{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Course website**: http://www.leouieda.com/geofisica1\n",
    "\n",
    "**Note**: This notebook is part of the course \"Geofísica 1\" of Geology program of the \n",
    "[Universidade do Estado do Rio de Janeiro](http://www.uerj.br/). \n",
    "All content can be freely used and adapted under the terms of the \n",
    "[Creative Commons Attribution 4.0 International License](http://creativecommons.org/licenses/by/4.0/).\n",
    "\n",
    "![Creative Commons License](https://i.creativecommons.org/l/by/4.0/88x31.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Prática 2 - Transformada de Fourier e transformações da anomalia\n",
    "\n",
    "Objetivos:\n",
    "\n",
    "* Visualizar os efeitos e resultados da Transformada de Fourier em sinais simples.\n",
    "* Aplicar a Transformada de Fourier para calcular derivadas da anomalia magnética de campo total.\n",
    "* Observar os efeitos do erro aleatório nas derivadas calculadas.\n",
    "* Aprender as utilidades e limites da continuação para cima e da redução ao polo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preparação\n",
    "\n",
    "Esse documento que você está usando é um [IPython notebook](http://ipython.org/notebook.html). É um documento interativo que mistura texto (como esse), código (como abaixo), e o resultado de executar o código (que pode ser números, texto, figuras, videos, etc). Esta prática usará a biblioteca [Fatiando a Terra](http://fatiando.org) de modelagem geofísica e também o módulo [numpy](http://www.numpy.org/).\n",
    "\n",
    "O notebook é divido em células (como esta). Para editar o conteúdo de uma célula, clique nela (clique nesta para editar esse texto). Para executar uma célula, aperte `Shift + Enter`. Execute as duas células abaixo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      ":0: FutureWarning: IPython widgets are experimental and may change in the future.\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from __future__ import division\n",
    "from IPython.html import widgets\n",
    "import numpy as np\n",
    "from fatiando.gravmag import prism, fourier\n",
    "from fatiando import utils, gridder, mesher\n",
    "from fatiando.vis import mpl\n",
    "import fatiando\n",
    "mpl.rc('lines', linewidth=2)\n",
    "mpl.rc('font', size=12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Versão do Fatiando a Terra: 0.3-237-g65602ba\n"
     ]
    }
   ],
   "source": [
    "print('Versão do Fatiando a Terra: {}'.format(fatiando.__version__))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tarefa 1: Tranformada de Fourier de funções simples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A primeira tarefa servirá para ilustrar o conceito de [Transformada de Fourier](http://en.wikipedia.org/wiki/Fourier_transform) que vimos na aula teórica.\n",
    "\n",
    "Como vimos em aula, uma função contínua pode ser escrita como:\n",
    "\n",
    "$$\n",
    "h(t) = \\int\\limits_{-\\infty}^{\\infty} H(f)\\ e^{i 2 \\pi f t}df\n",
    "$$\n",
    "\n",
    "em que $f$ é a frequência e $e^{i 2 \\pi f t}$ são \"coisas\" que oscilam com frequência $f$.\n",
    "A equação acima nos diz que a função $h(t)$ pode ser escrita como uma \"soma\" infinita de oscilações de frequências diferentes.\n",
    "A oscilação de frequência $f$ é ponderada por uma amplitude $H(f)$. A função que descreve as amplitudes, $H(f)$, é a Transformada de Fourier:\n",
    "\n",
    "$$\n",
    "H(f) = \\int\\limits_{-\\infty}^{\\infty} h(t)\\ e^{-i 2 \\pi f t}dt\n",
    "$$\n",
    "\n",
    "$H(f)$ é um número complexo. É comum fazer um gráfico do módulo da transformada ($|H(f)|$) pela frequência. Esse gráfico é chamado de **espectro de amplitudes**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Rode a célula abaixo para gerar uma figura interativa.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f87736b67d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def dois_senos(freq1, amp1, freq2, amp2):\n",
    "    sample = 1/200\n",
    "    t = np.arange(0, 10, sample)\n",
    "    n = t.size\n",
    "    s = amp1*np.sin(2*np.pi*freq1*t) + amp2*np.sin(2*np.pi*freq2*t)\n",
    "    f = np.fft.fftfreq(n, sample)[:n//2]\n",
    "    S = np.fft.fft(s)[:n//2]/n\n",
    "    fig = mpl.figure(figsize=(12, 5))\n",
    "    ax = mpl.subplot(211)\n",
    "    mpl.title(r'$h(t) = {:.0f}\\ \\sin(2\\pi\\ {:.0f}\\ t) + {:.0f}\\ \\sin(2\\pi\\ {:.0f}\\ t)$'.format(amp1, freq1, amp2, freq2),\n",
    "              fontsize=18)\n",
    "    mpl.plot(t, s, '-k')\n",
    "    mpl.xlim(0, 5)\n",
    "    mpl.ylim(-200, 200)\n",
    "    mpl.grid()\n",
    "    ax.set_xlabel('tempo (s)', labelpad=-15)\n",
    "    mpl.ylabel('Amplitude')\n",
    "    ax = mpl.subplot(212)\n",
    "    mpl.plot(f, np.abs(S), '-k')\n",
    "    mpl.xlim(0, 50)\n",
    "    mpl.ylim(0, 50)\n",
    "    mpl.grid()\n",
    "    ax.set_xlabel('frequencia (Hz)', labelpad=-10)\n",
    "    mpl.ylabel(r'$|H(f)|$', fontsize=14)\n",
    "    mpl.tight_layout(h_pad=0)\n",
    "widgets.interactive(dois_senos, \n",
    "                    freq1=widgets.IntSliderWidget(min=1, max=40, step=1, value=1), \n",
    "                    amp1=widgets.IntSliderWidget(min=0, max=100, step=10, value=0), \n",
    "                    freq2=widgets.IntSliderWidget(min=1, max=40, step=1, value=1), \n",
    "                    amp2=widgets.IntSliderWidget(min=0, max=100, step=10, value=100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sobre a figura\n",
    "\n",
    "* A figura de **cima** mostra a função $h(t)$ que é composta por dois senos.\n",
    "* Os **botões** controlam a amplitude e frequência de cada seno.\n",
    "* A figura de **baixo** mostra o espectro de amplitudes da função $h(t)$.\n",
    "\n",
    "### Tarefas e perguntas\n",
    "\n",
    "1. Deixe `amp1 = 0` e fixe `freq2` em algum valor. Varie somente `amp2`.\n",
    "    * Por que o espectro de amplitudes é zero em todas as frequências exceto uma? Qual é essa frequência?\n",
    "    * O que acontece com $h(t)$ e o espectro de amplitudes quando `amp2` varia?\n",
    "2. Deixe `amp1 = 0` e `amp2=100`. Aumente `freq2`.\n",
    "    * O que acontece com $h(t)$? A amplitude do seno muda?\n",
    "    * O que acontece com o especro de amplitudes? A amplitude do espectro aumenta?\n",
    "3. Deixe `amp2 = 100` e `freq2=1`. Aumente `amp1` e `freq1`.\n",
    "    * O que acontece com $h(t)$ e o espectro de amplitudes?\n",
    "    * Quais são as frequências que não tem amplitude zero no espectro?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tarefa 2: Continuação para cima\n",
    "\n",
    "A Transformada de Fourier é muito útil na geofísica para fazer transformações nos dados. Em gravimetria e magnetometria, uma transformação comum é a \"continuação para cima\". Essa transformação nos permite calcular como a anomalia medida seria se estivesse a uma altitude maior. Um dos jeitos de calcular a continuação é através da transformada de Fourier."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Rode a célula abaixo para gerar uma figura interativa.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "YII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f877419ad10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "shape = (100, 100)\n",
    "x, y, z = gridder.regular((-6000, 6000, -6000, 6000), shape, z=0)\n",
    "modelo = [mesher.Prism(-200, 200, -200, 200, 100, 500),\n",
    "          mesher.Prism(-4000, 4000, -4000, 4000, 1000, 5000)]\n",
    "inc, dec = 45, -10\n",
    "mag = utils.ang2vec(1, inc, dec)\n",
    "modelo[0].props['magnetization'] = 1*mag\n",
    "modelo[1].props['magnetization'] = 1*mag\n",
    "def continuacao(altitude, erro):\n",
    "    cubo_area = modelo[0].get_bounds()[:4][::-1]\n",
    "    tf = prism.tf(x, y, z, modelo, inc, dec)\n",
    "    if erro > 0:\n",
    "        tf = utils.contaminate(tf, erro, seed=0)\n",
    "    cont = fourier.upcontinue(x, y, tf, shape, altitude)\n",
    "    fig, axes = mpl.subplots(1, 2, figsize=(14, 5.5))\n",
    "    for ax, data, title in zip(axes.ravel(), [tf, cont], ['Anomalia em 0 m', 'Continuada para {:.0f} m'.format(altitude)]):\n",
    "        ax.set_aspect('equal')\n",
    "        mpl.sca(ax)\n",
    "        mpl.title(title)\n",
    "        scale = np.abs([data.min(), data.max()]).max()\n",
    "        mpl.contourf(y, x, data, shape, 30, cmap=mpl.cm.RdBu_r, vmin=-scale, vmax=scale)\n",
    "        mpl.colorbar(pad=0).set_label('nT')\n",
    "        mpl.m2km()\n",
    "    mpl.tight_layout(h_pad=0, w_pad=0)    \n",
    "widgets.interactive(continuacao, \n",
    "                    altitude=widgets.FloatSliderWidget(min=0, max=5000, step=50, value=0),\n",
    "                    erro=widgets.FloatSliderWidget(min=0, max=20, step=1, value=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sobre a figura\n",
    "\n",
    "* Os dados são a anomalia magnética de campo total gerada por dois corpos: um grande e fundo; outro raso e pequeno (no meio da figura).\n",
    "* A figura da **esquerda** mostra a anomalia calulada a uma altitude de 0 m.\n",
    "* A figura da **direita** mostra a anomalia continuada para uma altitude maior usando a Transformada de Fourier.\n",
    "* Os **botões** controlam a altitude da continuação e o erro (em nanoTesla) colocado no dado original (da esquerda).\n",
    "\n",
    "### Tarefas e perguntas\n",
    "\n",
    "1. Deixe o erro em 0 nT. Aumente a altitude de continuação.\n",
    "    * O que acontece com a forma e amplitude da anomalia continuada? Dica: observe a barra de cor.\n",
    "    * As duas anomalias (grande e pequena) sofrem a mesma influência da continuação?\n",
    "    * Se você quiser remover o efeito de uma das anomalias do seu dado, você poderia subtrair os dados da direta dos dados da esquerda? Ou seja, subtrair os dados continuados dos dados observados?\n",
    "2. Aumente o erro e varie a altitude de continuação.\n",
    "    * Como o erro afeta o resultado da continuação?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tarefa 3: Derivada total\n",
    "\n",
    "Outra operação que a Transformada de Fourier nos permite fazer é calcular derivadas dos nossos dados. Uma transformação muito utilizada na magnetometria é a Derivada Total (DT):\n",
    "\n",
    "$$\n",
    "DT = \\sqrt{\\left(\\frac{\\partial T}{\\partial x}\\right)^2 + \\left(\\frac{\\partial T}{\\partial y}\\right)^2 + \\left(\\frac{\\partial T}{\\partial z}\\right)^2}\n",
    "$$\n",
    "\n",
    "A DT é famosa por concentrar a anomalia magnética de campo total sobre o corpo causador da anomalia.\n",
    "\n",
    "As derivadas parciais na equação acima podem ser calculadas da anomalia de campo total ($T$) usando a Transformada de Fourier.\n",
    "Derivando a equação de $h(t)$ acima:\n",
    "\n",
    "$$\n",
    "\\frac{\\partial h}{\\partial t} = \\frac{\\partial}{\\partial t}\\left(\\int\\limits_{-\\infty}^{\\infty} H(f)\\ e^{i 2 \\pi f t}df\\right)\n",
    "$$\n",
    "\n",
    "Como a única coisa que depende de $t$ é $e^{i 2 \\pi f t}$, podemos passar a derivada para dentro da integral:\n",
    "\n",
    "$$\n",
    "\\frac{\\partial h}{\\partial t} = \\int\\limits_{-\\infty}^{\\infty} H(f)\\ \\frac{\\partial e^{i 2 \\pi f t}}{\\partial t}df\n",
    "$$\n",
    "\n",
    "e aplicando a [regra da cadeia](http://en.wikipedia.org/wiki/Chain_rule)\n",
    "\n",
    "$$\n",
    "\\frac{\\partial h}{\\partial t} = \\int\\limits_{-\\infty}^{\\infty} H(f)\\ i 2 \\pi f\\ e^{i 2 \\pi f t} df\n",
    "$$\n",
    "\n",
    "$\\frac{\\partial h}{\\partial t}$ também é uma função e pode ser escrita como a soma de infinitas oscilações. Se chamarmos a transformada de Fourier de uma função $h$ de $F\\{h\\}$, podemos escrever a derivada de $h$ como:\n",
    "\n",
    "$$\n",
    "\\frac{\\partial h}{\\partial t} = \\int\\limits_{-\\infty}^{\\infty} F\\left\\{\\frac{\\partial h}{\\partial t}\\right\\}e^{i 2 \\pi f t} df\n",
    "$$\n",
    "\n",
    "Se compararmos as duas equações acima, vemos que\n",
    "\n",
    "$$\n",
    "F\\left\\{\\frac{\\partial h}{\\partial t}\\right\\} = H(f)\\ i 2 \\pi f\n",
    "$$\n",
    "\n",
    "Ou seja, a transformada de Fourier da derivada é $i 2 \\pi f$ vezes a transformada de Fourier da função. Uma vez tendo a transformada da derivada, podemos calcular a derivada."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Rode a célula abaixo para gerar uma figura interativa.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8773853f10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "shape = (100, 100)\n",
    "x, y, z = gridder.regular((-5000, 5000, -5000, 5000), shape, z=0)\n",
    "dx, dy = 500, 5000\n",
    "cubo = mesher.Prism(-dy/2, dy/2, -dx/2, dx/2, 400, 4000)\n",
    "cubo_area = cubo.get_bounds()[:4][::-1]\n",
    "def derivada_total(inc, dec, erro):\n",
    "    tf = prism.tf(x, y, z, [cubo], inc, dec, pmag=utils.ang2vec(1, inc, dec))\n",
    "    if erro > 0:\n",
    "        tf = utils.contaminate(tf, erro, seed=0)\n",
    "    total = fourier.ansig(x, y, tf, shape)\n",
    "    fig, axes = mpl.subplots(1, 2, figsize=(14, 6))\n",
    "    for ax, data, title in zip(axes.ravel(), [tf, total], ['Anomalia', 'Derivada Total']):\n",
    "        ax.set_aspect('equal')\n",
    "        mpl.sca(ax)\n",
    "        mpl.title(title)\n",
    "        mpl.square(cubo_area)\n",
    "        scale = np.abs([data.min(), data.max()]).max()\n",
    "        mpl.contourf(y, x, data, shape, 30, cmap=mpl.cm.RdBu_r, vmin=-scale, vmax=scale)\n",
    "        if title != \"Anomalia\":\n",
    "            mpl.colorbar(pad=0).set_label('nT/m')\n",
    "        else:\n",
    "            mpl.colorbar(pad=0).set_label('nT')                \n",
    "        mpl.m2km()\n",
    "    mpl.tight_layout(h_pad=0, w_pad=0)    \n",
    "widgets.interactive(derivada_total, \n",
    "                    inc=widgets.IntSliderWidget(min=-90, max=90, step=5, value=45),\n",
    "                    dec=widgets.IntSliderWidget(min=-90, max=90, step=5, value=0),\n",
    "                    erro=widgets.FloatSliderWidget(min=0, max=20, step=1, value=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sobre a figura\n",
    "\n",
    "* A figura da **esquerda** mostra a anomalia magnética de campo total causada por um paralelepípedo (contorno preto).\n",
    "* A figura da **direita** mostra a derivada total (DT) da anomalia da esquerda calculada com a Transformada de Fourier.\n",
    "* Os **botões** controlam a inclinação e declinação do campo da Terra (`inc` e `dec`) e o erro colocado nos dados originais (da esquerda).\n",
    "\n",
    "### Tarefas e perguntas\n",
    "\n",
    "1. Mantenha a inclinação em 45º e declinação em 0. Aumente o erro aplicado aos dados.\n",
    "    * O que acontece com a Derivada Total quando o erro aumenta? Por que acontece isso? Dica: olhe as equações acima.\n",
    "2. Varie a inclinação e a declinação.\n",
    "    * A DT varia com a inclinação?\n",
    "    * A DT continua concentrada em cima do corpo para qualquer inclinação?\n",
    "    * Se a inclinação for 0, o que acontece com a DT quando a declinação varia até $\\pm90$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tarefa 4: Redução ao pólo\n",
    "\n",
    "Vimos na [prática 1](http://www.leouieda.com/geofisica1/lessons/magnetometria/pratica1.html) que a anomalia magnética é muito complicada de interpretar. A melhor situação é quando a inclinação do campo magnético é $\\pm90^\\circ$ (nos polos). \n",
    "\n",
    "Felizmente, há um jeito de calcular como a anomalia que medimos ficaria se estivesse nos polos. A técnica chama-se (numa explosão de criatividade) \"Redução ao Polo\". Um dos jeitos de se calcular a redução ao polo é usando a transformada de Fourier.\n",
    "\n",
    "**É necessário conhecer a direção de magnetização do corpo para aplicar a redução ao polo**. Isso é fácil se o corpo tiver somente magnetização induzida pelo campo geomagnético. A magnetização é paralela ao campo da Terra. A situação complica quando há magnetização remanente, aquela que os minerais ferromagnéticos guardam quando se resfriam abaixo da [temperatura de Curie](http://en.wikipedia.org/wiki/Curie_temperature). Por isso a redução ao polo não é tão facilmente utilizada quanto a Derivada Total."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Rode a célula abaixo para gerar uma figura interativa.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8773105910>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "shape = (100, 100)\n",
    "x, y, z = gridder.regular((-5000, 5000, -5000, 5000), shape, z=0)\n",
    "dx, dy = 500, 5000\n",
    "cubo = mesher.Prism(-dy/2, dy/2, -dx/2, dx/2, 400, 4000)\n",
    "cubo_area = cubo.get_bounds()[:4][::-1]\n",
    "def reducao_polo(inc, dec, erro):\n",
    "    tf = prism.tf(x, y, z, [cubo], inc, dec, pmag=utils.ang2vec(5, inc, dec))\n",
    "    if erro > 0:\n",
    "        tf = utils.contaminate(tf, erro, seed=0)\n",
    "    rtp = fourier.reduce_to_pole(x, y, tf, shape, inc, dec)\n",
    "    fig, axes = mpl.subplots(1, 2, figsize=(14, 6))\n",
    "    for ax, data, title in zip(axes.ravel(), [tf, rtp], ['Anomalia', 'Reduzida ao polo']):\n",
    "        ax.set_aspect('equal')\n",
    "        mpl.sca(ax)\n",
    "        mpl.title(title)\n",
    "        mpl.square(cubo_area)\n",
    "        scale = np.abs([data.min(), data.max()]).max()\n",
    "        mpl.contourf(y, x, data, shape, 30, cmap=mpl.cm.RdBu_r, vmin=-scale, vmax=scale)\n",
    "        mpl.colorbar(pad=0).set_label('nT')\n",
    "        mpl.m2km()\n",
    "    mpl.tight_layout(h_pad=0, w_pad=0)    \n",
    "widgets.interactive(reducao_polo, \n",
    "                    inc=widgets.IntSliderWidget(min=-90, max=90, step=5, value=45),\n",
    "                    dec=widgets.IntSliderWidget(min=-90, max=90, step=5, value=0),\n",
    "                    erro=widgets.FloatSliderWidget(min=0, max=20, step=1, value=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sobre a figura\n",
    "\n",
    "* A figura da **esquerda** mostra a anomalia magnética de campo total causada por um paralelepípedo (contorno preto).\n",
    "* A figura da **direita** mostra a anomalia reduzida ao polo (calculado com a Transformada de Fourier).\n",
    "* Os **botões** controlam a inclinação e declinação do campo da Terra (`inc` e `dec`) e o erro colocado nos dados originais (da esquerda).\n",
    "\n",
    "### Tarefas e perguntas\n",
    "\n",
    "1. Mantenha a inclinação em 45º e declinação em 0. Aumente o erro aplicado aos dados.\n",
    "    * O que acontece com a redução ao polo quando o erro aumenta? O efeito é tão grande quanto na Derivada Total?\n",
    "2. Varie a declinação.\n",
    "    * A anomalia reduzida ao polo continua concentrada em cima do corpo?\n",
    "    * A qualidade da redução ao polo varia com a declinação?\n",
    "3. Varie a inclinação\n",
    "    * Como a qualidade da redução ao polo varia com a inclinação?\n",
    "    * Quais limitações essa variabilidade impõe na aplicabilidade da redução ao polo?  "
   ]
  }
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