球谐分析教学提纲

1、球谐分析球谐分析，带谐，田谐，瓣谐球谐函数是拉普拉斯方程的球坐标系形式的解。球谐函数表示为：疗（仇以:W）CC汽仇卬=2 YnlgM =n=0球谐分析（如重力场）是将地球表面观测的某个物理量f（theta,lambda）展开成球谐函数的级数:(dj,rcosmti? + 办上” sin Pnni(cas3)其中，theta为余纬，lambda：经度如重力位可表示为:根据球谐函数的性质可知，K表示了在纬度方向的节点粒,2册表示在经度方向的节点数“当 布=0时，邛（8S&）退优为二（8劣），因该函数在n个穿度圈上为零，而与经度无关，类似于地球 上五带的划分，故称为带状球谓函数：当阳关0时，在球面t

2、icosmX和虹nmA在加个子午圈上为5加）个纬度网上为零，则和比；（cosC）在沿球面上直角正方形的边上等于零.它称为田状球谐函数或块状球请函数.当=用时，球谐函数把球面分成从一个极到另一个极 的很多扇形.成为扇形球请函数或瓣谐函数.带谐系数：coefficient of zonal harmonics地球引力位的球谐函数展开式中次为零的位系数。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 ,不随经度方向变化)扇t皆系数：coefficient of sectorial harmonics地球引力位的球谐函数展开式中阶与次相同的位系数 田谐： coefficient of tesseral harmonics地球引力位的球谐函数展开式中阶与次不同的位系数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阶（EGM96）分辨率为0.5分的来历：纬向180、360=0.5因止匕，different spherical harmonic degrees corresponds to different