2018 MCM Problem A notes

每年一见的美赛赛题点评又见面啦~

这次工作太忙，有点迟到，实在抱歉~

还是老规矩，先走三类信息标注法，再看点评哦~

本篇是A题，后两题会延后发布，敬请期待~

2018 MCM Problem A: Multi-hop HF Radio Propagation(bk1)

Background: On high frequencies (HF, defined to be 3 – 30 mHz), radio waves can travel long distances (from one point on the earth’s surface to another distant point on the earth’s surface) by multiple reflections off the ionosphere and off the earth.(bk1’) For frequencies below the maximum usable frequency (MUF), HF radio waves from a ground source reflect off the ionosphere back to the earth, where they may reflect again back to the ionosphere, where they may reflect again back to the earth, and so on, travelling further with each successive hop.(bk1’’) Among other factors, the characteristics of the reflecting surface determine the strength of the reflected wave and how far the signal will ultimately travel while maintaining useful signal integrity. (imp1)Also, the MUF varies with the season, time of day, and solar conditions.(imp2) Frequencies above the MUF are not reflected/refracted, but pass through the ionosphere into space. In this problem, the focus is particularly on reflections off the ocean surface. It has been found empirically that reflections off a turbulent ocean are attenuated more than reflections off a calm ocean.(imp3) Ocean turbulence will affect the electromagnetic gradient of seawater, altering the local permittivity and permeability of the ocean, and changing the height and angle of the reflection surface.(imp3’) A turbulent ocean is one in which wave heights, shapes, and frequencies change rapidly, and the direction of wave travel may also change.(imp3’’)

Problem:

Part I: Develop a mathematical model for this signal reflection off the ocean.(spm1) For a 100-watt HF constant-carrier signal, below the MUF, from a point source on land, determine the strength of the first reflection off a turbulent ocean and compare it with the strength of a first reflection off a calm ocean.(rsc1) (Note that this means that there has been one reflection of this signal off the ionosphere.) If additional reflections (2 through n) take place off calm oceans, what is the maximum number of hops the signal can take before its strength falls below a usable signal-to-noise ratio (SNR) threshold of 10 dB? (mss1)

Part II: How do your findings from Part I compare with HF reflections off mountainous or rugged terrain versus smooth terrain? (spm2)

Part III: A ship travelling across the ocean will use HF for communications and to receive weather and traffic reports.(bk2) How does your model change to accommodate a shipboard receiver moving on a turbulent ocean?(spm3) How long can the ship remain in communication using the same multi-hop path? (mss2)

Part IV: Prepare a short (1 to 2 pages) synopsis of your results suitable for publication as a short note in IEEE Communications Magazine. (mss3)

Your submission should consist of:

· One-page Summary Sheet,

· Two-page synopsis,

· Your solution of no more than 20 pages, for a maximum of 23 pages with your summary and synopsis.

· Note: Reference list and any appendices do not count toward the 23-page limit and should appear after your completed solution.

这道题属于“经典物理学”范畴，设定在“无线电传播”这一经典领域，考察的是仔细理清题意所指明的条件和问题以及查阅应用已知模型公式的能力，这一切都依赖于对题目信息的准确定位和解读，用三类信息标注发标注的相关信息标注已经列在题干中。

通过bk1s的信息和资料查阅，我们基本可以确定可以使用的无线电传播模型（任选一经典公式作为参考），后面的imp1,2,3则切实地给出了本题需要给定公式的自变量，参数，以及可能的函数形式和构成要素：

A. 我们关注的wave strength(WS) = f(surface characteristic(SC))；

B. SC = (permittivity, permeability, height, angle)

C. MUF = f(season, time of day, solar conditions)

影响传播的关键现象turbulence也逃不过上面的要素，现在spm1中的模型需要建立的就是这样一套WSn = f(WSn - 1, SC)模型，即给定传播条件下，强度的变化情况，这里又涉及三种情况：

1. 空间传播衰减；

2. 在电离层削弱；

3. 在海洋表面削弱；

其中1,2均可假设以固定衰减系数分别建立连续和离散的模型，3则需要把海洋的特殊情况一并考虑进去，对海洋表面要素对无线电强度减小方式细致分析建模；

关于信噪比SNR，可以假设噪音以一定能力混入原信号，并以同样的机理和有用信号一同传播，最后可算出SNR值随n次海面传播的结果，题目中mss1解决，注意对未知参数带入已知值给出参考值并作灵敏度分析；

Spm2要求比较崎岖和光滑地形的传播强度影响，同样从SC的四个参数角度入手建模即可；

Spm3和mss2要求对移动船只和turbulence建模并回答问题，在前面模型基础上这些都是计算问题而已；

Mss3是摘要，对完整模型以及特点，适用范围，灵敏度，鲁棒性都需要有重点地阐述。

综合看，本题审题难度不大，关键是要完成每一步合理而可行的假设恰好把题目提到的影响参数纳入其中，即为所求，对物理原理理解及公式推导有着一定的要求，有此方面特长的同学可选此题。