WooCommerce passing custom filter parameters

I'm working on a custom filtering to pass the parameters to the page url when user selects a value and clicks the search button. However I can't get the proper output parameters.

in order for the filtering to work i need a format which looks something like this:
shop/?filter_tyre-width=22

What I get currently looks like this:
shop/?tyre_width=22

How do slot in the prefix filter_ after the ? symbol?

<?php 
  $terms_width = get_terms("pa_tyre-width");
  $terms_height = get_terms("pa_aspect-ratio");

  $filter_width = isset( $_GET['tyre_width'] ) ? esc_attr( $_GET['tyre_width'] ) : '';
  $filter_height = isset( $_GET['aspect-ratio'] ) ? esc_attr( $_GET['aspect-ratio'] ) : '';


  if ( '' == get_option( 'permalink_structure' ) ) {
    $form_action = remove_query_arg( array( 'page', 'paged' ), add_query_arg( $wp->query_string, '', home_url( $wp->request ) ) );
  } else {
    $form_action = preg_replace( '%\/page/[0-9]+%', '', home_url( trailingslashit( $wp->request ) ) );
  }

  $new_link = add_query_arg( 'filtering', '1', $form_action );
  //."filtering=1" 
  echo $new_link;

  echo '<form method="get" action="' . $new_link . '" class="col-md-12 clearfix">';

  echo '
          <div class="col-md-3">
          <label for="">Width</label>
          <div class="input-group select-large">
            <select id="tyre_width" name="tyre_width">
              <option>Select Width</option>
  ';
  foreach ( $terms_width as $term_width ) {
  echo '
          <option value="' . $term_width->name . '"> ' . $term_width->name . '</option>
  ';
  }
  echo '</select></div></div>';

  echo '
          <div class="col-md-3">
          <label for="">Height</label>
          <div class="input-group select-large">
            <select id="aspect-ratio" name="aspect-ratio">
  ';
  foreach ( $terms_height as $term_height ) {
  echo '
          <option value="' . $term_height->name . '"> ' . $term_height->name . '</option>
  ';
  }
  echo '</select></div></div>';

  echo '
  <div class="col-md-3">
        <div class="input-group search-btn">
        <label for="submit">Start Search!</label>
        <input type="submit" placeholder="Go" value="Search">
  </div>
  </form>';

via Chebli Mohamed