I am planning to do an advanced program in spatial science and engineering. It states as a requirement in the program, that the student should have a through understanding in math and statistics. So far I have done only up to undergraduate level (i.e. only A/Ls) in both the fields and am baffled as to how much of understanding I should have in the fields. Also I my BS degree is a special in Geology. As the program is basically an engineering graduate level one, what do you recommend? Some good books, would be highly appreciated.
In the US, most programs that explicitly involve engineering would, at a minimum, expect you to be prepared for or have passed the EIT (or FE) exam. Requirements in the UK (and most western countries) are probably similar. You can read the NCEES requirements for mathematical and statistical knowledge online (pdf format). "Spatial science" looks like it fits within the "other disciplines" category. Its math/stats requirements are (with the amount they count towards the exam):
I. Mathematics 15% A. Analytic geometry B. Integral calculus C. Matrix operations D. Roots of equations E. Vector analysis F. Differential equations G. Differential calculus II. Engineering Probability and Statistics 7% A. Measures of central tendencies and dispersions (e.g., mean, mode, standard deviation) B. Probability distributions (e.g., discrete, continuous, normal, binomial) C. Conditional probabilities D. Estimation (e.g., point, confidence intervals) for a single mean E. Regression and curve fitting F. Expected value (weighted average) in decision-making G. Hypothesis testing
Although this is what is covered in strong programs in the first year of math (differential and integral calculus) and one semester of statistics, study beyond these levels, especially involving applications, is probably helpful. In weaker programs, the matrix operations, vector analysis, and differential equations would usually be covered in second-year math courses and some of the statistical material (especially distributions and conditional probabilities) would also be topics in a second-year course.
The afternoon session of the exams includes engineering versions of all these subjects (totaling 19% of that exam). The new topics include
Mathematics Partial differential calculus Numerical solutions (e.g., differential equations, algebraic equations) ... Statistics Design of experiments Goodness of fit (coefficient of correlation, chi square) ...
There's little here in addition to the previous topics: some exposure to "advanced" (multidimensional) calculus and a course in numerical methods would be useful.
There are many good books on these subjects. A great place to start, though, would be to review the syllabi of undergraduate courses offered in the department to which you are applying. The books they use would be most relevant (and, as a bonus, are likely available, used, in great numbers on campus :-).
If you're planning on taking the course at the university you got your BSc from, then tutors should be happy to discuss the requirements in more detail. If not, then an email to the department's admin office saying you're interested in signing up for the course, and would it be possible to discuss it with the admissions tutor.
As for the amount of maths and stats you need, I scraped a pass at A-Level in maths and stats and have rarely needed anything more than that in day-to-day work I can't get from a book. If you have it at UG level, then you should have all the grounding you need - it's the foundations they're looking for rather than any domain-specific knowledge, although YUMV (you university may vary).
I presume also that you have looked at the published course outline to see what is taught, which can give you an idea, then do a few searches to see what's been written about the different aspects. Some universities put their course content online publically which can be a big help.
Here in the UK, The Guardian newspaper publishes the rankings of all the universities based on research and student satisfaction, categorised by subject. This can be a useful tool if something similar exists where you are, in determining if the course you want to apply for is taught well. In my experience, it's more about who teaches and how a module is taught, rather than the content.
GIS programs in and of themselves don't usually require so much math. I had to take two classes more or less of my choosing. As far as discrete math it seems like a class that is always different - loosely defined subject so a professor can more or less talk about what they want. To me discrete wasn't really something I used but a class that helped me to understand other things way better.
It is usually a base level class required for computing / computer science type programs. So if you plan on going in a coding direction with your spatial science then discrete math would be a good idea. And statistics are always good. "Geographic Information Analysis" by OSullivan and Unwin was the book we used and it has sections on general stat with the primary emphasis on spatial statistics.
I am not sure how rigid the university would be in relaxing the norms for deserving candidates, but personally I don't see Math/Stats are a mandate for a GIS course.
With Geology/Geography knowledge already in place (for you), you'd be a good candidate to get GIS working.
They might be requiring it for the "engineering" aspect of it...not sure what they would be covering there...