Square root, least squares estimation algorithms are examined from the viewpoint that square root arrays are arrays expressing relevant problem random variables in terms of basis sets of orthonormal (identity covariance) random variables. With this viewpoint, projection-theorem arguments can be expressed directly in the square root arrays, giving simple, geometric derivations of square root algorithms. A unifying framework is provided for previously proposed square root algorithms (and square-root-free factorization algorithms), and several new square root estimation procedures are derived. Square root techniques for determining the actual error covariance of a state estimate based on an incorrect model are extended. The ease with which either of two previously proposed techniques may be used with either the covariance or the information form of the estimator is enhanced. The two techniques are compared, and some guidance is provided as to when the use of each is advantageous. Square root error analysis is extended to the smoothing problem.