球諧分析教學(xué)提綱

1、球諧分析球諧分析，帶諧，田諧，瓣諧球諧函數(shù)是拉普拉斯方程的球坐標(biāo)系形式的解。球諧函數(shù)表示為：療（仇以:W）CC汽仇卬=2 YnlgM =n=0球諧分析（如重力場）是將地球表面觀測的某個(gè)物理量f（theta,lambda）展開成球諧函數(shù)的級(jí)數(shù):(dj,rcosmti? + 辦上” sin Pnni(cas3)其中，theta為余緯，lambda：經(jīng)度如重力位可表示為:根據(jù)球諧函數(shù)的性質(zhì)可知，K表示了在緯度方向的節(jié)點(diǎn)粒,2冊(cè)表示在經(jīng)度方向的節(jié)點(diǎn)數(shù)“當(dāng) 布=0時(shí)，邛（8S&）退優(yōu)為二（8劣），因該函數(shù)在n個(gè)穿度圈上為零，而與經(jīng)度無關(guān)，類似于地球 上五帶的劃分，故稱為帶狀球謂函數(shù)：當(dāng)陽關(guān)0時(shí)，在球面t

2、icosmX和虹nmA在加個(gè)子午圈上為5加）個(gè)緯度網(wǎng)上為零，則和比；（cosC）在沿球面上直角正方形的邊上等于零.它稱為田狀球諧函數(shù)或塊狀球請(qǐng)函數(shù).當(dāng)=用時(shí)，球諧函數(shù)把球面分成從一個(gè)極到另一個(gè)極 的很多扇形.成為扇形球請(qǐng)函數(shù)或瓣諧函數(shù).帶諧系數(shù)：coefficient of zonal harmonics地球引力位的球諧函數(shù)展開式中次為零的位系數(shù)。In themathematicalstudy ofrotational symmetry, the zonal spherical harmonics are specialspherical harmonicsthat are invariant

3、 under the rotation through a particular fixed axis.(故 m=0 ,不隨經(jīng)度方向變化)扇t皆系數(shù)：coefficient of sectorial harmonics地球引力位的球諧函數(shù)展開式中階與次相同的位系數(shù) 田諧： coefficient of tesseral harmonics地球引力位的球諧函數(shù)展開式中階與次不同的位系數(shù)The Laplace spherical harmonicscan be visualized by considering their nodal lines, that is, theset of point

4、s on the sphere where q =6Nodal lines of are composed of circles: some are latitudes and others are longitudes.inOne can determine the number of nodal lines of each type by counting the number of zeros ofthe latitudinal and longitudinal directions independently.For the latitudinal direction, the ass

5、ociated L egendre polynomials possess ?-| m| zeros, whereas for the longitudinal direction, the trigonometric si n and cos functions possess 2| m| zeros.nt = 2/-/w = 1When the spherical harmonic order m is zero (upperleft in the figure ), the spherical harmonic functions do not depend upon longitude

6、, and are referre d to as zonal . Such spherical harmonics are a special case ofzonal spherical functions.secWhen ? = | m | (bottomright in the figure ), there are no zero crossings in latitude, and the functions are referred to as toral .For the other cases, the functionscheckerthe sphere, and they are referred to as tesseral .More general spherical harmonics of degree ? are not necessarily those of the Laplace basis 1, andtheir nodal sets can be of a fairly general kind. 10 360階（EGM96）分辨率為0.5分的來歷：緯向180、360=0.5因止匕，different spherical harmonic degrees corresponds to different