorological element field; S2: calculating a total amount error and a total area error of the meteorological element; $\begin{array}{cc}{E}_R=\frac{\left|{\mathrm{RT}}^{\left(f\right)}\left(t\right)-{\mathrm{RT}}^{\left(o\right)}\left(\right)\right|}{\max \left({\mathrm{RT}}^{\left(f\right)}\left(t\right),{\mathrm{RT}}^{\left(o\right)}\left(\right)\right)}& \left(2\right)\end{array}$ $\begin{array}{cc}{E}_S=\frac{\left|{\mathrm{ST}}^{\left(f\right)}\left(t\right)-{\mathrm{ST}}^{\left(o\right)}\left(\right)\right|}{\max \left({\mathrm{ST}}^{\left(f\right)}\left(t\right),{\mathrm{RT}}^{\left(o\right)}\left(\right)\right)}& \left(3\right)\end{array}$ wherein, RT.sup.(f)(t) denotes a total amount of a forecasted meteorological element field at a time t; RT.sup.(0)() denotes a total amount of an observed meteorological element field at a time ; ST.sup.(f)(t) denotes a total area covered by the forecasted meteorological element field at the time t; and ST.sup.(0)() denotes a total area covered by the observed meteorological element field at the time ; S3: calculating a structural error of the meteorological element field; firstly, calculating, based on a probability density function, a structural feature p.sup.(f)(.sup.f,t) of the forecasted meteorological element field and a structural feature p.sup.(0)(.sup.0,) of the observed meteorological element field according to Eqs. (4) and (5), respectively; $\begin{array}{cc}\{\begin{array}{c}{p}^{\left(f\right)}\left({}^f,t\right)=H^{\left(f\right)}\left({}^f,t\right)\\ {}^f=\left(R^f-R_{\min }^f\right)/\left(R_{\max }^f-R_{\min }^f\right)\left[0,1\right]\end{array}& \left(4\right)\end{array}$ wherein, H.sup.(f)(.sup.f,t) denotes a probability density function for the forecasted target element field at the time t; .sup.f denotes a coefficient of the meteorological element after the forecasted meteorological element field is normalized; and .sup.f takes a value of 0-1; $\begin{array}{cc}\{\begin{array}{c}{p}^{\left(0\right)}\left({}^f,t\right)=H^{\left(0\right)}\left({}^0,t\right)\\ {}^0=\left(R^0-R_{\min }^0\right)/\left(R_{\max }^0-R_{\min }^0\right)\left[0,1\right]\end{array}& \left(5\right)\end{array}$ wherein, H.sup.(0)(.sup.0,t) denotes a probability density function for the observed target element field at the time t; .sup.0 denotes a coefficient of the meteorological element after the observed meteorological element field is normalized; and .sup.0 takes a value of 0-1; and secondly, measuring, based on a Kullback-Leibler (KL) divergence algorithm (6), a similarity E.sub.H between the structural feature of the forecasted meteorological element field and the structural feature of the observed meteorological element field; $\begin{array}{cc}{E}_H={.Math.}_{{}^f}{p}^{\left(f\right)}\left({}^f,t\right)\log \frac{p^{\left(f\right)}\left({}^f,t\right)}{p^{\left(o\right)}\left({}^o,t\right)}& \left(6\right)\end{array}$ S4: calculating a similarity between the target element field and the observed element field; calculating the similarity E between the forecasted meteorological element field and the observed element field according to Eq. (2); $\begin{array}{cc}E=E_R+E_S+E_H& \left(7\right)\end{array}$ wherein, E.sub.R denotes the total amount error of the meteorological element; E.sub.S denotes the total area error of the meteorological element; and E.sub.H denotes the structural error of the meteorological element.

2. The method for quantifying the structural feature forecast error of the meteorological element based on the graphical similarity according to claim 1, wherein the method further considers a time error E.sub.t between the forecasted meteorological element field {right arrow over ()}.sup.(f)(t) and the observed meteorological element field {right arrow over ()}.sup.(0)() at consecutive times, specifically as follows: firstly, calculating, within a time range of t=2t, the similarity E(t,) between the forecasted meteorological element field {right arrow over ()}.sup.(f)(t) at each time and the observed element field {right arrow over ()}.sup.(0)() at the time according to steps S1 to S4; calculating the time error as t, wherein when a similarity E(t,) reaches a maximum value, the time corresponding to the forecasted meteorological element field is t; and normalizing according to Eq. (8) to acquire a normalized time error E.sub.t; $\begin{array}{cc}{E}_t=\frac{\left|-t\right|}{2t}& \left(8\right)\end{array}$ wherein, E.sub.t denotes the time error between the forecasted meteorological element field {right arrow over ()}.sup.(f)(t) and the observed meteorological element field {right arrow over ()}.sup.(0)() at the consecutive times; and t denotes a time interval of the forecasted meteorological element field; and secondly, acquiring an overall error E between the forecasted meteorological element field and the observed meteorological element field, wherein E=E.sub.R+E.sub.S+E.sub.H+E.sub.t.

3. The method for quantifying the structural feature forecast error of the meteorological element based on the graphical similarity according to claim 1, wherein the meteorological element comprises rainfall, radar reflectivity, temperature, visibility, and wind speed.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0030] FIG. 1 shows a value-coverage area curve of a meteorological element according to the present disclosure;

[0031] FIGS. 2A and 2B show an example of a meteorological element field according to the present disclosure, where FIG. 2A is an example of forecasted meteorological element field {right arrow over ()}.sup.(f)(t) at time t; and FIG. 2B is an example of observed meteorological element field {right arrow over ()}.sup.(0)() at time ; and

[0032] FIG. 3 shows original numerical value-area distribution curves and normalized distribution curves of the forecasted meteorological element field and the observed meteorological element field according to the present disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0033] The following further describes the present disclosure with reference to the drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present disclosure, not to limit the scope of the present disclosure. Those skilled in the art should understand that any equivalent modifications to the present disclosure shall fall within the scope defined by the claims.

[0034] The quantitative calculation of weather forecast error has always been a research topic, and its essence is to discuss the distribution similarity of the target element in the forecasted field and in the observed field. To discuss the distribution similarity of the target element, it is first necessary to define the distribution feature of the target element. The present disclosure proposes a method for quantifying a structural feature forecast error of a meteorological element based on a graphical similarity. The present disclosure provides an evaluation technique for the forecast error of the meteorological element based on the concept of graphical similarity. This technique aims to evaluate the forecast error of a scalar meteorological element. For concise expression, the scalar meteorological element that needs to be evaluated is referred to as the target element, and the evaluation process is detailed as follows.

[0035] As shown in FIGS. 1 to 3, in the present disclosure, the method first defines an overall distribution feature of the target element (i.e. target meteorological element to be analyzed, such as rainfall, radar reflectivity, temperature, visibility, or wind speed) as total area S of the target element, total amount R of the target element, and structure H of the target element. Therefore, the overall feature of a target element field is expressed as structure {right arrow over ()}:

[00008]$\begin{array}{cc}\stackrel{.fwdarw.}{}=\left[S,R,H\right]& \left(1\right)\end{array}$

[0036] In the equation, the area S of the target element is defined as the area of a spatial range covered by the target element, and the total amount R of the total target element is defined as a total size of the target element within a target region. The structural feature H of the target element reflects the size and spatial distribution of the target element, and is expressed as a numerical size-area graph of the target element, meaning the area covered by the target element of a certain size. As shown in FIG. 1, S.sub.i denotes the area covered by the target element of size R.sub.i, and R.sub.max denotes a maximum value of the target element.

[0037] In addition, in the present disclosure, the method further considers a time error of the forecasted field, which is defined as follows. In forecasted target element field {{right arrow over ()}.sup.(f)(t), t=t.sub.1, t.sub.2, . . . } at consecutive times and observed target element field {{right arrow over ()}.sup.(0)(), =t.sub.1, t.sub.2, . . . } at the consecutive times (as shown in FIGS. 2A and 2B), observed target element field {right arrow over ()}.sup.(0)() with the greatest similarity to the forecasted target element field {right arrow over ()}.sup.(f)(t) at time t is found. Since the time of the forecasted field is t, the corresponding time of the observed field with the greatest similarity to the forecasted field is t, so the time error is t. Starting from the actual situation, it is agreed that |t|2t (i.e., the time range of t=2t), where t denotes a time interval of the forecasted field, that is, the time error between the forecasted target element field and the observed target element field does not exceed 2 forecast time intervals. The time error is normalized as follows:

[00009]$\begin{array}{cc}{E}_r=\frac{\left|-t\right|}{2t}& \left(2\right)\end{array}$

[0038] It is necessary to calculate the similarity between the forecasted target element field {right arrow over ()}.sup.(f)(t) and the observed target element field {right arrow over ()}.sup.(0)(). According to the meaning of components of feature quantity of the target element field, the similarity between the two target element fields refers to the similarity of the total area S, the total amount R, and the distribution pattern H of the target element. Therefore, by calculating the similarity of these three parameters step by step, a quantitative expression of the forecast error of structural feature of the meteorological element can be acquired as follows.

[0039] Step 1. The total amount error E.sub.R of the target element is calculated

[0040] according to Eq. (3):

[00010]$\begin{array}{cc}{E}_R=\frac{\left|{\mathrm{RT}}^{\left(f\right)}\left(t\right)-{\mathrm{RT}}^{\left(o\right)}\left(\right)\right|}{\max \left({\mathrm{RT}}^{\left(f\right)}\left(t\right),{\mathrm{RT}}^{\left(o\right)}\left(\right)\right)}& \left(3\right)\end{array}$

[0041] The total area error E.sub.S of the target element is calculated according to Eq. (4):

[00011]$\begin{array}{cc}{E}_S=\frac{\left|{\mathrm{ST}}^{\left(f\right)}\left(t\right)-{\mathrm{ST}}^{\left(o\right)}\left(\right)\right|}{\max \left({\mathrm{ST}}^{\left(f\right)}\left(t\right),{\mathrm{RT}}^{\left(o\right)}\left(\right)\right)}& \left(4\right)\end{array}$

[0042] where, RT.sup.(f)(t) denotes a total amount of a forecasted target element field at time t (a cumulative numerical value of the target element at each point in the region); RT.sup.(0)() denotes a total amount of the observed target element field at time ; ST.sup.(f)(t) denotes a total area covered by the forecasted target element field at the time t; and ST.sup.(0)() denotes a total area covered by the observed target element field at the time t.

[0043] Step 2. Structural error E.sub.H between the forecasted target element field {right arrow over ()}.sup.(f)(t) and the observed target element field .sup.(t)() is calculated.

[0044] In FIG. 3, the left figure shows an original numerical value-area distribution of the observed field and the forecasted target element field, where R.sub.min.sup.0 denotes a minimum value of the target element in the observed field, R.sub.max.sup.0 denotes a maximum value of the target element in the observed field, R.sub.min.sup.f denotes a minimum value of the target element in the forecasted field, and R.sub.max.sup.f denotes a maximum value of the target element in the forecasted field. In FIG. 3, the right figure shows a normalized numerical value-area distribution of the observed target element field and the forecasted target element field.

[0045] In the present disclosure, the structural feature of the target element is expressed by a probability density function of the target element. The structural feature of the normalized forecasted target element field is expressed by Eq. (5), and the structural feature of the normalized observed target element field is expressed by Eq. (6):

[00012]$\begin{array}{cc}\{\begin{array}{c}{p}^{\left(f\right)}\left({}^f,t\right)=H^{\left(f\right)}\left({}^f,t\right)\\ {}^f=\left(R^f-R_{\min }^f\right)/\left(R_{\max }^f-R_{\min }^f\right)\left[0,1\right]\end{array}& \left(5\right)\end{array}$$\begin{array}{cc}\{\begin{array}{c}{p}^{\left(0\right)}\left({}^f,t\right)=H^{\left(0\right)}\left({}^0,t\right)\\ {}^0=\left(R^0-R_{\min }^0\right)/\left(R_{\max }^0-R_{\min }^0\right)\left[0,1\right]\end{array}& \left(6\right)\end{array}$

[0046] where, H.sup.(f)(.sup.f, t) denotes a probability density function for the forecasted target element field at the time t; .sup.f denotes a coefficient of the target element after the forecasted field is normalized; .sup.f=(R.sup.fR.sub.min.sup.f)/(R.sub.max.sup.fR.sub.min.sup.f); H.sup.(0)(.sup.0, ) denotes a probability density function for the observed target element field at the time ; .sup.0 denotes a coefficient of the numerical value of the target element after the observed field is normalized, .sup.0=(R.sup.0R.sub.min.sup.0)/(R.sub.max.sup.0R.sub.min.sup.0); and .sup.f and .sup.0 take a value of 0-1.

[0047] Step 3. A similarity between the structural feature of the forecasted target element field and the structural feature of the observed target element field is measured based on a Kullback-Leibler (KL) divergence algorithm:

[00013]$\begin{array}{cc}{E}_H=KL\left(p^{\left(f\right)}\left({}^f,t\right)|p^{\left(o\right)}\left({}^o,\right)\right)={}_{{}^f}{p}^{\left(f\right)}\left({}^f,t\right)\log \frac{p^{\left(f\right)}\left({}^f,t\right)}{p^{\left(o\right)}\left({}^o,\right)}& \left(7\right)\end{array}$

[0048] where, KL(p.sup.(f)(.sup.f,t)|p.sup.(0)(.sup.0,)) denotes a relative entropy of p.sup.(f)(.sup.f,t) and p.sup.(0)(.sup.0,), which is calculated according to the right-hand side of Eq. (7).

[0049] Eqs. (3), (4), and (7) are used to calculate the similarity in the total amount, total area, and structure of the target element field. E.sub.RE.sub.SE.sub.H take values of [0,1], which are logically comparable. Therefore, the overall similarity between the forecasted target element field {right arrow over ()}.sup.(f)(t) at the time and the observed target element field {right arrow over ()}.sup.(0)()at the time is defined as follows:

[00014]$\begin{array}{cc}E=E_R+E_S+E_H& \left(8\right)\end{array}$

[0050] Under the framework of Eq. (2), when E(t, ) is minimized, E.sub.t is the time error between the forecasted target element field and the observed target element field, and the overall error is:

[00015]$\begin{array}{cc}E=E_R+E_S+E_H+E_t& \left(9\right)\end{array}$

[0051] The above are merely preferred specific embodiments of the present disclosure, but the protection scope of the present disclosure is not limited thereto. Any equivalent replacement or modification made by a person skilled in the art according to the technical solutions of the present disclosure and inventive concepts thereof within the technical scope of the present disclosure shall fall within the protection scope of the present disclosure.