Unit 11 Least-Squares Adjustment （最小二乘平差）
Whenever the surveyor conducts a field survey, no matter how simple or complex, he invariably makes more measurements than are absolutely necessary to locate the points in the survey.（无论何时测量人员进行【conduct操作、管理】外业测量，不管【no matter 不论】怎样简单或是复杂，他总是【invariably总是】进行多于绝对必要的观测来定位点 在测量中）
A line taped in two directions introduces one measurement more than is necessary to establish the length of the line.（一条线由两个方向丈量，产生一个观测值，要多于确定【establish确定】直线长度所需的观测值）【原句more that大概写错了】
Measuring all three angles of a triangle introduces one superfluous measurement.（测量一个三角形中所有三个内角产生一个多余【superfluous多余的】观测值）
These extra measurements are termed redundant measurements.（这些额外的【extra】观测值被称为多余【redundant多余的】观测值）
Least-squares adjustment is a mathematical and statistical technique for dealing with the optimal combination of redundant measurements together with the estimation of unknown parameters.（最小二乘平差是一个数学的和统计学的技术用以处理多余观测值的最优【optimal最佳的】联合，和【together with】未知参数的估计）
The least-squares adjustment is rigorously based on the theory of mathematical probability, whereas in general, the other methods do not have this rigorous base.（最小二乘平差严格地【rigorously严格地】基于数学概率理论，然而【whereas】通常【in general】，其它方法并没有这样严格的基础）
In a least-squares adjustment, the following condition of mathematical probability is enforced: the sum of the square of the errors times their respective weights are minimized.（在最小二乘平差中，下列数学概率的条件被要求【enforce强迫、坚持】：误差平方
与其各自的权的乘积【time乘v.】的和最小）
In surveying, errors in measurements conform to the laws of probability, and they follow the normal distribution theory.（测量中，测量误差遵照【conform to】概率的法则，它们遵循正态分布【normal distribution】的理论）
Thus they should be adjusted in a manner that follows these mathematical laws.（这样，它们将以这些数学法则的方式被平差）
A mathematical model for adjustment is composed of two parts: a functional model and a stochastic model.（一个平差的数学模型由两个部分组成【be composed of由...组成】：函数模型【functional model】和随机模型【stochastic model】）
Mathematical Model （数学模型）
A functional model describes the geometric or physical characteristics of the survey problem.（一个函数模型描述了测量问题的几何或是物理特征）
In adjustment computations a functional model is an equation that represents or defines an adjustment condition.（在平差计算中一个函数模型是一个方程，表现或定义了一个平差条件【condition】）
It must either be known, or assumed.（它或者是已知或者是假定的）
If the functional model represents the physical situation adequately, the observation errors can be expected to conform to the normal distribution curves.（如果函数模型充分【adequately充分地】表现了物理情形【situation】，观测误差可以被认为【be expected】遵照【conform to】正态分布曲线【normal distribution curve】）
For example suppose that we are interested in the shape of a plane triangle.（例如假如【suppose that】我们对一个平面三角形的形状感兴趣【be interested in对...感兴趣】） All that is required for this operation is to measure two of its angles, and the shape of the
triangle will be uniquely determined.（所有所需的操作只是测量两个角，该三角形的形状就会唯一地【uniquely唯一地】确定了）
However, if we were to decide, for safety’s sake, to measure all three angles, any attempt to construct such a triangle will immediately show inconsistencies among the three observed angles.（然而，如果我们决定，为了安全的缘故【sake原因、缘故】，测了所有三个角，任何试图【attampt】建立【construct】这样一个三角形将立即在三个观测角之间出现矛盾【inconsistency矛盾】）
In this case the model simply is that the sum of the three angles must equal 180o.（这种情况下，这个模型仅仅是三个内角之和必需为180度）
If three observations are used in this model, it is highly unlikely that the sum will equal exactly 180.（如果三个观测值在这个模型里使用，非常【highly】不可能【unlikely未必的、不可能的】其和正好等于180）
Therefore, when redundant observations, or more observations than are absolutely necessary, are acquired, these observations will rarely fit the model exactly.（因此，当多余观测，或者多于绝对必要观测，被获得【acquire获得】，这些观测将很少【rarely】与模型完全【exactly完全地】吻合）
Intuitively, this results from something characteristic to the observations and makes them inconsistent in the case of redundancy.（直观地，这是由观测的某些特性【characteristic特性n.；特性的】产生的【result from由……产生】，并使得它们在冗余【redundancy】情况下【in the case of在……情况下】不一致【inconsistent不一致的】）
Of course, we first need to be sure of the adequacy of the model (it is a plane triangle and not spherical or spheroidal, for example).（当然，我们首先需要确认模型的适当【adequacy适当n.】（这是个平面三角形而不是球面或类球面，例如））
Then, we need to express the quality of the measurements before we seek to adjust the
observations to fit the model.（然后，我们需要表示【express】出观测值的质量，在我们试图【seek试图；寻找】平差观测数据以适合模型之前）
So from above, a well-known mathematical model states that the sum of angles in a plane triangle is 180.（这样由以上，一个众所周知的数学模型规定【state声明、规定v. 当然翻译时可以灵活些】，平面三角形内角和为180）
This model is adequate if the survey is limited to a small region such as the plane survey.（这个模型是适当的【adequate】如果测量局限于一个小范围内就如同平面测量一样）
The determination of variances, and subsequently the weights of the observations, is known as the stochastic model in a least-squares adjustment which describes the statistical properties of all the elements or represents a way to enter information about the precision of the observations involved in the functional model.（方差的测定【determination决定、测定】和其后【subsequently其后、接下来】的观测值的权，被认为是【is known as】最小二乘平差中的随机模型【stochastic model】，描述了所有元素的统计特性【statistical property】或者表达【represent表达】了将关于观测值精度的信息加入进函数模型的一个方法）
The importance of the stochastic model is often overlooked and undervalued.（随机模型的重要性经常被高估和低估）
As a general rule, if the stochastic model contains misleading information, the adjustment and conclusions drawn from the adjustment can be unreliable.（作为一个一般的惯例，如果随机模型包括误导信息，平差和平差的结果会是不可靠的【unreliable不可靠的】） The stochastic model is represented by the variance-covariance matrix (weighting matrix) of the observations.（随机模型由观测值的方差－协方差矩阵（权阵）表现） It is crucial to the adjustment to select a proper stochastic (weighting) model since the
weight of an observation controls the amount of correction it receives during the adjustment.（选择一个适当的【proper】随机（加权）模型对平差来说是至关重要的【crucial至关重要的】，因为一个观测值的权控制了它在平差时【during the adjustment】所能收到的改正值的大小【amount原意为数量，这里只能译为大小】 However, development of the stochastic model is important not only to the weighted adjustment.（然而，随机模型的发展不只对加权平差有重要意义）
When doing an unweighted adjustment, all observations are assumed to be of equal weight, and thus the stochastic model is created implicitly.（当做一个未加权平差时，所有观测值被假定为等权，因而随机模型暗含地【implicitly含蓄地、暗中地】被创建）
Adjustment Methods（平差方法）
There are two adjustment methods: the conditional and parametric adjustments.（平差方法有两种：条件平差和参数【parametric参数的】平差）
In the conditional adjustment, geometric conditions are enforced, upon the observations and their residuals.（条件平差中，在观测值和它们的余差【residual剩余的、残留的；这里的意思是 余差 可以理解为改正值或误差】上，对几何条件【geometric condition】提出了要求）
So the conditional adjustment is called direct adjustment.（因此，条件平差被称为直接平差。）
Examples of conditional adjustment are: ⑴ the sum of angles in a polygon is (n- 2)*180, where n is the number of angles in the polygon; ⑵ the sum of the angles in the horizon at any station equals 360; ⑶ in a closed traverse, the algebraic sum of the departures should equal the difference between the X coordinates at the beginning and the ending stations of the traverse, similarly, the algebraic sum of the latitudes should equal the difference
11最小二乘平差
