单项污染指数 内梅罗综合污染指数 因子分析法 高斯模型 GM(1,1)模型

承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。我们知道，抄袭别人的成果是违反竞赛规则的,如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。如有违反竞赛规则的行为，我们将受到严肃处理。我们参赛选择的题号是（从A/B/C/D中选择一项填写）：A我们的参赛报名号为（如果赛区设置报名号的话）：3979所属学校（请填写完整的全名）：广东金融学院参赛队员（打印并签名）：1.蔡宗奇曾晓骏陈友辉指导教师或指导教师组负责人（打印并签名）：周雪刚日期：2011年9月12日赛区评阅编号（由赛区组委会评阅前进行编号）：编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：赛区评阅记录（可供赛区评阅时使用）：全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：#5.4针对问题四的建模与求解对于解决问题三时所建立的模型，该模型的优点是所需要的数据量相对较少，而缺点是该模型建立的前提是忽略了天气、地貌和水流等影响。所求的的结果是在相对理想化情况才存在的，在实际中借鉴意义不大。地质环境是自然环境的一种，是指由岩石圈、水圈和大气圈组成的环境系统。在长期的地质历史演化的过程中，岩石圈和水圈、岩石圈和大气圈之间、大气圈和水圈之间进行物质迁移和能量转换，组成了一个相对平衡的开放系统。人类和其他生物依赖地质环境生存发展，同时，人类和其他生物又不断改变着地质环境。而地下水［7］作为水圈中极其重要的组成部分，由于水量稳定，水质好，一直是农业灌溉，工矿和城市的重要水源之一。随着经济的发展，人类生产活动中排出的有毒有害物质使得地下水质持续恶化。所以我们选择地下水污染作为研究城市地质环境演化模式的一个侧面，对地下水进行水质模拟，建立模型以便对地下水域内的地下水域状况进行了解。为了对地下水水质进行模拟，我们需要采集该城市历年来的垃圾掩埋量，生活废水排放量，工业废水排放量，工业废气排放量，地下水总硬度超标率，各种重金属离子含量浓度超标率和浓度，地下水污染成因分析由于地下水水质受到气候，地下水位，污染源的存在以及互相影响非线性的关系。所以这里我们采用灰色关联度分析法来分析产生地下水污染的成因。建立参考数列和比较数列设历年的地下水总硬度数据为参考数列y,设垃圾掩埋量数据y,生活废水排放量o1数据y，工业废水排放量数据y和工业废气排放量数据y作为比较数列。234对各类列进行初值化处理y(k)y=丿/八(y=i,2,3,4)jmaxy(k)|y(k)|y(k)—y(k)1y：(k)0-y(k)4,ky0(k)-y.(k)|+pp为分辨系数，具体取值规则如下［11］：minmin|y(k)-y(k)+p*maxmax£(k)=—i4,ky0(k)-y.(k)|+pp为分辨系数，具体取值规则如下［11］：i记A为所有差值绝对值的均值，即vA二一*-埜"|y0(k)-y.(k)|TOC\o"1-5"\h\zvm*n0ii=1k=1记£=上一，则p的取值为：当A>3A时，£<p<1.5£；AAmaxvAAmaxA<3A时，1.5£<p<2£。maxvAA该模型侧重于总体分析，p不仅可以调节£(k)的大小，而且可以调节它的变化区i间，£(k)的下界值随p增大而增大，下界值增大说明区间变小，分辨率变低，分辨效i果则不明显。根据因素见的关联分析可选择不同的分辨系数，在计算时，一般取p=0.5便可得到满意的分辨率。当pts则无法进行关联分析，此时所有£(k)T1，即关联i系数蜕变为一个点［8］。

r二1工e(k)(i=l,2,3,4)iNik=1(5)排关联序在得出各个序列的灰色关联度排序后我们就可以比较出城市地下水总硬度这一指标与垃圾掩埋量，生活废水排放量，工业废水排放量和工业废气排放量中那一个指标关联系数较高。从而得出地下水总硬度超标率的主要原因。依照上述(1)-(5)的做法我们同样可以用灰色关联度分析法对造成城市地下水各种重金属离子超标的成因做出判断。5.4.2地下水污染预测地下水中的污染是一个包含了物理作用，化学作用和生物作用的过程。由于作用复杂，我们可以利用灰色系统理论，把地下水环境视为灰色系统，他污染物看作是在一定范围内变化的灰色变量。从而我们可以建立起GM(1,1)模型［12。(1)GM(1，1)模型建立原理。设原始数列为Q(o)={Q(0)(1),Q(0)(2),,Q(0)(n)}为原始非负时间序列，预测的结果就是想知道Q⑼(n+1),Q(0)(n+2),Q⑼(t)为累加声称序列，即Q(1)(t)=工Q(0)(i-1),i=2,3ndQ⑴dQ⑴

dt+aQ(1)=u具体地说，就是:m=1|Q⑴(1)=Q(o)(1)(Q(1)(i)=Q(0)(i)+Q(应一1),i=2,,nGM(1,1)模型的白化微分方程为:(a\a为待辨识参数；u为待辨识内生变量；设待辨识变量a=，按最小二乘法得:Iu丿a=(BtB)-1BTy，_1(Q(1)(1)+Q(1)(2))1厶Q(0)(2)_其中B=1-2(Q(1)(2)+Q(1)(3))11，y=Q(0)(3)1-2(Q⑴⑺-1)+Q⑴(n))1_Q(0)(n)_于是我们可以得到灰色预测模型为:Q(1)(t+1)=(Q(0)⑴--)e-at+-,t=1,2,3nza(2)残差检验和后残差检验:模型建立后，我们还应该对模型进行残差检验和后验差检验。残差检验有两种，绝对误差和相对误差检验:绝对误差A(0)匚(0)(i)=Q(0)(i)-Q(i),(i=1,2,n)，相对误差a吨=號*100%，(i=…………“),n)。其中Q(0)(i)二Q⑴(i)-Q⑴(i-1),Q(0)(1)二Q⑴(1),(I二1,2,n)。后残差检验：首先计算原始数列X⑼的均方差S，其定义为：0.S2，—0n-1s2=£Q(0)(i)-q⑼.0i=1Q(0)=—£Q(0)(i)，ni=1然后计算残差数列:⑼的均方差S，其定义为：1S二1—1n-1S2二工匕(0)(i)-匸⑼」,1i=1—1_匚⑼=1£匚(0)(i),ni=1S由此计算方差比C=-1，最后计算/小误差概率p,S0f(0)-匚(0)<0.6745S:根据下面的预测精度等级划分表，如表10所示表10预测精度等级划分表小误差概率P值方差比c值预测精度等级>0.95<0.35好>0.80<0.5合格>0.70<0.65勉强合格<0.70>0.65不合格最后我们将各种重金属离子含量浓度的超标率代入模型中，对将来该城市地下水的污染进行预测。六、模型评价本文通过建立单项污染指数法模型和内梅罗综合指数评价模型，对该地区土壤中的重金属元素进行评价分析，从而客观地了解到土壤中的8种重金属元素的分布、超标率以及各重金属原子分布来说明不同区域的重金属污染程度。由于题目所给数据较少，因此拟合出的曲线和污染源点的位置并不是其准确位置，而在实际中，突发事件也是必须考虑到的，情况将更加复杂，而该模型并不能很好地解决突发事件对该地区影响，因而这也是该模型需要改进的方向。七、参考文献[1]佚名，重金属污染来源、分布、治理方法，第一页，出版年不详/view/39f4f1fd910ef12d2af9e77f.html,2011-9-11.内梅罗综合指数分析法，百度百科，/view/1693176.htm,2011-9-11．HJ/T166-2004,土壤环境监测技术规范.姜芝萍，杨俊衡，城市重点污染区土壤重金属污染评价标准探讨——以衡阳市某区为例，安全与环境工程，第17卷第1期:第59-60页.蔡信德，郭杨，不同标准对城市土壤重金属质量分数的评价,环境科学研究，2005，26（5）:第191-197页.王雄军，基于因子分析法研究太原市土壤重金属污染的主要来源，生态环境，200817（2）:第671-676页.地下水，百度百科，/view/36813.htm，2011-9-11.吕锋灰色系统关联度之分辨系数的研究,系统工程理论与实践，1997.李明，吴超，粉尘点污染扩散模型的可视化研究，中南大学资源与安全工程学院.黄金杰，杨桂花，马骏驰，基于高斯大气污染评价模型，计算机仿真，2011，28（2）:第101-106页.吕锋,灰色系统关联度之分辨系数的研究系统工程理论与实践，1997，第51页.温丽华，灰色系统理论及其应用，计算机应用技术，2003,第15-17页.附录附录一TOC\o"1-5"\h\zqqu1=[40431895242739714777489765345641459246032486599935736213537586437304523093284311807764017056834815467865814844551916569605514298741821418107212645355772641665085101408053823012550311275636133660537411649351512591106313855334514862252415387729158102307172036218163018299179048287103951120315467120801642890691628910072121531233611958133131080013282927716148111211643212625162591798118449];qu2=[017871647272823833692474272934948729355676782642324065735325998703227177777126073880724631942295222546198342608611094260151207827700116096423240657353259987032271777771260738807246319422952225461983426086110942601512078277001160939941273440151489616031694774875006884610981844611200761211938886613143947512000120867776106137106946764238831745889209560140008970142079980151401110116440132322267412173543863951044374059212113058629116469381126411002212204933314631108561059917980];qu3=[1432586661704410691184132366497901899312371199682536124702147271264414943903617538117211900711488187381092112961225351129327816558127696116212734613331265911371527823147372723214482245801331924153124502296513535246851427828654875524003152862168413101221931218517079589415255511015007553515801396670084775147585402026175861956973481941169342335953253238650222624481821703659112734103441440518032140741651614262151292059113549209831586220177176421890616346184671700116607173651595218397226051430123146153822204617634237851764325981180512738018202233251670126852161141068555281064344721170244801063010685552810643447211702448010630877411678861811902770912746845012855894513797962115658759414177668412778579917814107071813410046171981173055321148263541070081841324470561244243291309343391392053541708711933170751292417962128239810171449081183939183197678810];qu4=[74781137373113211791104921272883361727082295293317673526435750624339586849045481600432996018563579655394863152917349700462267048460081804496909053658049543980177210686972867747826084578991946083119062763993196799106316472210068819210919482228469149223041052721439113832055411228201011077421072104042021599512176612348252215795264248639248131079923198135234020299040263913531420607093138124813107992319813523402029904026391353142060709313817100244968373490790639788045305283942035840310758079096631288946922869178329990953975102253821102102789103401369418954199098622176423574874530011557160321801221018106389237158130614414576498721141525851687227981907285192217654928307972612696302417734362910340136941895419909862217642357487453001155716032180122101810638923715813061441457649872114152585168722798190728519221765492830797261269630241773436292028285905734890496598868124002060170057212214507555791212840929613102137651353184386539195016091105479591103981036011529112431156310298140651098712727769115198101001524891061626711058164401206815412129821426912877132771320413175122381263217949146241400416629144811847014411190411576917414154761574815728250211629025675985];qu5=[4233895474164341638766091606173521509269363518257134695495183031431810142];2308376221703927211041532299326779346841364120556641653554120935451275715087351216823420773851855613569103521662177655588205826548141731194115517170341448212692171339095164141051015314139545615356169245696467837656182200576531952qq1=qqu1(:,1);qq2=qqu1(:,2);qq3=qu2(:,1);qq4=qu2(:,2);qq5=qu3(:,1);qq6=qu3(:,2);qq7=qu4(:,1);qq8=qu4(:,2);qq9=qu5(:,1);qq10=qu5(:,2);plot(qq1,qq2,'o',qq3,qq4,'*',qq5,qq6,'+',qq7,qq8,'s',qq9,qq10,'p')title('o-生活区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