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Автокорреляционные функции и энергетические спектры погрешностей наблюдений

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Кафедра общей и прикладной геофизики

Курсовая работа

на тему:

Автокорреляционные функции и энергетические спектры погрешностей
наблюдений

Выполнил: студент группы 3151

Климов Ю. С.

Проверил: профессор

Серкеров С. А.

Дубна, 2005

Содержание

Введение

Теоретическая часть

Расчётная часть

Заключение

Список литературы

Введение

В данной работе рассматриваются элементы теории случайных функций и их
применение для интерпретации гравитационных и магнитных аномалий.
Аппарат теории случайных функций и основанный на нём статистический
подход можно применять в различных ситуациях. Во-первых, когда мало
известно о параметрах аномалий или геологических объектах, которыми они
вызваны. Во-вторых, когда поставленную задачу гравиразведки и
магниторазведки можно решить только с применением аппарата теории
случайных функций и, наконец, в-третьих, при решении задач различными
детерминированными методами.

Получаемые данные, корреляционные функции и связанные с ними
энергетические спектры аномалий имеют следующие свойства: малая
чувствительность к погрешностям наблюдений; взаимозаменяемость; чётность
получаемых выражений.

В работе также приведены примеры применения теоретического материала к
практике. Представлены расчёты для бесконечной горизонтальной
материальной линии, бесконечной вертикальной материальной полосы и
бесконечной горизонтальной полосы.. Для исследуемых функций построены
графики при различных исходных данных.

Теоретическая часть

Автокорреляционные функции и энергетические спектры погрешностей
наблюдений

При решении различных задач грави- и магниторазведки почти всегда
возникает необходимость учета влияния погрешностей наблюдений. Поэтому
очень важно выяснить законы изменения их автокорреляционной функции и
энергетического спектра. Необходимо также выяснить чувствительность
вычислительных схем к погрешностям наблюдений и получить формулы,
позволяющие оценить их точность. Существующие формулы оценки их
погрешности дают только предельное, следовательно, во многих случаях и
завышенное значение погрешности.

Не менее важным является выяснение возможности корреляции погрешностей
наблюдений с аномалиями. Обычно полагают, что они не коррелируются, но
это не всегда так. Во многих реальных случаях и, особенно, когда искомая
аномалия небольших размеров, погрешности наблюдений могут
коррелироваться с аномалией. И тогда неучет коррелируемости может
привести к значительным погрешностям в решаемой задаче. В таких случаях
необходимо пользоваться способами, учитывающими корреляцию.

Под погрешностями наблюдений понимаются сумма случайных погрешностей
наблюдений и влияний самых верхних плотностных неоднородностей.
Рассмотрим основные энергетические характеристики погрешностей
наблюдений [38].

Высокочастотные случайные помехи можно аппроксимировать белым шумом с
ограниченной полосой частот, для которой

Bп(?) = Bп(0)[sin(рф / Дx)](рф / Дx), (3.70)

где Дx – расстояние между пунктами наблюдений; Bп(0) -максимальное
значение автокорреляционной функции – средний квадрат ошибок наблюдений.
Это для случая, когда радиус корреляции погрешностей наблюдений r = Дx.
В остальных случаях, т.е. когда r > Дx, автокорреляционную функцию
погрешностей наблюдений можно выразить функциями

Bп(x)=Bп(0)exp[-(ф / d)2], (3.71)

Bп(x)=Bп(0)exp[-ф / d1], (3.72)

где d и d1 – постоянные, зависящие от радиуса корреляции ошибок
наблюдений r.

Если ошибки между пунктами наблюдений взаимонезависимы, то

r = Дx.

Но обычно r > Дx, и это происходит из-за наличия в погрешностях
наблюдений, кроме некоррелируемых между соседними точками измерений
помех (ошибка в отсчете, ошибка в нивелировке и др.), случайной
составляющей, коррелируемой между несколькими пунктами наблюдений.
Последняя может быть обусловлена неравномерными в течение рейса
условиями транспортировки, неравномерным изменением температуры,
неравномерными атмосферными условиями (ветер, дождь), ошибками учета
нуль-пункта и другими причинами. Для определения более правильных
законов изменения автокорреляционной функции, энергетического спектра
ошибок наблюдений и оценки соотношения между r и Дx были получены
экспериментальные данные погрешностей наблюдений с гравиметрами (выборка
из 400 значений).

Анализ этих данных показал, что их наилучшим образом можно
аппроксимировать выражением

Bп(ф) = Bп(0)exp(-бф)cosвф (3.73)

при значениях постоянных б = 0,80 / r, в = р / 2r.

Значения радиуса корреляции погрешностей наблюдений r, найденные по этим
экспериментальным данным, колеблются от l,3Дx до 2,0Дx (при разных
выборках из 400 – при 50, 100, 200 и 400 значениях). При этом среднее и
наиболее вероятное значение r=1,6Дx (это значение соответствует кривой
автокорреляционной функции, построенной по всем 400 значениям
погрешностей наблюдений). Поэтому здесь и в дальнейшем в качестве
радиуса корреляции ошибок наблюдений г будет принято это уточненное
значение r = 1,6Дx. Что же касается систематических ошибок, то для
определения их радиуса корреляции можно воспользоваться формулой для
определения радиуса корреляции суммарного поля, полагая, что

fп(x) = fс(x) + fо(x),

где индексы “c” и “о” указывают соответственно на случаи систематических
ошибок и инструментальных ошибок, не коррелирующих между двумя соседними
пунктами наблюдений. При значениях их средних квадратов Bс(0), Bо(0)
значение rп можно определить из равенства
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Данные анализа наблюденных погрешностей показывают, что Bc(0) ? Bo(0).
Поэтому

rп = (rc+Дx) / 2.

Отсюда rc = 2rп – Дx.

Подставляя в это равенство вместо rп предельные значения rп = 1,ЗДx и rп
= 2Дx, найдем rc = 1,6Дx и rc = ЗДx, т.е. можно принять, округляя до
целых, что rc меняется примерно от 2Дx до ЗДx.

Некоторые авторы полагают, что rc должен меняться от Дx до 2Дx. Более
уточненные данные показывают, что rc меняется от 2Дx до ЗДx (значению rп
= 1,6Дx соответствует величина rc = 2,2Дx). Эти же данные, как более
обоснованные, можно принять за основу при исследованиях в дальнейшем.

Учитывая изложенное выше, в дальнейшем в зависимости от решаемой задачи
в качестве автокорреляционной функции ошибок наблюдений будем принимать
выражения, определяемые равенствами (3.70) и (3.73). Им соответствуют
следующие выражения энергетических спектров.

Для равенства (3.70)
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(3.74)

Для равенства (3.73)

picscalex100010009000003f302000004001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02c004201e1200000026060f001a00ffffffff000010000000c0ffffff
a7ffffffe01d0000670400000b00000026060f000c004d61746854797065000000010800
0000fa0200001000000000000000040000002d010000050000001402ba03160605000000
13029e03470608000000fa0200002000000000000000040000002d010100050000001402
a603470605000000130228048e06040000002d0100000500000014022804960605000000
1302a402f406050000001402a402f406050000001302a402d6080500000014026002a705
05000000130260023e090500000014026002280a0500000013026002d012050000001402
600256140500000013026002f81c1c000000fb0280fe0000000000009001000000cc0402
001054696d6573204e657720526f6d616e000c020d7e15020d7e20d00f7e000030000400
00002d01020008000000320ac002a01d010000002e7908000000320af303a41b01000000
297908000000320af3039c1701000000287908000000320af3037c110100000029790800
0000320af3036e0d01000000287908000000320a12040c0701000000327908000000320a
c101ac0801000000297908000000320ac101ec0701000000307908000000320ac1016807
01000000287908000000320ac002850301000000297908000000320ac002f30101000000
28791c000000fb0220ff0000000000009001000000cc0402001054696d6573204e657720
526f6d616e000c020d7e15020d7e20d00f7e00003000040000002d01030004000000f001
020008000000320a4703431c01000000327908000000320a47038d150100000032790800
0000320a47031b1201000000327908000000320a47035f0b0100000032791c000000fb02
80fe0000000000009001000000020002001053796d626f6c007e400000005e090a500c02
0d7e15020d7e20d00f7e00003000040000002d01020004000000f001030008000000320a
2b03101d01000000fa7908000000320a4904101d01000000fb7908000000320abb01101d
01000000f97908000000320a2b03860901000000ea7908000000320a4904860901000000
eb7908000000320abb01860901000000e97908000000320af3037619010000002d790800
0000320af3037c16010000002b7908000000320ac0022a13010000002b7908000000320a
f303480f010000002b7908000000320af3034e0c010000002b7908000000320ac0026904
010000003d791c000000fb0280fe0000000000009001010000020402001053796d626f6c
007e40000000330d0a4f0c020d7e15020d7e20d00f7e00003000040000002d0103000400
0000f001020008000000320af3039c1a01000000627908000000320af3030e1801000000
777908000000320af303521401000000617908000000320ace010e180100000061790800
0000320af303741001000000627908000000320af303e00d01000000777908000000320a
f303240a01000000617908000000320ace01e30d01000000617908000000320a1204ba07
01000000707908000000320ac00265020100000077791c000000fb0220ff000000000000
9001010000cc0402001054696d6573204e657720526f6d616e000c020d7e15020d7e20d0
0f7e00003000040000002d01020004000000f001030008000000320a2102bd0601000000
ef7908000000320a2003480101000000ef791c000000fb0280fe00000000000090010100
00cc0402001054696d6573204e657720526f6d616e000c020d7e15020d7e20d00f7e0000
3000040000002d01030004000000f001020008000000320ac101cd050100000042790800
0000320ac0022e000100000051790a00000026060f000a00ffffffff0100000000001c00
0000fb021000070000000000bc02000000cc0102022253797374656d000000000a002100
8a01000000000200000094f312000efe0c7e040000002d01020004000000f00103000300
00000000 (3.75)

Можно также пользоваться выражениями (3.71) и (3.72), но только для
получения отдельных прикидочных оценок или с целью получения менее
громоздких выражений из интегралов.

Для трехмерных аномалий, предполагая значения ошибок наблюдений
симметричными относительно вертикальной оси, автокорреляционные функции
ошибок наблюдений для законов, аналогичных равенствам (3.70) и (3.73)
двухмерного случая, опишем соответственно формулами

Bп(ф)н = 2J1(eф) / eф, (3.76)

Bп(ф)н = exp(-pф)J0(tф) (3.77)

при следующих наиболее вероятных значениях постоянных [38]:

e = 2,4 / Дx, p = 0,5 / Дx, t = 1,5 / Дx.

Для энергетических спектров ошибок наблюдений, определяемых равенствами
(3.76), (3.77), получим соответственно следующие выражения [для
равенства (3.77) из-за громоздкости выражения приводим значение Qп(0)]:
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 (3
78)
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 (3.79)

Связь между энергетическими характеристиками исходных и
трансформированных аномалий

Выразим энергетический (взаимный энергетический) спектр или
корреляционную функцию трансформированной аномалии через энергетический
(взаимный энергетичекий) спектр или корреляционную функцию исходной
аномалии.

Так как спектр трансформированной аномалии ST выражается через спектр S
[см. равенство (2.5)], то на основании формулы (3.16) для
энергетического спектра трансформированной аномалии получим

QT(u,v) = Q(u,v)|Ф(u,v)|2, (3.80)

где Q — спектр исходной аномалии, а Ф – частотная характеристика
преобразования. Из этой формулы можно получить энергетический спектр
трансформированной аномалии, зная энергетический спектр исходной и
частотную характеристику преобразования. Пользуясь равенствами (2.5),
(3.17), такое же соотношение можно написать и для взаимных
энергетических спектров:

QT(u,v)12 = Q(u,v)12|Ф(u,v)|2. (3.81)

Что же касается корреляционных функций (автокорреляционной и взаимной
корреляционной), то, как видно из равенств (3.12), (3.15), (3.20),
(3.80), (3.81), для получения корреляционной функции трансформированной
аномалии по известной корреляционной функции исходной необходимо
последнюю подвергать трансформации с частотной характеристикой
|Ф(u,v)|2. Все это верно и для двухмерного случая. Рассмотрим несколько
частных случаев.

1. Аналитическое продолжение на уровень Н аномалий в области верхнего
или нижнего полупространства

Как известно, в этом случае

Ф(u,v) = ехр(±сH),

где знак “минус” относится к аналитическому продолжению в области
верхнего полупространства, знак “плюс” – в области нижнего. Тогда

|Ф(u,v)|2 = ехр(±2сH).

Отсюда видно, что для получения корреляционной функции, аналитически
продолженной на уровень Н в области верхнего или нижнего
полупространства аномалии, необходимо корреляционную функцию исходной
аналитически продолжить на уровень 2Н. На основании этого положения для
корреляционных функций можно записать интеграл Пуассона, заменив в нем
значение Н на значение 2Н.

2. Вычисление n-й горизонтальной производной

В этом случае (рассматриваем произвольную по оси x)

Ф(u,v) = (iu)n. (3.82)

Следовательно,

|Ф(u,v)|2 = u2n = (-1)n(iu)2n. (3.83)

Аналогичный результат получим и при дифференцировании по направлению оси
у. Из последних двух равенств видно, что для получения корреляционной
функции аномалии n-й горизонтальной производной необходимо
продифференцировать корреляционную функцию исходной аномалии по
направлению соответствующей оси 2n раз и умножить полученный результат
на (-1)n. Например, для оси x верно равенство
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 (3.84)

где B(о, з) и Bn(о, з) – автокорреляционные функции исходной аномалии и
аномалии n-й производной по направлению оси х.

3. Вычисление n-й вертикальной производной

Так как для данного случая

Ф(с) = (-с)n, (3.85)

то

|Ф(с)|2 = (-с)2n. (3.86)

Отсюда видно, что вывод такой же, как и в предыдущем случае, только
результат не нужно умножать на (-1)n. На основании этого положения в
двухмерном и трехмерном случаях для автокорреляционных функций получим
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 (3.87)

где B(ф) и Bn(ф) – автокорреляционные функции исходной аномалии и
аномалии n-й вертикальной производной (здесь учтено, что в выражение
B(ф) глубина залегания аномального тела входит в виде 2h).

В двухмерном случае из-за равенства автокорреляционных функций аномалий
горизонтальных и вертикальных производных следует, что

picscalex100010009000003b902000003001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c028004e0171200000026060f001a00ffffffff000010000000c0ffffff
a9ffffffa0170000290400000b00000026060f000c004d617468547970650000f0000800
0000fa0200001000000000000000040000002d0100000500000014024002530605000000
130240022e0c0500000014024002501105000000130240022b171c000000fb0280fe0000
000000009001000000cc0402001054696d6573204e657720526f6d616e000c020d7e1502
0d7e20d00f7e00003000040000002d01010008000000320aa0025517010000002e790800
0000320aae012d1602000000295d08000000320aae01e91401000000285d08000000320a
ae015e13010000005b5d08000000320aa002d00f01000000295d08000000320aa0022e0f
01000000315d08000000320aa002d20d01000000285d08000000320ad303cf0902000000
295d08000000320ad303430801000000325d08000000320ad303b90701000000285d0800
0000320ad3036906010000005b5d08000000320aae01300b02000000295d08000000320a
ae01ec0901000000285d08000000320aae016108010000005b5d08000000320aa0024503
01000000295d08000000320aa002010201000000285d1c000000fb0220ff000000000000
9001000000cc0402001054696d6573204e657720526f6d616e000c020d7e15020d7e20d0
0f7e00003000040000002d01020004000000f001010008000000320a2703871401000000
325d08000000320a02014b1201000000325d08000000320a2703da0a01000000325d0800
0000320a02014e0701000000325d1c000000fb0220ff0000000000009001010000cc0402
001054696d6573204e657720526f6d616e000c020d7e15020d7e20d00f7e000030000400
00002d01010004000000f001020008000000320a27030715010000006e5d08000000320a
0201cb12010000006e5d08000000320af4016f10010000006e5d08000000320a27035a0b
010000006e5d08000000320a0201ce07010000006e5d08000000320a0003e70301000000
5a5d08000000320af4015601010000006e5d1c000000fb0280fe00000000000090010100
00cc0402001054696d6573204e657720526f6d616e000c020d7e15020d7e20d00f7e0000
3000040000002d01020004000000f001010008000000320aae01f41301000000425d0800
0000320ad303090901000000685d08000000320aae01f70801000000425d08000000320a
a002460001000000425d1c000000fb0280fe000000000000900101000002040200105379
6d626f6c007e400000002a0b0a2a0c020d7e15020d7e20d00f7e00003000040000002d01
010004000000f001020008000000320ad303881301000000745d08000000320aae014915
01000000745d08000000320aae014c0a01000000745d08000000320aa002610201000000
745d1c000000fb0280fe0000000000009001000000020002001053796d626f6c007e4000
0000ab0a0a1c0c020d7e15020d7e20d00f7e00003000040000002d01020004000000f001
010008000000320ad303ce1201000000b65d08000000320aae01641101000000b65d0800
0000320aa0025c0e010000002d5d08000000320aa002a00c010000003d5d08000000320a
d303ed0601000000b65d08000000320aae01670601000000b65d08000000320aa0021505
010000003d5d0a00000026060f000a00ffffffff0100000000001c000000fb0210000700
00000000bc02000000cc0102022253797374656d000000000a0021008a01000000000100
000094f312000efe0c7e040000002d01010004000000f0010200030000000000
(3.88)

Усреднение и применение вычислительных схем

При усреднении (например, по двум точкам, на отрезке профиля, по
окружности, по площади круга) также верно равенство

|Ф(u,v)|2 = Ф2(u,v).

Поэтому во всех этих случаях для получения корреляционной функции
усредненной соответствующим образом аномалии необходимо корреляционную
функцию исходной аномалии усреднить дважды.

Вывод о применении трансформации дважды относится и к преобразованиям с
помощью различных вычислительных схем, основанных на усреднении по
точкам или по окружности. Полученные соотношения в двухмерном и
трехмерном случаях позволяют определить автокорреляционные функции и
энергетические спектры трансформированных аномалий через
автокорреляционную функцию и энергетический спектр одной исходной
аномалии, минуя процесс самой трансформации. Приведенными равенствами
широко пользуются на практике (см., например, работы К.В. Гладкого, В.Н.
Глазнева, В.Н. Луговен-ко и других исследователей).

Расчётная часть

Возьмём нормированную автокорреляционную функцию погрешностей
наблюдений. Рассмотрим ёе поведение для радиуса корреляции погрешностей
наблюдений r = Дx, для r > Дx.

1. Радиус корреляции погрешностей r = Дx.

Bн(t)=exp[-(t / d)2],

Рассматриваем для значений d = 1, 5, 10.

График изменения автокорреляционной функции при различных d
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0000

Bн(t)=exp[-t / d1],

Рассматриваем для значений d = 1, 5, 10.

График изменения автокорреляционной функции при различных d

picscalex1030100090000037126000008001610000000001610000026060f002220574d
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2. Радиус корреляции погрешностей r > Дx.

Bн(t) = exp(-бt)cosвt; где б = 0,80 / r, в = р / 2r;

Рассматриваем для значений r = 1, 5, 10.

График изменения автокорреляционной функции при различных r

picscalex1030100090000035828000008001610000000001610000026060f002220574d
46430100000000000100a8a9000000000200000000200000201400002034000001000000
6c00000005000000050000009a010000d800000000000000000000009829000009160000
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1a040000ae0100000c010000000000000000000000000000b08f0600e016040046000000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00000000a0010000dd000000260000001c00000001000000000000000000000000000000
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00000000000000000000000000000000000000000000000000f000000000000000000000
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68ed1300d2d6850573ebab5124ec1300d2d6850500000000000000008ceb1300771f0830
000000000000000068613a0500000000c8000000000000002e00000020ec130054e28505
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1e00050000000102ffffff0005000000090200000000040000002701ffff030000001e00
050000000102ffffff0005000000090200000000040000002701ffff030000000000

Заключение

С помощью графиков можно судить о поведении значений автокорреляционной
функции. Очевидно, что при малых d функции для аномалий более пологие.
Видно, что при ф = 0 функции имеют все общую точку равную 1. Графики
функций для выбранных тел имеют относительное сходство.

Список литературы

1. Серкеров С. А. Спектральный анализ гравитационных и магнитных
аномалий. — М.: Недра, 2002.

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